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C =εSm

d=

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C(t)=

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wIH Q = Q0 + q cgibdceifgahkjg^~Cl"#Æmlcà<gei~Cgiij[c]RVydg¡dy²aw<hzdgahdflj[giydcaÊDfw<jºlny²bzcga¹qCgicwncdcgdg~ClRVymlnhkj>j[e ξ/d0Ê

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q

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d0ξ ,

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u(t) =1

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i(t) +U0

jω d0

v(t) , Ð O?PQO Ó

wIH v(t) =∂ ξ

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− Q2

2εSm= − Q2

2Cd= −(Q0 + q)2

2C0d0≈ − Q2

0

2C0d0− Q0q

C0d0= − Q2

0

2C0d0− U0q

d0,

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F =U0

jω d0i+ (Zm + S2

mZL)v .

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p = − F

Sm+ ZLSmv ,

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Sm− ZLSmv = −p =

U0

jω d0Smi+

Zm

S2m

Smv .Ð O?P »<Ó

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u = −ZeiÐ O?P ékÓ

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v =jω d0

U0

1

Ze

[

Ze +1

jωC0

]

u Ð O?P Z Ógij

Smp = −[

(

U0

ωd0

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Ze + 1/(jωC0)+ Zm

]

v . Ð O?P ½<Ó

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p= −U0Sm

jω d0

Ze

Zm[Ze + 1/(jωC0)] + [U0/(ωd0)]2 ,

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Za = − p

Smv=

1

S2m

[

(

U0

ωd0

)21

Ze + 1/(jωC0)+ Zm

]

. Ð O?P Y Ó

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Mp =

(

u

p

)

i=0

= − U0Sm

jω d0Zm

Ð O?P[\ Ó

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pp = ZLSmv .

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u

i=

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+

(

U0

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p = − U0

jω d0Sm

ZL

ZL + Zm/S2m

i =ZLZm/S

2m

ZL + Zm/S2m

Mpi ,Ð O?PVO »<Ó

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Smv = − Zm/S2m

ZL + Zm/S2m

U0Sm

jω d0Zmi =

Zm/S2m

ZL + Zm/S2m

Mpi .Ð O?PVO ékÓ

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Smv ZL Zm/S2

m−Ck −Ck C0 i

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fw<hkjdw<hdhdeigafbmlnc

uTh =−piSmU0

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gij ZTh =1

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+

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U0

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ZL + Zm/S2m

, Ð O?PVO=Z Ó

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u

Cref

ie

ur Ze

ir: 2b0* !$C

; 5 GHKV

. *Z>F#%5L?/5-D#%C&>F&L=!$&G# _>2>F* !$C

Mpr

. *Z>F#%5L?/5-]CHKF!$!$&G# _>2>F* !$C

Mpe

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haw<ydbz~Cgayzclaw<ydfÀj[QR yzg_ÐÜ)l) >j[e!dg!<wn~Cyd¹qg VcÓtgifj@ÛÜgac¹¡e q fgaf·gt jcea¹q>jeaf@bmlnc@dgiyBt¹qCacw i

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gihkj[cg~Cgafdgay.t ¹¡>acw<bdÆdw<hzgafaÊ9c[lnbzbxw<cÀjdg ~Il^j[gahzfCwnhqiCciyd>j3w<yzngacj ur0

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Êmiw<hdhmln>fflnhkj~Cl3ln~Cgiydcdg@~Il)lbmlnaj[e CrefP

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gafj~Ò >¹¡bQeamlhdag@lniw<ydfÀj[]RVydg@dy^¹qCacw<bdÆdwnhdgtceiagibzj[giydcdwnhdhdeag@bmlc~Ò g tzbzcgaff>w<h(Ð O?P Y Ó

Zar =1

S2r

[

(

U0r

ωd0r

)21

Ze + 1/(jωC0r)+ Zmr

]

, Ð O?PVO ½<Ó

Segj Sr

ej#lnhkjÏ~CgifÏfydcÛëlagafÏ¹qw<|dC~>gaf/zgaf±¹¡gi¹º|dclnhdgaf±dgÅ~Ò ei¹¡gjj[giydc±gijÏdyqceiagab1j[gayzcaÊVgijÏwIHq~CgiCciyd>jei¹¡gjj[giydcÅdg@agjj[g mà<yzcgqÐ O$P ÞkÓåceabQw<hdqq~Ò e4RVymlj[>w<hÐ O$PQO é<Ó P

Mpeie Zme/S2

e ZLe pe pi Zar pr

qe Seve Srvr ZLr

%'&#f#()&#$

WAÂ O?P Þ É1#Ædei¹l!ea~>gaj[cQRVydg@eRVyd>ln~>gahkjÅdy^dCfbxwnf>j>Û6bQw<ydcÅ~ eij[ln~Cwnhdhmlnà<gbmlncÅceia>bdcw aj[e P3 lnbdcâifÅ~ eRVymljCw<hÐ O$P ÞkÓÖÊd~Ilºbzcgaff>w<h pr

lny1j

pr = − 1

Ze

[

(

U0r

ωd0r

)2

+ Zmr

(

Ze +1

jω C0r

)

]

jω d0r

U0rSrur .

Ð O?PVO ÞkÓ

Ú lncln>~C~>gaydcfaÊ ~ g4uq)lni>je Mprzy^¹qCicw<bzÆdw<hdgRceaagibzj[giydcfa eaac>j3Ðëe4R PO?P \ Ó

Mpr =

(

ur

pr

)

ir=0

≈ ur0

pr

,

ÐÜ~ lnbdbdcwt1>¹ljCw<h (pr)ir=0 ≈ prgafÀj-¾yzfj[meaglnyqbmlnclnà<c[lbdÆdg O?P » P é RVydEfyz>jÖÓÖÊkgijÏagi~C~>g Mpe

dgÅ~ ea¹qgij ijgaydcÐdà<ydcg O?P ÞkÓ

Mpe =qeie

=Seve + pe(Zme/S

2e )

−1

ie,

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gij Mprzw<hdhdgihVj~ gt1bdcgaffCw<h"dyZbzcw dyd>jdgif

gu¡alnaj[eif

MpeMpr ≈ur0

ie

Seve + pe(Zme/S2e )

−1

pr

= ZEYT ,Ð O?PVOY Ó

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fydc~Cl^¹qga¹3|dc[lnhzgqzyceiagibzj[giydc

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j[eicCfÀj[QR yzgafÅdgaf¹¡>acw<bdÆdw<hzgafÐ åcdga~ ¾c ZFO Þn¼qgj Z5O\ ¼kÓgijdy"bdcealn¹qbd~Cm)ljgaydcy1j[C~>CfeqÐ åcdga~ ¾c5Ì»nÞ Y é j Þ/lNÓ@ÐÜj[ln|d~>g)lny O?PVO Ó P

ÑGÉ O Ú ÑGÉd» ÚÐ O=Z Þ<¼kÓ Ð ZFO\ ¼kÓ

áGgahdfCw<h^dg@bQw<~IlncCflj[>w<h U0Ð@Ó »¼<¼ »n¼<¼

Çlnbdlnaj[e mnlny^cgibxwnf n C0ÐÜb+X Ó ½<½ OY Ê½

Äl)¿<w<h^Chkj[eic>gaydc a Ðë¹q¹Ó ¸zÊé Z Ê Þk½±w<~>yd¹qgteRVydnl~Cgahkj Veq

Ðë¹q¹ 3 Ó OZ<\ ¸zÊ é>fj[lnhdag@¹qga¹º|zc[lnhdg i ei~Cgiij[cw1zg d0 = εSm/C0

Ðë¹q¹Ó ¼zÊ O\ ¼zÊ OZ

è8DÂ O?PVO Ïln~>gaydcfTm jß¿ bxgif n^zgaf)lnclnijeac>fj]RVydgafÅdg@¹qCicw<bdÆzw<hdgaf ZFO Þn¼!gj ZFO=\ ¼ PÇåw<hdhmln>fflnhkjÅ~IlÛÜceRVydgahzagnÊEigafÅdw<hdhzeagafÅbQgac¹¡gjj[gihkjÅdgR)ln~>ayd~>gac~Ò >¹¡bQeadlnhdaglnawnydfj]RVydgUqºa>c i

aydjÅw<yzngacjdy^¹qCacw<bdÆdwnhdg (Za)i=0 = γRPR/(jω Veq)Ð » P »nÞ<Ó P

Ú w<ydcD~CgÏbdcealn¹¡bz~Cd)ljgaydcGyzj[>~C>fenÊç~Ò >¹¡bQeadlnhdagÏ5 gahkj[ceag ZegifjDdw<hdhdeigbmlnc 1/Ze = jωCe+1/Rel)<ga Ce ≈ ¼1Ê¼k½ºb+XÊ Re ≈

O ΩP

Çågafdw<hzhdeagifqgj~Cgifºcgi~IljCw<hzf Ð O$P Y Ó@gij"Ð O?P ¸<Ó·bQgac¹qgijj[gihVj!zg)ln~>ayd~>gac ÐÜiÛ P j#ln|d~>g)lny.t O?P »"gj O?P ébxwnydcdgay.t nl~CgayzcfÅgt j[c a¹qgafdg@~IlºÛÜce4RVydgahdigçÓÅ~ w<cdcgRdg@à<c[lhddgayzczg~Ò >¹¡bQeamlhdagRlaw<ydfÀj[QR yzggiha>cayz>j@w<yzngacÀjºdy¹¡>acwnbdÆdw<hdg (Za)i=0 = Zm/S

2mgij~ eaalncjSq agjj[gqln~Cgiydc@f±~Cg!¹qCacw<bdÆdwnhdg!gafj

#Æmlncà<e@bmlnc~CgRbzce)l¹¡bd~>malj[giydc@ÐÜagafdgiyBtRVymlnhkj[j[eiffw<hkjiw<hkj[>h z¹¡gihVjÅzeaacw<Cff[lnhkjgaf#Ó PÇågij@eaalncj@gifj@hdeià<~C>à<g)l|d~CglnChzfhRVydgÊ5bmlncfydj[g^Ðëzy_Ûëln>j R yzgagijj[g!C¹qbQeamlnhzag Zar

h5 gifj·lny1j[cgRVydgº~ C¹qbQeamlnhzag@j[gac¹¡>hmln~>gSRVydg3nw<>j~Ilal)1j[edgºawnydbd~Ilà<gçÓÖÊx~Ò >hda>dgahdig3dg3agijj[g!lnbzbdcw=t1C¹¡lj[>w<hfydc!~Òlnz¹¡jj#lhdag law<ydfÀj[QR yzg^zg"#Æmlcà<g dy ¹¡>acw<bdÆdw<hzg ea¹qgijjgaydc YLe = 1/ZLe

Ê ag RVyd9¾ydfjmg

'<' ()*+#,=# !"$-. 8/, 0.2&-B"$=/,4=*5%36 #-.,D0.:;-<=-. 8/,D-A 0."?-A@#-. 8/,$C 0<*+2&!8*FEG-L@#*Fk2&*

:;0<J88-L8-K0."<%3@#"<%M0

"/0.6 8-.!"?- _ _

Zm/S2mÐÒÈ P f P ¹ −5 Ó i ¾ O Ê » PQO ¼ 11 i ¾ O Ê» PQO ¼ 8

(

U0

ωd0Sm

)21

Ze + 1/(jωC0)ÐëÈ P f P ¹ −5 Ó Ð »1Ê éM¾ O Ê O Ó PVO ¼ 7 Ðë¼zÊ» \ ¾ O Ê \ Ó PVO ¼ 2

èÏ8 9Â O?P » Ëß¹qbQeamlnhzag (Za)i=0 = Zm/S2mgijeaalncj Za − (Za)i=0

bQw<ydcydh^¹qCacw<bdÆdwnhdg Z5O\ ¼ P"/0.6 8-.!"?- _ _

Zm/S2mÐÒÈ P f P ¹ −5 Ó i ¾ Y Ê Þ PVO ¼ 9 i ¾ Y Ê Þ PQO ¼ 6

(

U0

ωd0Sm

)21

Ze + 1/(jωC0)ÐëÈ P f P ¹ −5 Ó Ð O Ê ¼M¾¼zÊ O ÞkÓ PVO ¼ 5 O Ê O M¾ÅÞzÊ Þ

èÏ8 9Â O?P é Ëß¹qbQeamlnhzag (Za)i=0 = Zm/S2mgijeaalncj Za − (Za)i=0

bQw<ydcydh^¹qCacw<bdÆdwnhdg Z5O Þ<¼ P

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gij Zarfw<hkjbzc[lj]RVydga¹qgahkj·Czgahkj[QR yzgaf@bxwnydcR~Cg¹qCicw<bdÆzw<hdgceiagibzj[giydc

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% % 8&/7/B*- 52198&: 5 6*/ 1 >?* :68&/ A (19, * Ñ6gifRea#Æmlhdà<gaf@j[Æzgac¹q]RVydgifRmlnhdfR~>gmyd>dg^Ðëzw<hkj@~Il¡jga¹qbxeic[ljydcge<w<~>ydgmlnhdfR~>g3j[gi¹¡bzf@lçnga!~Il

bdcgiffCw<hEÓGgijÏl)<gi~Ilbmlcw<1dg~Ilal)1j[e@Ð&RVyddcgifjg R ydlnf i Cfwnj[Æzgac¹qgçÓ6¹¡w dmgahkj/~>g#Æmln¹qbdgbdcgaffCwnhgahcgiàklnczgag RVy5 C~/fgac[l>j@fDagaf@g gj[f@h5 gt1CfÀj#ln>gahkj@bmlnf P Ñ6l iw<ccgaj[Cwnhbdcw<bQw<feagbmlncR~Clhdw<c¹¡ggafÀjlffw a>eagTqydh^¹¡w dâa~>gtdg@)l) >je@eij#l|d~C5bdlnc@gic|QgacTj O »/l Ptlnhdf@ag!bmlc[lnà<clnbdÆdgÊ5~ gt1bdcgaffCwnhdg!~Ò >¹¡bQeamlhdag5 yzhdg¡al)1j[e^ÐÜc[lnbdbQw<cÀj@dg!~IlbdcgaffCw<h q ~Il

fw<cÀj[>gdg~Il!)l) >je3fyzc~Cg·dei|d>j q¡~Ò gihVjceigçÓg t1a>jeagºbdlncyzh_à<eahzeac[l j[gayzc m>de)ln~ n fga~>w<hLRVydg3 gifjRyzhà<eahzeac[l j[gayzcdg@dei|d>jw<y"dg·bdcgiff>w<h^gafjeij[ln|d~>CgR5 lnbdcâaf Rgac|xgicÐÒl P ÓÊm~Ò g tzbzcgaff>w<hZdg·~ C¹qbQeamlnhzag5 ydhdgåal)1j[egt1aj[eagbmlncGydhdgfw<ydcagåceagi~C~CgÏgah3gafjDgahdfyd>jgådeidydj[gRÐÜ| P ÓÊçj[w<yH¾w<ydcf95 lnbdcâifWRgac|xgic P Ñ6g#Ædw<gt3 ydhzggt1bdcgiffCw<h!dg~Ò >¹¡bQeadlnhdag5 ydhdgalç j[eej#ln|d~>CgbmlncWRgac|xgic lnbdbz~C]RVyml|d~CgÏmlnhdf ~>gåalndcgdg·~ eij#l~Cw<hdhdlnà<gRdg·¹¡>acw<bdÆdw<hzgafÅbmlncÅceia>bdcw aj[e@gafÀjgihdfydj[g·gt1bd~>]RVydeÐë P Ó P Ì h mhÊx~Ò g tzbzcgaff>w<hZdg~Il^aw<ccgaj[>w<hj[Æzgac¹q]RVydghdw<c¹¡ln~>CfeigKj »<»/låln>hdf R yzg~Il"Ûël w<h(zw<hkjga~>~Cggifjy1j[C~>Cfeaggifjg t1bd~C>a>jeagÐë P Ó P* 4:;@W0.8*+!"?-*+"? !,%76 #- eV ##-k"?* %&0|-8"<%M0A-@#*F !8-k, 8/"?- &F%780A*+2&-)(x#- #0 !%&U

8-L@#-B,,=%3@gic|QgacaÊ dlnhdfq~ lncjCa~>gChkj>j[yz~Ce "! "! " " "!

# "!" "! j O »/lÒÊ1¹qw<hkj[cg RVydgt~Ò >¹¡bQeadlnhdaglniw<ydfj]RVydgdgj[clnhdfÀÛÜgacj5 ydhdgal)1j[eqdg¡nw<~Cyz¹¡g V gjºdgqfydcÀÛëlnag A ÐÜc[lnbzbxw<cÀj3zg~ eaalncj·>hdfj[lnhkj#lnhde¡dgbzcgaff>w<h(mlnhzf~Cl)l) >je fydc~>g dea|dj!dg ~Il_fw<ydcagçÓÖÊ wIH C~h5 ¿²lbdlnfdg bdcwnbmlnàkljCw<hÊ9gifjqdxeacgahkj[gÊ fgi~Cw<h1RVydg ~Cg#Æmln¹qb_lniw<ydfÀj[]RVydg3gafjà<eahzeace3bmlncyzhdg3fw<ydcigm>de)ln~>gn dg3dei|d>jÐëà<eihdeaclj[giydczg3dei|d>jtlç¿<lnhkjRydhdgC¹qbxeimlnhdigtCh mhdCg)Óåw<y dg@bdcgiffCw<hÐëà<eihdeaclj[giydcdg@bdcgiff>w<h"lç¿<lnhkjydhdg@C¹qbxeimlnhdigRhVyd~C~>gçÓ Ptlnhdf@~Cgalnf@5 ydhdg!fw<yzcag!dg!dei|d>jaÊ5~ C¹qbQeamlnhzaglniw<ydfÀj[]RVydg ZV

zgq~Cl)l) >jeqfi eiac>j)Ê5lbdcâif@gic|QgaciÊ

ZV =γP0

jω V

1 + (γ − 1)EV

γ, Ð » P ½<Ó

gijÏbQw<ydc ydhdgfwnydcigdgÅbdcgaffCwnh5ÊV~ C¹qbxeimlnhdigdg~IlR)l) >jenÊkhdwnjeag ZPgifj)Ê<jw<y ¾w<yzcf±5 lnbdcâif Rgac|xgicaÊ

ZP =γP0

jω V

1

γ − (γ − 1)EP, Ð » P ÞkÓ

wIH ~CgtÛëlnj[giydc γP0/(jω V ) cgabzceafgahkj[g~ lnd¹q>jj#lnhdig5 ydhdg·)l) >jeTq!bdlncw<>fc>à<Cdgif PÑ6giftaw g4uqiCgahkjf EVgj EP

fw<hkjRej[cw<>jga¹qgahkj~>Ceif q~Cl¡à<eiw<¹qeij[cCg·dg3~Cl¡alç j[e P Ëß~Cffw<hkjtzw<hdhdeifmlnhdf~Òlcj[>a~>g¡dgRgac|xgicj O »lgij!aln~Ciyd~Ceif5 lnbdcâif~Il_ceiÛÜeacgahdig j O ¼/l·Ðë~>gdeij[lnC~dg agif)ln~>ayd~>fgafjdw<hdhze·dlnhdfÅ~CgRbmlc[lnà<clnbdÆdg@é P »<Ó P

_G__ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F

0."<%3@#"<%M0

Ì hfgtceiÛÜeic[lnhkjrq3igafaln~Ciyd~CfiÊ1>~Qlnbdbmlncl Cj RVydgtigafaw g4uqiCgahkjfbxgiyz<gihkj ij[cgteaicj[fåfw<ydf~Cl·ÛÜw<c¹¡gdy^de<ga~>w<bdbQga¹qgahkj¹¡w mln~xfydnlhVjUp

EV,P =

∞∑

n=n0

DNn

1 +MnX2V,P

, Ð » P Y Ó

l)<giXV =

A

V

1 − j√2

√

γαT

ωgj XP =

A

V

1 − j√2

√

αT

ω, Ð » P \ Ó

wIH αT = λ/(ρ0CP ) = c0`hÐëbdlncRde dhd>jCw<h_zg `h Ógifj@~Cg3aw g4uqa>gahkj@dgdydf> >jejÆdgac¹¡QR yzgdymyzCdg P ÇågRdeinga~>w<bdbQga¹qgahkjgafjÅzeadydjdgR~Ò g tzbzcgaff>w<hÐÒé P » Z Ê é P »nÓÊzzgR~ ea)lcj¹qwç¿<gah dgtjga¹qbxeic[l i

jydcg@mlnhzf~Cl)l) >j[e PÑ-gaftj[gic¹qgaf Nn

Ê MnÊ D gj n0

deibxgihddgahkj·dg~Clà<eaw<¹qeijc>g3dg~Cl)l) >je P Ëß~>fRfw<hkj@dw<hzhdeafRmlhdf~>g·j[ln|d~Cgalny" PQO dg3~Òlhdhdgt1g¤bQw<ydcdxeacgahkj[gifÛÜw<c¹¡gifÅdg3alç j[e P Ñ Chdd>ag n bxgiyzjtcgabdceafgihkj[gacydhfgayd~6ChzdCignÊdydh^dw<yz|d~CgjÅw<y ydh^jc>bd~Cgj5 Chdd>agif PÇågde<ga~>w<bdbQga¹qgahkj!awnhVngacà<g^lnffg ~Cgihkj[ga¹qgahkj^ÐÜfydcÀj[w<y1j!gah²¹qwH¿ngahdhzgaf!gijÆdlnyzj[gifºÛÜce4R yzgahdigaf#ÓÖÊ

igDRVyd9cgahdZfw<h_y1j[C~>Cflj[>w<h bxgiy_bdclj[QR yzg P h_de<gi~Cw<bdbQga¹qgahkjsqaw<yzcjj[gac¹¡gbmlncÅj[clnhdfÛÜwnc¹qeag3zgÑ6lnbd~Clnagdeazyd>jzgt~ gt1bdcgaffCw<hÐÜeR P é P é<¸1Ê Åé P »<Ó±dgt~Ò ei)lncÀj¹qwH¿ngah dgjga¹qbxeic[ljydcgmlnhdfå~Il3alç j[enÊgifjbdcw<bQw<feRbmlnc @gic|QgacbQw<ydcÅgt1bdcC¹qgac EV

gj EPp

EV,P = 1 −XV,P +G1X2VP

+3

4

√πG2X

3VP, Ð » P ¸kÓ

wIH^~Cgifaw g4uqa>gahkj[f G1gj G2

fw<hkjdeijgac¹¡>hdeafbmlc~Ilà<eawn¹¡ej[c>gtdg@~Ilalç j[eqÐÜj[ln|d~>g)lny^ PVO Ó P

4:;@W0.#*+#"?-L*+"? #,4%36 8-LzeQ #8-x"?*8%&=0K-8"<%M0A-@#*F ##-L, #/"?-K0A-.272&-Ñ-gaf±C¹qbxeimlnhdigaf ZV

gij ZPaln~Ciyd~Ceigaf±hdgaw<ccgafbxwnhddgahkj RVy5 q·dgafÏ)lfÏ>de)lyBtxÊ ~ClRfw<ydcagaw<¹qbQw<c i

j[lnhkjjw<y ¾w<ydcfÅgahZbzc[lj]RVydg·ydhdg·C¹qbxeimlnhdig·>hkj[gachdg mhd>g P Ý>hdfÒÊdaw<hzfCzeac[lhVj RVydg3~>g@#Æmln¹qbZlniw<ydf ij]RVydg·mlnhdfÅ~Il!)l) >je@gafjaceae·bmlncÅydh"à<eahdeic[ljgaydcdg@jß¿ bxg·áÆdeingahd>h_5 C¹qbxeimlnhdig Zd

ÊRgac|xgicSj O »/lgih"deidydj~Ò >¹¡bQeadlnhdagtlnaw<yzfj[QRVydg Zc

dg@~Ilºalç j[e@mlnhdfÅig·alnfÅhdw<h^Cdealn~ p

Zc =ZP + Zd

1 + Zd/ZV

. Ð » PVO ¼kÓ

Ý hzwnj[gic RVydgfE~ C¹qbxeimlnhdig Zdgifjà<c[lhddgnÊ Zc ≈ ZV

Ðëàneahdeic[ljgaydcdgzea|djÖÓ gijf Zdgifjbxgj[j[gnÊ

Zc ≈ ZPÐëà<eihdeaclj[giydcÅdg@bdcgiff>w<hEÓ P

" @!@!27%3"?*5%& 2feV0<*F23##*FEG-K#-x:;%7"?G@hJ8#-.,D@#*Fk0."<%3@#"<%M0tlnhdf~>g)lfqzg ~ eij#l~Cw<hdhdlnà<ggih bdcgiff>w<h dg ¹qCacw<bdÆdwnhdgafbmlcºceiaCbzcw a>jenÊ9~CgfÀ¿1fÀj[âi¹¡g dg ~Cl

mànydcg O?P ½@gifjydhdg)l) >j[egt1i>j[eigbdlnc±ydh¹qCicw<bdÆzw<hdgyzjC~C>fegihea¹qgijj[giydc q·~Ò ydhzgzgfgafåg t j[cei¹¡j[eifgjÛÜgic¹qeagTq~ lnyzjcgRgt jcea¹q>jeRbmlncydh^¹qCacw<bdÆdwnhdgRceiagab1j[gayzc P

lnc >fgah^deazyd>jln~>w<cf)R yzgnÊmf~CgR#Æmln¹qb dgRbdcgaffCw<h^gafÀjÅydhdÛÜw<c¹qgtmlnhzf~Cl3)l) >jenÊm~Ò g tzbzcgaff>w<hdg¡~ C¹qbxeimlnhdigqlnaw<yzfj[QRVydg¡zg~Il )l) >je¡fgayd~>g_ÐÜgah(fydbdbQw<f[lhVj3bmlncÀÛëln>jga¹qhdgij·c>à<Czg~Il fydcÀÛëlnag¡dyceaagibzj[giydc#Ógifjdw<hdhdeig_bmlnc¡~ gt1bdcgaffCw<hÐ » PVO ¼kÓÖÊÏw H Zd = 1/Ye

gifj ~Ò >¹¡bQeadlnhdag^lniw<ydfÀj[]RVydgZdy¹qCicw<bdÆzw<hdgRea¹qgijj[giydcTj O ½l P3 lnyzj[cgbdlncjaÊ~ C¹qbxeimlnhdiglnawnydfj]RVydgdgaf^¹¡>acw<bdÆdw<hzgaf eij[ln~Cw<hzf^zg~Cln|Qw<c[ljw<Ccg_eij[lnhkj j[câif

à<clnhddgÊ lc Cfq<eacmgRVydg_~ gaccgiydcfydc~>gafceafyd~>j[lj[f¡gahdà<gihddceag_bmlc~ lnbdbzcw=t1C¹¡lj[>w<h Zc ≈ ZV

" ! "# "#$ _Gv

gafÀj>hzÛÜeacCgayzcgTq¡¼1Ê¼<¼ O ¶gah"|mlnffgafÛÜceRVydgihdagif P Çågjjgºlnbdbzcw=t1C¹¡lj[>w<h bxgiyzjdw<hd ijcg·aw<hdfCdeiceigaw<¹q¹qgtfy5uqflnhkj[g·aw<¹qbzj[gRjgahVy"zg·~ClbdceiaCfCw<h cga#Ædgic#Ædeig3mlnhzf~>gafej#ln~>w<hdhmlnàngaf PÚ lncGln>~C~CgiydcfiÊi~Ò g t1bdcgiff>w<h·dg EV

Êçlnbzbmlnc[lCfflnhkjGmlnhdf-iga~C~>g dg ZVÐëe4R P » P ½<ÓgifjG#Ædw<>f>g gah@ÛÜw<hdijCw<h

dgÅ~Iltàkl¹¡¹qgdgÅÛÜce4R yzgahdigafåg t1bd~Cw<j[eiggijÏdgafÏd>¹¡gihdf>w<hdf/dgifiw<ydbd~>gaydcf±yzj[>~CCfeaf P ±y¡~>gaf d>¹¡gihdf>w<hdfdgafaw<ydbz~Cgayzcfy1j[C~>CfeafgijR~>gafà<ln¹¡¹qgafdg·ÛÜceRVydgihdagif@gt1bd~Cwn>j[eigafiÊx~>g3de<ga~>w<bdbQga¹qgahkjR¹qw1dln~Ð » P Y Ólydhdg·aw<hkngacàngahdig!lnffg º~>gahkj[g Egih_g xgijaÊq O ¼<¼<¼ ê Êx>~5Ûëlnyddcln>jàklcdgic O ¼n¼ O ¼<¼ºjgac¹qgafbxwnydcl)nw<Ccydhdg@bdceaa>f>w<h qqÞ#ÆdcgafÅf>à<hdm)lj>ÛÜffydcÅ~Il3ln~>gaydcdg EV

j » Z l PÑGl ln~Cgiydc@w<|zjgahVydgl)nga~>g!deinga~Cwnbdbxgi¹¡gihkj q aw<ydcÀjjgac¹¡g!bmlncRjc[lnhdfÀÛÜw<c¹qeagqdg¡Ñ6lnbd~Ilag"Ð » P ¸kÓ

gafÀjj[câifÅbdcw #Ædg@dgRagi~C~>gt)ln~>ayd~>eagUqbmlncÀj[>cådy^deinga~>w<bdbQga¹qgahkj¹¡w mln~ P Ì h^àklncmlnhkj O ¼<¼ O ¼<¼jgac¹qgafmlnhdfå~Cgzei<gi~Cw<bzbxgi¹¡gihVjÅ¹qw mln~9Ð » P Y ÓÖÊ1iga~Cyz i ixgifjjcâifbdcw #Ædgtdy zei<gi~Cw<bzbxgi¹¡gihVjÐÒ» P ¸kÓ q½3#Æzxcgaff>à<hdd)lj>ÛÜfaÊzlny^dgaffydftzg O ¼ ê @bQw<ydcdgaf¹qCacw<bdÆdwnhdgafjß¿ bxgÑGÉ O Ú gjén¼ ê bQw<ydcÅdgifÑGÉd» Ú P Ñ6gdeinga~>w<bdbQga¹qgahkj ÐÒ» P ¸<ÓRbQgay1jºdw<hzj[cg¡yzj[>~C>femlnhzfj[wnyzj[gq~Il àkln¹q¹¡gzg¡ÛÜce4RVydgahdigafºg t1bd~Cw<j[eig¡~>w<cfdgafÅej#ln~>w<hdhmlnàngaf P Çågijjgeiac>j[yzcgRdg EV

fgi¹º|d~Cg@~Clº¹¡>gayBt lnmlnbzjeag·lny bdcw<|d~Câi¹¡g|j » Z l PÌ±h mh5ÊÅ~Il²)l) >jeÛÜw<c¹qeagbmlnc ~>gaf ¹¡>acw<bdÆdw<hzgaf gij"~Cgiw<ydbd~>gaydciÊaw<yzbd~Cgiydc d>j^dgmà<clnhd´<w i

~Cyd¹qgn(w<y§aw<yzbd~Cgiydc¡z>jKmgah w<hddgifbd~Clnhdgafon<ÊÏbQgay1j j[cgâwnhdf>deaceagZiw<¹q¹¡g ydhdg")l) >je"dg ÛÜw<c¹¡gm`RVymlnf i ¿ ~CChzdcQR yzgn_ÛÜgac¹qeag q^fgafgt j[cea¹q>jeaf P Ñ-gafaw g4uqa>gahkj[f G1

gij G2#Ædw<>fCf@fw<hkjºdw<hz¡igayBt

aw<ccgifbQw<hdmlnhkj qagjj[g·à<eiw<¹¡ej[cCgtmlhdfÅ~Cgtj#l|d~Cgalny^ PQO p

G1 =8R + πR2

π(2R + 1)2gij G2 =

R3 − 6R2

3√π(2R + 1)3 ,

Ð » PVO?O Ó

wIH R gifj~CgRclnbdbQw<cjÅ~>w<hdà<ydgiydcfyzczIln¹qâijcgtdyî¿ ~>Chdzcg P S-."$%&1J8-./:9%36 8-@#0."?#%3,0A-K@#*F 2&* 8G/:;- w 8' FF _3lbdcâifå~Cgc[lCfwnhdhdga¹qgahkjdg lc CfÏgt1bQw<fetdlnhdfå~>gbdlnc[lnànc[lnbdÆzgbdceaieadgihVjaÊz~Ò >¹¡bQeamlhdaglniw<ydf i

j[QR yzgdgj[clnhdfÛÜgicj zg~Cl)l) >jegt1aj[eigbdlnc/ydh!¹qCacw<bdÆdwnhdgea¹qgijj[giydc 5 lnd¹q>jj[lnhdig YegafÀjfgahdfC|d~>g i

¹¡gihkjåeiàkln~Cg q ZVP ÝChdf Êk~ lnd¹q>jj#lnhzaglnawnydfj]RVydgtdgj[clnhdfÛÜgicjådyf¿ fÀj[âa¹qgt¹qCicw<bdÆzw<hdgaf i iw<ydbd~>gaydc

bdcgihmlnhkjºgihiw<¹¡b1j[gq~Il"zCffCbml j[Cwnhzydglny.tg gj[f·j[Ædgic¹q]RVydgaf·mlnhzf3~Il )l) >jenÊGhdwnjeag YT1fi eiacj)Ê

mlnhdfÅ~>gR)lnfÅwIH ~CgR#Ædln¹¡b^dg@bdcgaffCw<h^gafÀjydhd>ÛÜwnc¹qgnÊYT1 =

1

ZV+ Ye + Yr ,

fw<j)Ê15lbdcâif~Ò g tzbzcgaff>w<h"zg ZVÐÒ» P ½<Ó

YT1 = jωV

γP0CTH + Ye + Yr ,

Ð » PVO »<Ól)<ga

CTH =γ

1 + (γ − 1)EV, Ð » PVO ékÓ

wIH EVgafjdwnhdhde@bmlnc·ÐÒ» P ¸<Ó P

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YT1 = j sin k0`

[

S

ρ0c0CTH +

ρ0c0S

YeYr

]

+ cos k0` (Ye + Yr) .Ð » PVO=Z Ó

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YT1 ≈ j k0`S

ρ0c0CTH + j k0`

ρ0c0S

YeYr + Ye + Yr ≈ jωV

P0γCTH + Ye + Yr ,

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pte

qte= Zt0 tanh j kt`t ,

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Zt0 =ρ0c0

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kh =1 − j√

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√

k0

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2

√

k0

` ′v, Ð » PVO ¸kÓ

`h = λ/(ρ0c0CP ) gj ` ′v = µ/(ρ0c0)

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γP0

CTH + Ye + Yr +n

Zte

.

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YT2 = YT1 +n

Zte

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CP (Ω) =CP250

PP (Ω250)PP (Ω) , CT (Ω) =

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2mÐëe4R P O?P ¸<ÓÊ wIH~Il

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bxwnydc@~ eij#l~Cw<hdhdlnà<gºgihbdcgaffCw<hbmlnc·ceiaCbzcw a>je P Ì ~>~CgcgibdceifgihVjg~ C¹qbQeamlnhzag!lniw<ydfj]RVydgqdgq~ gah ifgi¹º|d~CgÅ¹qga¹º|zc[lnhdg i #Æmlncà<glnccCâacg P Çågjjg#Æmlncà<glnccCâacggafÀjawnhdfj>j[yzeagÅmlnhdfÏ~ ln>c aw<h dhdegihkj[cg~Il¹¡gi¹º|dclnhdggij~Ò ei~Cgaj[cw1dglnccCâicgnÊVmlhdf~Cgifåj[cw<ydfmlnhdfigijj[gtei~Cgaj[cw1dggij~Cl3)l) >jetlncc>âacg P Ñ-gj[cwny5 eaàkln~>Cflj[>w<hdgbzcgaff>w<hÐNj[yd|QgalnbdC~>~Iln>cg)Óh5 l"ydhc<~>gRVydg¡mlnhdf~Cgaf@j[câaf·|mlnffgafÛÜceRVydgihdagif^Ðëgihdgaffw<yzfdg3»n¼ ê çÓÖÊEÛÜceRVydgihdagiftgah"dgaÆdwncfzg~Cl!à<ln¹¡¹qg@5 yzj[>~CCflj[>w<hÆml|d>jydga~>~CgRdgif¹qCacw<bdÆdwnhdgafeijyddCeif¡>aÐÒÑGÉ O Ú ÊÏÑGÉd» Ú Ó ÏlnChzf Ê ~Cliw<hkj[cC|dyzjCw<h dg^ag^j[cwny mlnhzf~Il#Æmlcà<gZlnccCâacg bxgiyzj j[cghdeaàn~CCàneag P tlnhdf~Cg@)ldcg·5 C¹qbxw<cÀj#lnhkjfÅjc[l)lnyBt^dg·¹qw1dei~C>f[ljCw<hxj » Y ÊÀ» \ l ÊEÄlnf¹ºydffgah_bzcw<bQw<fg·ydhdg¹¡w dei~CCflj[>w<hlnffgawn¹¡bd~>âijgzgagjågihdfgi¹º|d~>g¹qga¹º|zc[lnhdg i #Æmlncà<glcc>âacgfw<ydfÏÛÜw<c¹qgÅ5 yzha>ciyd>jea~>gaijcQR yzgReRVyd>ln~>gahkj PÚ w<ydcDydhdgÅlnbdbz~CCalj[>w<h@j[ga~>~Cg R yzg~ eij[ln~Cw<hzhmlnà<gbmlncDceaiCbdcw1i>j[eÊ~Ò >¹¡bQeadlnhdag (Za)i=0

bQgayzj/fC¹qbd~>g i¹¡gihkj ijcgÅlnff>¹¡>~Ceag q~Ò >¹¡bQeamlhdag5 yzhqiCcaydjDceafwnhmlnhkj±feicCgRÐëa>ciyd>j9jß¿1bQgTmnÄÑGÇ ndÓ9gij/aw<hdfCdeiceigaw<¹q¹qg>hddeabQgahzmlnhkj[gRdgifaw<hzd>jCw<hdfln¹º|zIlnhkj[gifRÐÜmlnhdf~Cl¹qgafyzcgtwIH igafiw<hddj[CwnhdflncCgahkjåbxgiyEÓ PÌ ~C~Cgbxgiyzjln~Cw<cf ijcg@eaicj[g@fw<yzf~ClºÛÜw<c¹qgDp

(Za)i=0 =1

jωCa+ jωMa +Ra ,

Ð » P »<½<Ófw<j

(Za)i=0 =1

jωCa(1 − ω2MaCa + jωRaCa) .

Ñ± lnd¹q>jj#lnhzaglnaw<yzfj[QRVydg qa>ciyd>jw<yzngacj Ya = 1/(Za)i=0 ydh¹qCicw<bdÆzw<hdg·bxgiyzj ijcgºeiac>j[g

fw<yzf~Cl3ÛÜw<c¹qgRfyz>lnhkj[g¡Ðëhzw<c¹¡ln~C>feigDj »<»/lIÓ

Ya = jωVeq

γRPR, Ð » P »nÞkÓ

wIH PRgafÀj3~Cl bdcgiff>w<hfÀj#lj]RVydg!mlnhzf~>gaf@aw<hzd>jCw<hdfRfÀj#lnhzmlncdgqceÛÜeacgihdagqgij γR

~Cg!c[lbdbxwncj@dgif#Æmln~>gaydcf¹¡lnff]RVydgif qbzcgaff>w<hgj q·nw<~Cyz¹¡gÅawnhdfj[lnhkjmlnhzfåagifå¹ a¹qgafÏaw<hzd>jCw<hdfiÊkgijwIH~>gnw<~Cyz¹¡g

Veq =CaγRPR

1 − ω2MaCa + jωRaCa

, Ð » P » Y ÓgafÀj lnbdbQga~>e_~Cg^<w<~>yd¹¡gZeRVyd>ln~>gahkj dy¹qCicw<bzÆdw<hdg P Ëß~tcgabdceafgahkj[g~Cg^<w<~>yd¹qg_aw<¹qbd~>gt1g_5 ydhàkl gahzÛÜgic¹qe mlnhdfqydhdg^)l) >je"cCà<>dgnÊ/fyd|d>ff[lhVj¡dgifawn¹¡bdcgaffCwnhdf i zeij[gihkj[gaflndIl|mlj[QRVydgafiÊ dg^jga~>~Cg

v _ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F

0."<%3@#"<%M0

fw<cjgkRVydg~ C¹qbQeamlnhzag!dgig<w<~>yd¹qg¡iw<¹¡bz~Cgt1g¡dgàkl fwn>jeaàkln~>gkqZ~Ò >¹¡bQeadlnhdagqlniw<ydfÀj[]RVydgdy¹qCicw<bdÆzw<hdg P Ú lc3awnhVngahkj[>w<hdj»n»/l ÊGag¡<w<~>yd¹¡gqgafÀjgt1bdcC¹qegih(ÛÜw<hdijCw<hdgfl"ln~Cgiydcºlny.t|dlnffgafÛÜceRVydgahzagaf Vbf = CaγRPR

gijdg_~Ilbdyd~>f[ljCw<h§dgZceafwnhmlnhdig ω0 = 1/√

MaCadgZ~Ò >¹¡bQeadlnhdag

(Za)i=0bQw<ydcÅ~Il`R yzga~C~>g (Za)i=0

gafÀjceagi~C~>g p

Veq =Vbf

1 − ω2

ω20

+ jωRaCa

. Ð » P » \ Ó

Ñ-g@nw<~Cyz¹¡gRe4R yz>ln~Cgihkj Veq ydh^¹qCacw<bdÆdwnhdgRgafÀjdwnhd@gahkj[>âacgi¹¡gihkjzeij[gic¹qChde@bmlcfw<h^<w<~>yd¹¡g

lny.t|mlnffgaf3ÛÜceRVydgihdagif VbfÊ-~Cl^bdyz~Cf[l j[Cwnhzgceifw<hdlnhdag ω0

gij~ ln¹qw<cjCffga¹qgahkj Ra P Ñ g t1bdcgiff>w<h

ÐÒ» P »ÞkÓfyzbdbxwnfg RVydg~ lnd¹q>jj#lnhzag YagafÀj@ChddeibxgihdmlnhkjgdgafRiw<hddj[Cwnhdftln¹3|dIlnhkjgafaÊxga~>~Cg3gifj@g tzbzc i

¹qeag bmlcaw<hk<gihkj[CwnhHj »<»/liw<¹q¹¡g ~Òlnz¹¡jj#lhdaglnaw<yzfj[QRVydg dy <w<~>yd¹¡gaw<¹qbd~Cg t1g VeqÐë~Cyz i ¹ a¹qg

>hddeabQgahzmlnhkjdgafaw<hzd>jCw<hdfln¹º|dClnhkj[gaf[Óåmlnhdf~Cgifaw<hzd>jCw<hdfåfj#lhdmlnc"dg@ceiÛÜeacgahdig PÇågijj[g!gt1bdcgaffCw<hdg!~ C¹qbQeamlnhzaglniw<ydfÀj[]RVydg ydh¹qCacw<bdÆdwnhdgºh5 gafÀjRVy5 ydhdgqlnbdbzcw=t1C¹¡lj[>w<h5Ê

jw<yzj[gÛÜw<Cf6fy5uqf[lnhkj[gåbxw<yzc9~>gÏaln~Ciyd~VdgÏ~ g4uq)lni>je5 ydh¹¡>acwnbdÆdw<hdg±bmlc9ceaa>bdcw aj[e P Ëß~ gifjDagabQgahzmlnhkjq hdwnjgacRVydgigijj[glnbdbdcw=tz>¹l j[Cwnhgafj Czgzgfgahdf"mlnhdf igacj[lnChzf^alnf P Ì h g xgij)Êej#lnhkj <eicmeRVydg·~ g4uq)la>jeRgah"bdcgiff>w<h^5 yzh"¹qCicw<bdÆzw<hdgRgafÀjzeabQgahdmlhVjgzgafaw<hdz>j[>w<hdfÅln¹3|dIlnhkjgafdgRÛël w<hfCà<hdm)l j[>ngj » Y Ê »n¸`lÒÊ C~Qfgacln>jåln|dfydczgt5 ydh bxwnChkjåzg ydgj[Æzeaw<c]RVydgzgÛëlnCcgyzf[lnà<gdgt~Ò g tzbzcgaff>w<hÐÒ» P »n½<ÓDdg (Za)i=0

Ðë>hddeabQgahzmlnhkj[gÅdgifiw<hddj[Cwnhdf ln¹º|zIlnhkj[gif#ÓDmlhdf±~ gt1bdcgiffCw<hZÐ O?P[\ Ó9fga~>w<h¡~Cl$RVydga~>~Cg~Ò gu¡alnaj[e Mp

gifjt>hk<gacfgi¹¡gihVjtbdcw<bQw<cÀj[Cwnhdhdga~>~Cg qq~ C¹qbxeimlnhdig (Za)i=0P hdg·j[gi~C~Cg·lnbzbdcw #Ædg3bQgay1j

igabQgahdmlhVjijcg·gahk Cf[là<eag3l mh^5 wn|dfgic<gicsRVymln~>>j#l j[>nga¹qgahkjÅ~ ChBmydgahzag@dCcgaj[g·dgafbmlncln¹¡âj[cgafdy¹qCicw<bdÆzw<hdgRfyzc~Cl3ln~Cgiydcdg@fw<h^g4uqalnaj[e PÑ-gRcgabQw<cÀjÅdgafgt1bdcgaffCw<hzfºÐ O?P ¸<ÓÊ6ÐÒ» P »ÞkÓågijÐÒ» P » \ ÓåmlnhdfÅ~Ò g tzbzcgaff>w<h(Ð O?P[\ ÓådgR~ g4uqalnaj[eRdw<hzhdg

Mp = − U0

Sd0

Veq

γRPR=

Mpbf

1 − ω2MaCa + jωRaCa

, Ð » P »n¸kÓRVyd6h5 gafÀj RVy5 ydhdglnbdbzcw=t1C¹¡lj[>w<hgijwIH Mpbf

gafj~Il3ln~>gaydczg MplnyBt|mlnffgafÛÜce4R yzgahdigaf P

Ý>hdfÒÊzgah^w<|dfgacÀnlhVj~Il3ln~>gaydczg·~Ò guq)lni>j[e MpÊmC~5lbdbmlncln>j RVydg p

~Cl3ln~Cgiydcdy <wn~Cyd¹qg@lnyBt|mlnffgafÅÛÜce4R yzgahdigaf Vbf = CaγRPRgifjdw<hzhdeagRbmlc |Mpbf | ~Il3ln~>gaydclyBt |mlnffgifÛÜceRVydgahzagafdy^¹¡w dyd~>gtdg Mp

Ð mà<ydcg» P[Z ÓÊ ~Cl_ÛÜceRVydgihdag^dg ceifw<hdlnhdag F0 =

1

2π√

CaMa

gafÀjl jj[giChkj[gRVymlnhz§~Cl_bdÆmlnfg"dg MpgafÀjzg

¸n¼ EÐ mà<ydcg3» P ½<ÓÖÊ ~Òl¹¡w<cÀj[>ffgi¹¡gihkj Ra

Êziw<hdhmln>fflnhkjÅ~Cgafnl~Cgayzcfdg Vbfgij F0

Ê1gafÀjÅdeadyz>jdy clnbdbQw<cj |Mpbf ||MpF0

| =

ω0RaCaÊ |MpF0

| eij[lnhkj~>gR¹¡w dyd~>gtdg@~Ilºln~>gaydcdg MpbQw<ydc f = F0

Ðmànydcg·» P Z Ó P 0<J##-x8-x:;-.,= 8-L#%3-."$-Çågijj[g ¹¡ej[Ædw dgÊWqbdceifgahkj¡w<|dfw<~Câj[gnÊDgafjqfw<¹¡¹¡ln>cga¹qgahkj3deaic>jg mlnhdf!~Cl_hdw<c¹qg^ÇÌ ËºÞ O ¼<¸ Z`i

» j»<»l±gijeij#l>j·yzj[>~C>feag!bmlc·zghdw<¹3|dcgiyBt~Iln|Qw<clj[w<>cgiftdg¹qeijcw<~>w<à<>gnÊiw<¹q¹¡g!~ ËÍÈtÎkj é<¼`lÒÊ-bxw<yzc,2465#%27!$5*,#%&:=HKCF!%#$5 5+*, !1 d 274#%*Z>27-0!@L#$Cd #%-0!&;:HK2-*#%+C-/C#@27,;=>2#%2> !$C&#$*Z$#c b5,GHKC&I0G*

ba27,-0!]L627#D$2)ba27,G#D27G=` 4/2$$&de#%C&I0G-6>F&Vbf

;TZ2de#%C&I0G&-/>FE:=^#%C&$5-627-/>F0 = ω0/2π

F!D,Yd 2> !$&G#D:=EI0G/2 * !$CQ = ω0/RaCa = 1/Ra

√

Ma/Ca

M 5G Dd 2> !$&G#:=YL6&#"!%&1/Q

X n : !"!%i-57!%* 5- :=Dd 2>F!$G/#:=YI0G/2 * !$C;/>@?5*Z$* LB5G#:=&>5HKHK5=:=*S!%C&:=9-/57!%27!$*,5-;<- 3 2JL/2 b #@27*,HK-0!i:=9"&-/DL? 8="*ZI0GKL65G#i>2#%2> !$C&#$*Z$#G/-9* HKLBC&:2-/>F92>F5G6f!%*,I0G;>$3 &"!1L65G#@IG/5*<,,]- 3 f!1L/2#$!$&-0G/D:/27-/,L#%C&$-0!1:=5=>FGHK-0!n / -/"!$* !$G=! 2a!$*,5-627:= . C!$#%5,5+*

"# # " " " v '

102

103

104

−42

−41.5

−41

−40.5

−40

−39.5

−39

−38.5

−38

−37.5

−37

h/#$CI0G-/> M X

|Mpbf |

|MpF0|

|Mpbf |/|MpF0|

F0

Â » P Z ^ÎqÐ Û O Ú Ó!"$#%4u'& &)(*+,-#.0/"¹'(1&)32dÆ03/4*ß¿253ÑGÉd» Ú P

102

103

104

60

80

90

100

120

140

160

180

h#%C&I0G-6>F M6 X

789: ;

F0

Â » P ½ Ú Æ =< Ð0aàVÓ>"?#.u@& &A(*B+,-#%/"¹@(&A=2dÆ3/C*ß¿D25Ñ9Éz» Ú P

vG__ W()-."/J!#%36 #-., -<:;0<J##-., -.a !,*5E-K@W 8 2feg0A=*+23##*5E-K#-., :9%3"?/G@!J#8-.,|@#*F

0."<%3@#"<%M0

+ *B)¹'(1/) < 2 ¹¡â *B < CaÊ Ma

A Ra>?#.(>¹@25+A /&A q &A(1 &A(*63 ) * (Za)i=0

0 < ¹'(1&)32dÆ03/ </ 253&A" << )/ *B(1)1a¹'A/ * P( P #" #

Vbf " "!

/M¹'(1&)32zÆ3/ < * 0*B(1( < + A/ +i¹@ **B) / < / &A3201A0 2 + Iq &A * )* 3/ * ?# D*Bt*B +a¹'(*+ < *Û )¹'+A 2 1@¹'(1&A 32dÆ=/ 03/ *4(5Û 0*¹@ < A4 31z¹@"+R ( 1)/ *aÊ &AA( i &A(/-#%+)* / * 2 <C 1(C¹'A/ *+ P / *B=( < (>âa¹'t¹'(1&A 32dÆ=/!Ð)/"à=+A/+) /^¹'(1&A 32dÆ=/ @¹' < 0"21 <2 *B(* R "1 < A.t 0* < ÓÊ ) / *q 2 3( *B+) C < * 0*B(1( < + &A<¹q¹@4+A&)A20*AÐ mà3 » P5P Ó P Ñ #.(>¹@25+A /&A+A1)&)*(]R >* / < Û A * ZE = ur0/ie

¹' < 0+A>A/ *B !1 < )Bt¹'(1&)32dÆ03/ <2 )¹')* "+ *B)¹'(1/) 1,2 D0(* < u'& &)(*B+ < < A.t^¹@(&A=2dÆ3/ < Ð+4R PO$PQOY Ó p

MpeMpr = ZEYT .

Ñ à ¹q¹' 0Û +4R A/&) < + * / * &#Æ3( < (1 *B)11 R > 20 < < (3/ &)3 < *(]R < * < 202 < +A /0(>Û 3¹@)* 1 < )* < *[Æ0A¹'(]R < 2 *B(* < Ê ?# z¹@(** /&A &A3 < *B(QR0 * / < Û A * YT

25AD* 13 < *B 3//0+AnÊ # 20â < ?#. tD2 << (3/"Ð » P »<ÓÖÊ3&A<¹'20*>*B)/' < +R *(13/ < ÐÒ» P » Ó *Ð » P » \ ÓÖÊ < = < Û 3¹@

γP0YT

jω≈ Vc + Vf + Vfe + Vfr + (Vbf + Vbfe + Vbfr)

γP0

γRPR, Ð » P é<¼kÓ

A& < /0=* *(13/ < + /(1 < / < &) 2 à3 2dÆ * +0* 2 à3 2dÆ » P » Ê )* IH 1 < 3d¹' < Vf

)* Vbf

< 3/ *C A *B(Û < ¹@(&A 32dÆ3/0 m¹@ < +n P

. *,>#$5L?5- aHE"G/#$C

Vf

. *,>#$5L?5-/#%C&>L=!%G#

. *Z>F#%5L?5-CHKF!$!$&G#

ur0

ie

; *ZHK*,>#$5HKCF!$#%*ZIG/

Veff

[ *Zf!%5-

. *Z>F#%5L?/5-#%C&>F&L=!$&G#

. *Z>F#%5L/?5-C&HE!"!%G#

u′

r0

i′eY*B` Y 08`

Â » P Î < 3d¹', A&)*(>Û Bt =<< < Û +R )/&A < P / < / < )&A3/0 *Bi¹@2 < Ê 1¹'(1&)32zÆ3/ m¹' < +=n < * i¹@2 &A+ 2 / 2( < *B3/ ¹i¹@( ¹¡â *B )*·¹ /( #./0 ( < ¹'(1&)<¹'+)*(]R Ê &)ABt i &A( +)* / * + * 3//+ < -#.0/ 253(1/ *, ( i¹@)/ < (3//) P Ñ #.(>¹@25+A /&A!+)1)&)*B (]R *B / < Û ) * Z ′

E < * 3 < ¹@ < +A )* 1!2( < *3/ < * ¾ < *+

¹ /0(CâA rq + ( < A $#%+aà 1(*B+! < 0ABt@(C¹'25+A /0&A < ZE * Z ′

EÐ mà3 » P EÓ P Ñ =1d¹'Åà3+A i

¹@+ *B (]R Dq@$# / *4 2( < *B3/KR0( 25A¹')* &A)* *B +ià (*B+ < (C¹'2 +) /&) < +)1)&)*B (]R < < *,/=*+Veff

P Ñ ,2(*! < 4u'& &A(*+ < MpeMpr+)* / * (1/0+A25A/ / * < &)3/(*B(1=/ < / < 1 < R A1 < (1

< *C0+)*B)¹'(1/+Ê(1 (A/ *MpeMpr = Z ′

EY′T = ZEY

′T = ZEYT ,

"# # " " " vGv

&AUR (3//

γP0Y ′

T

jω= γP0

YT

jω≈ Vc + Vfe + Vfr + (Vbfe + Vbfr)

γP0

γRPR+ Veff .

Ð » P é O ÓÌ / +aà / * < tD2 < < (1=/ < ÐÒ» P é<¼kÓ>)*3ÐÒ» P é O ÓÊ( (1)/ *

Veff = Vf + VbfγP0

γRPR. Ð » P ék»<Ó

Ñ 31d¹' à3+)<¹@+ *B (]R Veff¹' < 0+ q ?# / * 2( < *B3/ )2+ < )/ *>=/& < <¹q¹' 31z¹@

Û 3/ * Vf0 ¹@(&A=2dÆ3/m¹' < +=n * < 3/ =1d¹' +R ( 1)/ * Bt =< < < Û +R A/0&A < Vbf&A3 (Cà=+ Û &)*A γP0/(γRPR)

P Ç> 3d¹'ºà3+An¹@+ *B(QR Veff < * 225A+ 1 31d¹' A& *B(Û

¹@(&A 32dÆ3/0 / < 1 < &)3/(*B(1=/ <4 ¹ ( / * < P Ñ 31d¹'Û 3/ * Vf+)* / * &A3// 202=tD( i

¹ *( a¹'A/ *^Ð !» P é PQO ÓÖÊ 3d¹' Bt =< < < Û +4R A/&) < Vbf ¹'(1&)32dÆ03/xm ¹@ < +n < *

13 < +A0(* " ¹@ < 0 31d¹'Rà3+)<¹@+ *B (]R Veffq $# / * 20( < *3/ P

(1( P #$ F0Ñ¹'(1&A 32dÆ=/ m¹' < 0+n < *2 &A+ q $#%t*B +a¹'(*B+ #./ *B 5 ¹ a¹' ( ¹¡â *BÊ!$# 0*

t*B +a¹'(*B+C * C+)* / *åÛ A¹@+)42 >/ * / < & *B)!+a¹')* *B)RÐ 4u'& &)(*+4&An¹@2tD4&)3// )Ó5tD+ < / 20( < *3/¶¹ /(,-#%/ ( < ¹'(1&)<¹'+)*(]R +)* 13//0+A P Ñ #%4u'& &A(*+ A/ 20 < < (3/ 0¹@(&A=2dÆ3/, +A&)A20*A3Ðë¹'(1&)32dÆ03/|m¹' < +=ndÓ < #%+A&)(* (&A(±Ð +R P8O?P \ Ó

Mp =ur0

pr=ur0

ue

ue

pe

pe

pr,

IH ue < * *B)/ < (3/ 22(]R +A .t =/ < *B / < &)*A+a¹')* *BA0 * pe

2 << (13/ &)3 < i*B(]R UR -#%(1-(>¹@25 < q'?#.)/ *B+)" * 5 PÑ * / < & *B) +a¹')* *BA0à3+A/dâ) /0 3/ &)3 < *B(]R q 0/Û +4R0A/&)2 D&#Æ0 Û + iR0A/&) 0 + < 3/ /&A < 225 < +) +)&AA2D*BA0 P Ñ ( < < * ¾ < *+A ¹ /(>âA Kq &)LR 2 << (3/ < ¹@i¹ / @ +A&)A20*AiÊ0)* =/&C *A/ < (13/q"&A(1 &A(* 3 ) *)q < < 3 / < ur0

Ê **A(Cà=/!/!¹ tD(C¹ d¹æÐ + < 3/ /&) *B 5çÓ P Ñ < 2 << (13/ < Bt tD*+i¹@(*B+ < 0 * 5 < 3/ * 3 <A/ R *B0 )* 2zÆ =< 0 pr/pe

< * Å¸n¼ P Ñ 2dÆ =< 22 = * ue/pe+)* / * &A3/0/0·ÐA1

< * +A0(*B>! 2dÆ =< !!?#.u@& &A(*B+!&)<¹'21 t0 *B / < &)*A +i¹@ **A < 225 < +) &A3/0/0çÓÊ 2dÆ =< C0 ?#.u'& &)(*B+ )/ 2 < < (1=/ Mp

< * 3 < +A0(*B ¹' < ur0/ueP Ì / +A25+)* / *

$#%32 +) *(13/ 253"( +)A/ * < Û +R )/&A < 3( < (/ < $#%/ @$# 0*B nÊ A ' Û +4R A/&) + < 3/ /&A ¹'(1&A 32dÆ=/nÊ62 = R0A1 Mp

/ 2zÆ =< ¸n¼ dÊ < * 0+)*B)¹'(1/+) 2 (1/ *B)253 *(13/ P

(1(( P " 9 RaÑ 1)C0 $# ¹'3 *( < < a¹'A/ * Ra

< * +)(*B ?#. t020 < < (3/ Ð » P »n¸<Ó $#%4u'& &A(*B+ MpP Ý

Û +R )/&A "+ < 3/ /&A F0Ê |Mp| = |MpF0

| *

Ra =1

ω0Ca

|Mpbf ||MpF0

| ,Ð » P é<ékÓ

1 < ¹@ < |Mpbf | * |MpF0| +)* / * +)*A¹@(/+ < à= 2dÆ0(]R a¹'A/ * Ð mà P » P[Z Ó)* A& Ca =

Vbf/(γRPR) Ð P Ó P( P " "! Ñ #%+A25.R0 &)*B0A1 < *km &#Æ /0(CâA n A/ *B)¹' -#.+ 30*B(3/ < *A&#Æ/(QR < !¹@ < < &)3 < i*B(]R < P Ý4(1/ < ( Ê < *A&#Æ/0(]R < 0¹@ < >&)(*B+) < 20+A&)+Ai¹¡¹'A/ * < 3/ * & *B)11i¹@)/ * /3/0/+A <& A1 <!< 30Câ A/ *C/&)A * (/ /0<¹ #.(/ *A <à *B(1=/ < P


0."<%3@#"<%M0

Ç>3/&)A/ / * + *BA¹@(/ *(13/" 3d¹@ .t =<< < Û +R )/&A < Ê==0*B ÏÛ (*8R > *A&#Æ/(QR < < *(*0*B(3/ < * / 2 D&)+A+,&) 0*B)Bt )/ *Bi¹@2 < Ê2 < (A < 2 Câi¹@ < -#%304*Æ+A3 (]R < 2 < A/ * P á630* -# 53Ê < 0ABt 3d¹' < &)<¹'2 + < / < 3/ * 2 < ¹ a¹'/ *B @Ðëà=+A<¹'+)*(]R )* &)3 < *B(]R )Ó P # D*B 2 *aÊ31¹'(1&)32zÆ3/Um ¹@ < +n /-#%+)* / * 2 < 1(>¹@)/ *+nÊ (1) <¹@i¹ / /-#% < *>2 =< t & *Ba¹'A/ *> ¹i¹@rR -#%A/ 0*(1( <B *B(1=/ /3¹ P Ì / /5ÊD?# 22=tD(C¹ *(13/< )13/ R )1 $#%(1/5 0A/&)'0 < * < *[Æ)¹'(]R < < * /+aà3(Cà3 nÊ-&)<¹@2D*B *B)/ ' à ¹q¹@ºÛ +4R A/&) < &#Æ3( < (1 )*" < 0(C¹'A/ < (13/ < <¹ (1/0nÊ5/-#. < * 2 =< *3 ¾ 30 < (A/ +)( +A / < 2 *B(QR nÊ 3d¹@,0 &)32A + * / * ( 1 PÚ 30 &ADR ( < *, ¹@ < Û +R A/0&A + < 3/ /&)nÊ- 2 D&)+A+ < * << ( *Bâ < 3 Ðë¹ *B+)(A Ê *a¹'2 < Ó q¹')* *B A/ " Ê--# 0* / *,21 < R -#.( < * =< < (Vu'&A(1 )/ 2 *(]R 0*B 3 )!/ * / < & *B) +a¹')* *BA0>3/ * < & &)*+A( < *(]R < 2 ) )/ * )*B < Bu < ¹q¹@)/ * (1)/&A3/0/0 < P Ì / /5Ê Bt A/ (1 3/ < Û +R A/0&A + < =/ /&) ÐNÛ +4R0A/&) << +A +)çÓÖÊ!1 <3/ < ( 1 < 25A A/ * 3(14/ (1/5A/&) /3/ /+ià31(>à3 01 < ,1 + < * *,)*")1 < / < 3/ *2 = * / * 2 =< 2 ( < < )/'&)<¹'20*B! / < ¹@ < P á630* &)A&A(0&A3/ *B ( 0rq + 1( < A < ¹' < < 0Û +4R0A/&) 4 + < 3/ /&) )*>-# ¹@= *B( << a¹'A/ *RÐ&R ( < *>+A0(*!0 F0

)* VbfÓ R (5( < R )/ *>-# )*B

=<< (C¹'2 +A&A( < < )* 3/0& 2 )m A/ * 1 < n &A<¹'20* *B)/ < ¹'ç¿3)/ < *BA&#Æ0/(]R < (>¹@253 * / * <¹@( < A/ " PÝ4(/ < (ÒÊ 5 &A302 53 *3(1 < ¹@+ *B=1<à3( 3/ * 20+iÛ +)+·¾ < R -#[q 2+ < A/ * < (C¹'(*B)Uq ¹@ < " 31d¹', A& *B(Û Bt =<< < Û +R )/&A < 2 ¹'+)*Æ" < < *B(*B0*(13/5Ê < A253 i* / * Bt 3//+) < &)3/ < *& *B),3 Bt 1) < *ß¿2(]R < 2 304 Û +R )/&A 0 + < 3/ /&A *$# ¹'3 *( << a¹'A/ * PÑ < *B)&#Æ/(]R < @¹' < -#%(C¹'2 +) /&),¹'(1&)32zÆ3/ < 2+ < A/ *+A < / < 1,2 à= 2dÆ0 < ( i / * < 3/ * & *B)11i¹@)/ * < *B)&#Æ/(]R < ¹'( < < A/ " ºÑ6ÈÌ )* A/ < à3 / < /<¹ ABt 53 *B3( < ¹'+)*3<à3(1 P Ç> < *B)&#Æ/(]R < ÊD21 < < (>¹@201 < q·¹@ **B C)/ " Ê < 3/ * < < (521 <2+)&A( < < P

" c0<J88-x%M0A/*B%-Ë < # à=(* 0 & &AAC2 202=tD(C¹ *B(3/ < < &)&A << ( < 1)4 < 2 ¹¡â *B < Vbf

Ê F0)* Ra-#%/¹'(1&)32zÆ3/ -# 2â < A < 3/@4u'& &A(*+C0+)*B)¹'(1/+)C253 < 1) < 220=tD(C¹ *B( <

VbfÊ F0

)* RaÐ 3//+) < &A3/ < *B &)*A 3 1) < 0*B)/0 < 3 < "2+)&A+A0A/ * < +)* 13// à3 < Ó P

Ñ < ¹'+)*Æ < 04+ *BA¹@(/ *(13/'+A&)(*B < / < &A 2 à3 2dÆCtD23(*A/ *>Û (* R0nÊ03 < -#%/+ * 3// à= 2 ! +A&)(12 D&)(*B+C *B 3( < ¹@(&A=2dÆ3/ < ÊD < ( *(13/ < < A < (1/ *(* < 2 30!1 <& & *B+)( < *B(QR0 < Ð Vbf

Ê F0Ê RaÓ -#%/¡¹@(&A=2dÆ3/ /¹'D( A/ * A/@(A/'$#%4u'& &)(*+ 0*B)/04253 1 <

)Bt D*B < ¹'(1&A 32dÆ=/ < D*B(1( < + < / < 2 *B&A3 -#%+)* 13/0/ à3 2 +A&A(2&A(*B+Ð&iÛ P // t0 Ê P » P »<Ó P( P # " #$

Vbf ""!

Ì / =< < < Û +R )/&A < Ê-?#.u'& &)(*B+ Mp < * R =< ( i (1/0+A25A/ / *B 0 F0

)* RaP Ñ 31d¹'"+R ( i

1)/ * Bt =< < < Û +R A/0&A < < *C03/&"+)*A¹@(/+,(1/0+A25A/ ¹¡¹'A/ * "&A < 2 ¹qâ)* < PË < # à3(* "+)*A¹@(/A ?#.u@& &A(*B+,,*3( < ¹@(&A=2dÆ3/ < 2 ¹@+ *[Æ,"+)&A(2&A(*B+ / < < & (*+ < 31z¹@ < ( +))/ * < P Ñ &)<¹'2 ( < =/@ < + < * * < 0*A/ < 253 &#Æ R C& (*+2 )¹')*" +)( 1 < 3d¹@ <" Bt =<< < Û +R )/&A < < *B=( < ¹'(1&A 32dÆ=/ < )/ < ( / * 12 *B&A3 +A&)(*C / < 1 < &#Æ+a¹ 0 mà3 » P Y.PÑ < & &A < /+A&) < <B ( < 253 &A **¹'+)*ÆD0qÐ& 1&A0- < YT

Ê MpdB)* DiffdB

Ó < =/ * )&)*B0+ <2 / 1<à3(&A(A=0& 1&)<R ( )*3 /)q $#%+A&) / / )2+ < )/ * *B(3/3à= 2dÆ0(]R < DiffdB

P Ñ < 1) < (/(*( < 0 < Vbf

< 3/ *6 < 1) < 22=tD(C¹ *( < Ð=//+A < &)3/ < *B0&)*B) = A <

"# # " " " v N

Vbf1AVbf2

AVbf3

ZE12aAZE13a

AZE23a ZE12b

AZE13b

AZE23b

M12a = ZE12a YT12a

M13a = ZE13a YT13a

M23a = ZE23a YT23a

M12b = ZE12b YT12b

M13b = ZE13b YT13b

M23b = ZE23b YT23b

Mp1dBaAMp2dBa

AMp3dBa Mp1dBb

AMp2dBb

AMp3dBb

Diff1dBADiff2dB

ADiff3dB

V ′bf1

AV ′

bf2

AV ′

bf3

; 2 &G#@* -* !$*Z27,&g2LL#%5`=* H92a!%* b&

. "G/#$:2-/ >5GL,G/#12 . &$G#%D:/27-/,i>F5GL,G#g4

>&27Z>FG >27Z>FG/

>&27Z>FG >27Z>FG/

* ABC&#$&-/>Fg:=&MpdBa

F!MpdBb

4/"&#$ba27!$*,5-+#@27L?*ZI0G

; 2 &G#%HK5=:=* ./Cg:=&Vbf

;>@?5*,$*,&27LL/#$5`=*,HK27!$*,b&HE&-0!LB5G#gHK*,-*,HE*Z$#1,&

DiffdB

:

9::

Vbf

: 0

f

Â » P Y Î+)*Æ < ( (,2 30 +)*A¹'(1/0A 1 < 3d¹@ < +R ( 1A/ * <C Bt =< < < Û +R A/0&A < P


0."<%3@#"<%M0

2+)&A+)A/ * + * 3// à=çÓ P Ñ < )Bt')/(>âA < + * 25 < < &#Æ+a¹ Ð < ) *(13/')*Ï¹'D( & *B(1=/ < Ó< 3/ * )&)*+A < ¹ / )11i¹@)/ * 2 / 325+A *B) P Ç>A10( i &)(Å¹@( 220=tD(C¹ *B( a¹'A/ *@1 < 1) < < Vbf

q +)(1/ ¾ )&)*B) / < 1@2<à= ¹q¹@' & 1&) A/(Û 3/& *B(3/ < 3/ < A *B(1=/ < A2 + < )/ * *(13/ < à3 2dÆ(QR < < DiffdB

P Ë !&A << -#% A& *B) < ¹'D( & *B(1=/ < < 1 <Vbf

R / (1 < *(C¹'DR 1 < DiffdB

< =/ *·¹'(1/(>¹ 1 < < 0 0/à ¹q¹@ Û +R )/&A < 21 <+)*A/2 << ( 1 PË + 1a¹'A/ *aÊ0&A **R¹@+ *[Æ ,0+)*B)¹'(1/ *B(1=/ /+)&A < < (*B"$#%0*B(1( <B *(13/ -# ¹'3(1/ < *B 3( < & (*B+ <( +A A/ *B < / -#.0*B)/(1 ¹'3(1/ < )Bt A < Vbf

253>&#Æ R t¹'(1&A 32dÆ=/C)/ < ( / *1 2 =*B&A=1 +)&A (*"2 , < &#Æ+i¹ mà=q» P Y Ê-&)R ( 25A¹@ *"-#% A&)*A /!¹'ç¿3A/0/< < 1) < < Vbf

0*B)/ < 2 = &#Æ R ·¹@(&A=2dÆ3/ P / < 2 *(]R Ê à ¹¡¹' "Û +R A/0&A < &#Æ03( < (1 2 30 2201(]R ) &A)* *B_¹'+)*Æ < * 0$#%30 O ¼<¼ »n¼<¼<¼ '2 = 0 < ¹'(1&)32zÆ3/ < ÑGÉ O Ú *!»n¼n¼ ½¼<¼<¼ '2 30 <ÑGÉd» Ú P Ú =4 < Û +4R A/&) < (/zÛ +A (1A0 < Ê $#%(1/5 0A/&) < Û (* <C &A= < *(]R < < ¿ < *[âa¹' < 1+ < * *2 )0* 2 A/0 < 2 32 = *B(3/ < /3/ /0+aà3(Cà3 1 < )*!253 < Û +R )/&A < 21 < +A +A < Ê$#%4u'& &A(*+,/-#% < * 2 < (1/0+A25A/ / *" < 1) < F0

* RaP

(1( P #$ F0

" 9 RaÑ < A < Û +R )/&A 0 + < 3/ /&A F0

)* ?# ¹'3 *B( < < i¹@)/ * Ra

< 3/ *|q + *BA¹@(/A&A3/H¾ 3(1/ *Bi¹@)/ * & &) < 2 ¹qâ)* < à3( << )/ * qÛ 3/ ( < ( 1' / < ¹i¹@qà ¹q¹@ qÛ + iR A/&) < Ð P Ó P / < &A **à ¹¡¹' @Û +4R0A/&) < Ê $#%4u'& &A(*+,+A25A/0 3/&RÛ 3 *Bi¹@)/ * &A <A.t 2 ¹¡â *B < q Û 3( < PË < # à3(* (1&)(@+)*A¹@(/A $#%4u'& &)(*+@@*B=( < ¹@(&A=2dÆ3/ < 2 ¹'+)*[Æ0D@ +A2(&A D&)(*B+ / < /à ¹q¹@ 0Û +R A/0&A < R ( (/&A0* 1 < Û +R A/0&A < 0C + < =/ /&)4 < ¹'(1&A 32dÆ=/ < P Ñ <4u'& &A(*+ < 0*A/ < 25A¹@ **B)/ * 13 < ÊA # 20â < $#% < A *B(1=/, < mà30 < » P ½!)*D» P Z )*$#%+R *B(1=/Ð » P »n¸kÓÖÊD0,+A0(14 A> Û +R )/&A"4+ < 3/ /0&A F0

* ?# ¹'3 *B( < < i¹@)/ * Ra2 30

&#Æ R ¹'(1&A 32dÆ=/ A/ < ( / * 20&A+A0 +)&A(*B / < 1 < &#Æ+a¹ mà=^» P[\5P Ñ <& &A <!< =/ * )&)*+ < A/ 2 A/ / * )/ &)<¹'20*B, < A < < Vbf

20+ a¹'A/ * + *B)¹'(1/+) < Pá =0*B &))* *B 2 D&)+A0 < * A& *B+) 2 0/ 1<à3(&A(A & 1&)Ð +)*A¹@(/ *B(3/ *B=* 1i¹@)/ * 0*<¹ *B( < +A)Ó P Ñ < A < (1/(*B( 1 < < F0

)* Ra

< =/ * < A < 220=tD(C¹ *B( < Ð=//+A <&A3/ < *B &)*A 3 1) < 2 +A&)+A)/ * +)* 13// à3çÓ P / < 2 *B(QR nÊ(1 < 5u * @*â < 2 )-#%(*B+) *(13/ < 2 = **A(/" < 1A0 < < * < 2 = 1 < F0

* RaP

Ç>)* *B¹@+ *[ÆnÊ &)*A1a¹'A/ * )/ < à3 tÚ Ñ6ÝHÊ 1¹'A(11) &)<¹@20<¹'( < )/ *B 0(11(*B+ * < (C¹'21(&A(*B+40·¹'( < A/ " 2 = 0*A/(!*B3 < < 2 ¹qâ)*B < =1d¹',+R ( 1A/ * PÑ #.(/&A3/ +A/(A/ *¹ ¾ ) ,&) 20&A+A0+ < *R " + *B)¹'(1/ *(13/ 0 < Vbf

< * A&)*+A 2 0/¹'+)*ÆD0xm ( +))/ *(1)11=n P Ñ < A ) < &)<¹q¹@( < < < 1 < ¹' < 0 < / < < ( +A A/ *B < & (*+ < < &)d¹ A/ *)Ê &)TR ( 25AD*C A/0"1 < + < * * < 25A )2& *B( 1 < )* (C¹'2+)&A( < P Ë < * =/&"(1/( < 25A/ i< 1 253 0*B(1( < A &))**¹'+)*ÆD0 R C1 < *B)&#Æ/(QR0 < ¹' < 0425A¹@ **A/ *> #. D*BA/0(1 /53//0 A2 D0&)*B( (1(*+ P

" F !#" %$

& &(' *) / ,+@ < .- +A&)(* < /10 243657-6 < * &A3/ < *B(*B+) -#%/ 202 )(1 +A&)(12 D&)(*B+8:9 5;5;< =>A ?>A@!B--#%/ / DC < ) )&)*3(A 8FE 2 9.G IHD&('J+ 5A 0 A(B> * -#.0/LKNMO3P 2-*Z"?J[ #%* H927#%8N 274B5#@2a!$5#$8K5dJQg>5G/"!$*Z>

! "#$%'&( *)+, - ,/.103254 687

ZE12 9 ZE13 9 ZE23

F01 9 F02 9 F03 Ra1 9 Ra2 9 Ra3

M12 = ZE12 YT12

M13 = ZE13 YT13

M23 = ZE23 YT23

Mp1 9 Mp2 9 Mp3

ϕ1 9 ϕ2 9 ϕ3 Mp1dB 9 Mp2dB 9 Mp3dB

F ′01 9 F ′

02 9 F ′03 R′

a1 9 R′a2 9 R′

a3

:-;"<=?>@;

ACBDE;"=?>@<GFH?FEIFJB%D;"<KBL?L?>@MN$FOPBI@FEQR;"< ACB%D;S=?>T<GFEHUFVI@FBDE;<KB%L?L?>@MN,FEOPB%IFQ;"<W B%D W =?D

W B%D W =?D

W BD W =?D W B%D W =UD

ACB%D;S=?>T<GOXM$Y?FVZU[";"<KY,;<F0 \ Y,[SI;">OXFH?[S;<Y] BL?>@^"<ZU_aÙb,`dc \ LUB>FHeI;">LMDJBI@FEMRHI;SDD;"<1fe=?;

ϕ(F0) = 90 g

ACBDE;"=?>@<OXM,Y,FEZU[S;<1Y,;"<Ra \Y,[SI;">OXFH?[S;<KY] BL?>@^"<["fàb?` hRhi;SIKZU_UÙb?` j

k l m m:9m : m:

F0

k l m m:

9m : m:

Ra

onPp 2436<qsr+)* 'D0 < ( (1,253 0+)*B) +'(1/) 1 <Xt +eu )/&A < " + < =/ /&) * +@3 *B( << +@)/ * < 3

6 ! #"$&%')(+*-,./0 1*$ #"$&234(%$ )5% 6(

(%(789

onPp 243653q MA' *) />0 +@ < !2 = $#%+)* 1=// .0 < +@(&A=2J'3/ < 2 +)&A(120&A(*+>0*(1( < +) &):<; 3

& < +@(&A=2J'3/ < &)32+ < (1/ < (Pu A < 2 + +'2( / & *B) <'< 3/ *@2 &A+ < / < /A/&)A(/ *&)1 < .->&) u (C2 ) +')* #.( < 3A'1 < C < *>= +@ 0(* &A3 < *B( u .3 &6 *BA/ < (1=/ 253 ( <B *(13/ < *2 D0(*,2 / 0 +A/0+A *BA0C@?D*A / * &A3/ *BA3+ADC $# (14-#.0/ 3* += *B*3

& 0 +A/+) *A '$# / C < A A&)*3(A 2 D0(* / *B)/ < (3/ ugC ?#.)/ *+)FE G ) $# 22 H

A(1 0 +A&)(12 D&)(*B+ 8 /10 3N24365 B3 M> < ( 0 / !-#%A/ *B +A < *'&)3/(*B(1=//+ 2 ?# 202 )(1 8 +'21( / & *B(1=/ -/ *B .0 .3 3 3 B * *B / < +'( < C < < < 3 *B( <I J )*2K 8 *A/ < (13/ uBB )1(+A < < 2 )&)*B( +@)/ * +'(1&) H

2J'03/ + +@ **B)- 2 >$#%(1/ *A +'+A0( (! < 3/ 20+ +@2( / & *B) 8 * C25ML ; GN 5 B - )*OC =(1QP 0?# / DC < )3 & < ( 0 / *B3 /+ 2 "1 +'(1&)327'3/ +A&A)20*B) < * +A&A02 +)+ < "$#%A/ *+A5R 6 0?# 202 )(1 +)&A(120&A(*+ u ( *B / <"t 3 +@ )*"1 *B / < +')*")2( <,<B < 3 *B(S 8 *B)/ < (3/ uA

BTC 3(1MPVU@0 ?# / DC < A A&)*3(A:3

W =0* 1 2&A << < +' < < * D*B;+ *B( < +*3 & # / DC < A A& *B3 (1A (1/ < ( u $# 22 )(1>0 +A&A(2&A(*B+ < 3/ * 2(1=*+ < 2 3(/ *A 0 >X=&)+C / 0 (&A(1) + 1( < +'A/3Y ( < A= < (1&*3 M> 1 0 (&A(A25A +@ * t (1 $# &eu ( < (*(13/ < + < 0* * < +' < ')* @ < A/ 0 ( < *A / < / t A(1'0& 1&) ; ?D&A) 3 & < &Ru ( < (*B(3/ << =/ * A/ < (** (*B+A < 2 / 2 0 + +@ )/ZY ( < 4= =< (1& )* r *B 3& 2 (1/&)(125 &A 2 0 +,+' A25 < < "1 +'D#=A1 2=2 < + 2 /3 +' M ;[ E G 5]\ H 2&^62;2_ *+)&A (* / < ,2 *0 27'243 243

& # 202 )(1 @+A&)(12 D&)(*+@25A +@ * -#%a` A& *B) 1'1(A/ )/ *B@1 < (b` +)A/ * < < ( 0 / G? /0+A&A << (1 <C +@ < .3 H3/ 2(/&A(2 t 3/& *B(3// +@)/ * (1/ < ( u < < +A++'A/ * < &)3/ < *B(*0*B( t < < 3/ * 3//+ < / <!<B/ &(' < CD/0320*B( u A/ //@?0 =O3Gc / / C&A3/0(*(13//0A +'(1A?'1 < (b` +)A/ * <!< ( 0 / G? - &A * 202 )(1 25A +')*ZC?#.D*B(1( < *B),'&('3( < (1 A/ *B (b` +))/ * <I/ *B < 2 =<< H ' 0* 8FE - 2 G - E G;G <d B *( < 25 < "-# +'21( / & *BA0 < 0 (1/ + 0 01 < 8:G C 9*G J= B 3

& < 2 + +'21( / & *BA0 < < +@(&A 32J'3/0 < &A=/ *(1)//A/ * &(' &A/ / + < ( < * /&A 253 ( <B *(13/-#%A/ (1 3/ Efe

Ω )/ 2 1g=)1 < 0 / +'21( / & *BA0 -#%( +'25+A /0&A> #.)/ *+)Pu =< ( H (1/ / /(1*3*M>A25A/ / *- < +'21( / & *A < /-#%+)* / * 2 =< 2 t (* < --(1 h?D( < *B' < & 2 &)(*B+ < 2 < (* < /=/ /+ 0 1( 0 1 < A/ *B

) ! "#$%'&( *)+, - ,/.103254 6

1) A/ *+A * + =<< .- &) <' +'21( / & *A < /-#%3/ * 3/& 2 =< / ( +'25+A /0&A -#%A/ *+A (1/ / /(1*3 [ < *4=/& /+)&A <<B (1 -#%+'21 C3)4 *A&('/( u *B)/ < (3/ (/ < +)+A 2 =C0+)*B) +'(1/) 1A0C$#%( +'2 +) /&)!+AA& *B( u *B / <"t ) *- 2253 * *B)/ < (3/ < 3 *B( @ +A&)A20*A C,&A(1 &A(* 3 A *ur0

< 1,&)3 / * ie A/ A/ *+A "$#%++')* *B)3& "2 + +'2( / & *B) L+'(1&)327'3/,+ +@ **B) &A3/ *B(A/ *CA/ 21 < & 2 &)(*B+" + t +A A/&) CrefC *B A < u A1 < *+' < 0+ 1 &)3 / * #.)/ *+) ie 8 /10 E 3 9 B3 & & 2 &)(*+ + t +A)/&A Cref0*B(1( < +A &:<; < * +)* 13//0+A"2 t (& / * *C + 0 01(g=) +@)/ *!+ H +)* 13/0/+A & M [-; 3& < +'(1&A 32J'=/ < + +@ **A *,+)&A)20*B) < 3/ *, A2 + < A/ *B+ < < /10 243 E G 2 C1) < &('0++ +eu ( 1A/ * W '+ A/(/ -5( +'2 +) /&) +)1)&)*B (u ZThe

253 $#%++')* *B) )* *B)/ < (3/ C &A(&A0(* 3 A *ur0

)*"(D+@25+A /&A ZThr253C1 +A&A)20*B) 8 & t 3 ) *(13/ E 3 E \ )*"2 .0 2J'0 E 3 E 3 9 B 3 & < & 2 &A(*B+ <

2 =< (*B < < 3/ * /=*B+) < CpA)* CpB

3 / < / 2 +'(1A *B +@2 < - *BA/ < (1=/ +AA& *B( u ug

< * 221( u +A < @$#%+AA&)* (g=A 0+@(&A 32J'3/0C+ +@ **B)3 & *B)/ < (3/ ur0

< * 0 +A/+)+),2 > 2 < < (1=/ 22(u +) < 0> +@ + /40+@(&A 32J'3/0 +A&)A20*A -D2 < < (1=/ / < & (*+C&)+A+)42 +'(1&)32J'03/ ++')* *BA03 A? *A/ < (13/ <uA

)* uB< 3/ * A +A <C G? < 3 *B(1 < S * K "$# 22 )(1 0 +A&)(12 D&)(*B+ 8 /10 0I243 E G B 3

ug

Yie ZThe

Cref CpB u

i ≈ 0gB FB hB

rB uB

ur0

ZThr

ur CpA

i ≈ 0gA

FA hA

rAuA

onPp 243 E G q/r < 0 "$#%( +'2 +) /&) +AA&)*( u0 0"* / <"t A * ZE 2 +'(g=A ,2 *B(1*3 # 2B= < 1 < 0AG? &A(&)(* < A2 + < A/ *B+ < /10 243 E G - < *B)/ < (3/ < uA

* uB25A A/ * *Bh?D2 ( +'+A <

A/ t 3/& *B(1=/ ie )* ur02

uB = hBFBgBu = hBFBgB−ie

jω (Cref + CpB)

8 243]\B)*

uA = hAFAgAur = hAFAgA1/(jωCpA)

ZThr + 1/(jωCpA)ur0 .

8 243 9 B& < u / *(*B+ < uA

- uB- ug

* Cref+)* / * <'< A01 <,0 /A0 < +' < 1 < - (1 < * /0+A&A << (1

-#%+A1(D+@(/A < @?020 < < (3/ < 8 243]\J-243 9 BC1 < 0*B < *A +' < 8 0 (1/ < - / *B .0 < --& 2 &)(*B+ < 2 =< (*B < B 3 "! B#aMR>@B%IMRFE>@;$8;"HRI@>@BDE; Y,;<&%oHUY,=U<!I>@F;"<&'CD; W I>@Ffe=?;<

! #"$&%')(+*-,./0 1*$ #"$&234(%$ )5% 6((%(789

K63"&A t (.- / < )&A3/ &A3021 +' < < < * h` )&)*B0+ < "1 +'3/ * .0 /10 243 E;E / <eu A + +@ *B)/ < (3/ +AA& *B( u 4-#%A/ *+A uref

< *!0*(1( < +)C253h?D&A(*B)>+AA& *B( u +'A/ *! < AG?+'(1&)32J'03/ < 3

u′

g uref u′ u′

BCpB rB

r Cref

YgB FB hB

ZThe

i ≈ 0

uref u′

r u′

ACpA rA

gA FA hAZThr i ≈ 0

onPp 243 E;E qsr < "$#%( +'2 +) /&) +AA& *B( u ",*B / <"t ) * ZE )G?D(g= +@"2 *B(.3& 0 (1/ gB

< *C0 G 1=O-?#.(D+@25+A /&A ZThe/-# 2253 *B,3/0& &A0/"&A3/ *B ( D*B(1=/ &A(&A0(* *

*A/ < (13/ u′B< #.+)&A (* 1= <

u′B = hBFBgBu′ = hBFBgB

Cref

Cref + CpBuref .

8 243 B& *B)/ < (3/ u′A < * h?D2 ( +'+A )/ t 3/& *B(1=/ *B)/ < (3/ uref

-# 2 = < 4&A(&)(*>+A1)&)*(u /10 243 E;E - < 3(*

u′A = hAFAgAu′r = hAFAgA

1/(jωCpA)

ZThr + 1/(jωCpA)uref .

8 243 N B; / / / - )/ t ( <B / * , 2253 * < ) *(13/ < 8 243 -1243 \B>)* 8 2 3 9 - 243 N B -0(1 (A/ *

u′BuB

=jωCref

−ieuref

)* uA

u′A=

ur0

uref,

&)u ( =//"2 = $#%h?D2 << (13/ "$#%( +'25+A /0&A +)1A& *B (u ",*B / <"t ) * ZE

ZE =ur0

ie=

−1

jωCref

uA

uB

u′Bu′A

.8 243;< B

& 1) ?#.(D+@25+A /&A +)1)&)*B (u * / <"t A * ZE < *'3/& )/ *B(g=) +@)/ *'+ *B) +'(1/+) 2

+' < " < 2253 * < uA/uB)* u′A/u′B - &)3// ( << / * 1) " & 2 &)(*B+ Cref

3


!"#%$& '!"' %()** + (),

&.-0/ '21436587:9 - 369:; / ;=< -?> 7 + - 7 -=> ;A@B58< -?>C/?- -CD7E9:1F@ - 9HGJI >6- ;A7EI6< - 65 6KL58M 0 9E143J'65uNI - -OKQPR5 > G:ICS'G:1 >6/?-< -T/?- 9H7U145 >6- G 61VG - GW7 '6;?MV9:5uNI - GXGHI69WK - GuNI - K8K - GY9 - 3MZG -?> 7[K - G\7 -?/ ' > 5 uNI - G]< P^;=7U14KLM >6> 1 0Z-_-?> 369 - G:GH5LM > 321V99:; / 5L3C9:M / 587:;.- +`;A7 '6Ma< - G-uaIC5\3 - IC@ -=> 7b1VK8MZ9:G =7E9 - 9 - +`58G - G -?> uNI - Gc7E5LM > 3MdP - GH7eK -J/ 1VG > M47U1.+ + -=> 7f< - G+`Ma<6==K - G[3J' CBGH5uNI - G > MZ9 +g14KL5LGH;?Gh< - KQP -=> G - + 6K - +`5 / 9:MV3J'6M >6- G Hi/ MZI63CK - IC93j 1 > G /?-k/ '21V365l7E9 - -B<C5 `;?9 -=> 7EGN+`Ma<6=?K - GW3J' CaG:5 uaI - G_<6; / 9:5l@41 > 7[K -k/ '61.+`3g1 / MVI6GH7:5uNI - <61 > G]Km1 / 1n@a587E;t MZ9 +'; - 361V9_K - G +`5 / 9:MV3J'6M >6- G - 7kK -o/ MZI636K - I69 -2G:M > 7k<6;A@ - K8MZ363;?G - 7 / M;+`361V9:;=G3

cd5 > G:5 -p1V369 =?G'1n@ZMZ589b3MZG:;#K - Gb;RuaI6147E58M > Gq< - KQPr1 / MZI6Gc7E5uNI -g- 70<6; /2> 5hK - G ' Ca3MV7 '6=?G - G`1V<J+'5LGHG:5 CK - G -I >6- ;A7EI6< - <6;=7E1V5LK8KL; - <6I / '21*+`3s1 / MZI6Gc7E5uNI - <21 > GfK -b/ 14GfMZtu5LK - Gc7oG:IC363MVG:;qI > 5 t MZ9 + - - / M.+`3C7 - 7 -?> I< - G - ` - 7:Gv7 ' - 9 +'5uNI - Ge7 - K8GuNI - 369:MZ3MZGH;g321V9 e - 9 - 9^ E 2_:- - GH7 - ?B3MVG:; -g- 7v<65LG / IC7:; - 3 &%- Gv9:;?GHI6K87E147EGM C7 -?> I6G CKwP^5LGHG:Ix< -`/=- 7:7 - ;A7EI6< - G:M > 7 -?> GHI6587 -g/ M.+`32149:;?Gv1VIG? +'MB<#=?K - Gy< -g/ '21*+`3zI > 5 t MV9 + -'- 7b14I+`Ma<6==K - < - 369HMZ321 0 147E58M > < P^M > < - Gk3CKm1 >6- GT369:; / M > 58G:;=G32149=h1VKLKL1 0 ' ^ _:3

|g=~ ' A I e [ ` = oC, 'Tg

& PR1.+'36KL5l7EI6< - <CI +`MVIC@ - + -=> 7b1 / MVI6GH7:5uNI -g- GH7qGHICS'GE1.+ + -?> 7 t 1V5 6K - 3MZI69v3MZIB@ZMZ589 -=> KL5 > ;n1V9H5LG - 9K - G_;euNI2147:5LM > G 3 &%- G_@41V9:5L16K - Gh<6; / 9:5l@41 > 7KQPR;A7U147h<7C > 1.+'5uNI -y- 7T7 ' - 9 +'Ma<7C > 1.+'5uNI - <6I2I658< - G:M > 7 qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - 369 - G:G:58M > p 321V9T9E1V3C3MZ9c7fCqKm1b369 - G:GH5LM > GH7E147E5 uNI - P0

-qKL1q@a5l7 - GHG - 321V9c7E5 / I6KL1V5L9 - ~v -qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - +g1VGHG - @ZMVKLIJ+'5uNI - ρ′ 321V9_9:1V363MZ9H7fCqKL1 +`1VGHG - @ZMVKLIJ+'5uNI - Gc7U147:5uNI - ρ0

-qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - 7 - +'3;?9E17EI69 - τ 361V9T9E1V363MZ9c7fC0KL1J7 - +`3;?9:147EI69 - GH7U17E5uNI - T0

-qKQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< P -=> 7E9:MV365 - +`1VG:GH5uNI - s 32149_9E1V363MZ9c7fC0KQP -=> 7E9:MV365 - +`1VG:GH5uNI - GH7E147E5 uNI - S0

3&%- GT321V9:1.+=A7E9 - GPuNI65 -?> 7:9 -?> 7<21 > GkKm1b<C; /2> 5l7E58M > < - Gk369HMZ369:58;=7:;?GT<6I#2I658< - G:M > 7kK - G_G:I65l@V1 > 7:G qK - G / M - S / 5 -?> 7EGT< - @a5LG / MZGH587:;e< -e/ 5LG:1V5LK8K - + -?> 7 µ - 7k< - @ZMZK8IJ+ - η -qK -o/ M - S / 5 -?> 7k< -v/ M > <6I / 7E58M > 7 ' - 9 +'5uNI - λ -qK -o/ M - S / 5 -?> 7k< -v/ M;+'369 - G:G:5 65LK8587:;d5LGHMV7 ' - 9 + - χT

-qK - G / '61VK - I69:G +`1VGHG:5uNI - Gf321V9I > 5l7E;J< - +g14G:G - Cg369 - G:GH5LM >u/ M > Gc7U1 > 7 -J- 7MC'@ZMZK8IJ+ -v/ M > GH7U1 > 7 Cp- 7 Cv

- - 7kK - IC9_9E1V363MZ9c7 γ -qK - 7U1VIG?u< Pr1 /?/ 9HMZ5LGHG - + -=> 7e< - 3C9 - GHG:58M > 361V9oI > 5l7E;q< PR1 /?/ 9:MZ58G:G - + -?> 7e< - 7 - +'3;?9E17EI69 - C#@VMZKLI7+ -/ M > GH7E1 > 7 β = (∂P/∂T )V3

\ E

' 6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

&%- Go;euNI217E5LM > GoKL5 > ;n1V589 - GO'6M;+'M 0 = >6- G 0 MZIC@ - 9 > 1 > 7eK - +`MVIC@ - + -=> 7y<CIs6I65L< - GHM > 7e1VI > M;+ 69 - < -7:9:MZ58G 8 ^6< _O243 9 3 E B

q *-,. G$ ) )/( 9 8 ;euNI2147:5LM >st M > <21.+ -?> 7U1VK - < - Km1q<7C > 1*+`5 uaI - B1

c0

∂ ~v

∂t+

1

ρ0c0~grad p = `v ~grad div ~v − ` ′

v~rot ~rot ~v +

~F

c0,

8 43 E B

MVt `v - 7 ` ′vGHM > 7k< - GTKLM >J0 I - I69:G / 149E1 / 7E;?9H5LGc7E5uNI - GT<6; /2> 5 - GT32149

`v =1

ρ0c0

(

4

3µ+ η

) - 7 ` ′v =

µ

ρ0c0,

- 7kMZt c0 =√

γ/ρ0χT

- GH7kKm1 / ;?KL;=9:5l7E;o1V<C5m1217E5uNI - <CIGHM > 3q *-,. G$ ) 4$ !/(# # $ V)2* 10! 8 uNI65B7E9E14<6I6587 Km1 / M;+'369 - G:G:5 65LK8587:;]<CI06I65L< - B<21 > GKL1uNI - K8K - KwP^; / 1V9H7J5 > Gc7U1 > 7E1 > ;'< - +`1VG:G - @ZMVKLIJ+'5uNI - ρ′ - GH7 - ?B369:5D+`; -?> t M >C/ 7E58M > < - Gv@41V9H5m1CK - G5 > <6;=3 -=> <21 > 7 - G p - 7 τ 361V9 ρ′ = (p− βτ)γ/c20

-

ρ0c0 div ~v +γ

c0

∂

∂t(p− βτ) = ρ0c0q ,

8 43 2;Bq *-,. G$ )7$)( /( %$( * 34$ ] ' )V* ) 1* G)( <21 > GKm1uNI - KLK - KQPR; / 1V9c7d5 > G H7E1 > 7U1 > ;x< P -?> 7E9HMZ365 - s - GH7 - ?B369:5D+`; -?> t M >6/ 7:5LM > < - G@41V9:5L16K - G5 > <6;=3 -=> <21 > 7 - G p - 7 τ 361V9s =

CP

Tτ +

βχT

ρ0p -

(

1

c0

∂

∂t− `h ∆

)

τ =γ − 1

βγ

1

c0

∂ p

∂t+

h

c0CP,

8 43 B

MVt `h = λ/(ρ0c0CP ) <6;?GH5 0Z>C- I >6- KLM >J0 I - I69 / 1V9E1 / 7E;=9:58GH7E5 uNI - < - <65 `I6GH5LM > 7 ' - 9 +'5uNI - 3&%- GY7 - 9 + - G ~F - q - 7 h 9 - 369H;?G -?> 7 -=> 7[KwP^5 > 6I -?>6/=- < - G:MZIC9 /?- G[1 / MZI6Gc7E5 uaI - G]9 - GH3 -=/ 7:58@ - + -=> 7[< -Pt MZ9 /?- -< - <6;6587 - 7T< -y/ '21VK - I69T<21 > GTK - <6M;+`1V5 >6-d/ M > GH5L<6;=9:;*3 ;W> < - 'CMZ9:GT< - GkGHMZI69 /?- G - /?- GT7 - 9 + - GTG:M > 7 > I6K8G3

|` . = A"! #%$ "& '( )*) ! + ,- . #/$ # 0. )1#2$435)6$ & )6$7 ~98;:=< $

e - 9 - 9^ E 2_:-O<CM > 7yK - Gd9:;?GHI6K87E147EGdG:M > 7DCDKm1 21VG - <CI +'Ma<6=?K - 9 - 7 -=> I <21 > GoKm1 > MV9 + - M ;[?> E G 5]\ H24-T369HMZ3MZG - < P^;=7:I6<65 - 9K -s/ '21.+'3 1 / MZI6Gc7E5uNI - <21 > GI >C-x/ 1n@a587:; -?> 3C9 -?> 1 > 7 -?>/ M.+`3C7 - K - G - ` - 7:G7 ' - 9 +`5 uaI - GD14G:G:M / 58;?Gg1VI +'MZIC@ - + -=> 7#1 / MZI6GH7:5uNI - 3AKMZI69 / M > Gc7E9:IC5L9 - G:M > 9E1V58G:M >6>C- + -?> 7 -]5LK >6- t 1V587I6G:1 0Z- uaI - < - KQPR;RuNI2147E58M > < -g/ M > <6I / 7E58M > < - Km1 / '21VK - I69 - 7b< - Km1KLMV5W7 ' - 9 +`Ma<7C > 1.+'5uNI - K85 > ;?1V9:58G:; -- ?C3C9:5 +`1 > 7TKm1 65l@41V9:5L1 >6/?- <6I+'5LK85 - I 8 ;RuNI2147E58M > G 73 B 3

& PRM @ - 7b< -g/?- 321V9E1 0 9E1V37' -`- GH7q< - 369HMZ3MVG - 9vI >6-`- ?B36KL5 / 147:5LM > <6;A7U1V58KLKL; - <6I 9E145LG:M >6>6- + -?> 7JG:I65l@B532149 e - 9 - 9 - 7k< - +'M > 7E9 - 9_K - GkK85 +'587 - G]< -y/?- 9E1V58G:M >6>C- + -?> 7 3

,+U +U )+e *P)+)+,, )#' & 4 6

! #"$ &%'()*+,.-$0/213%5467()*+89-7;:'+,7-$67:<*+,=-$>/(?:&@7(,/56$ &.- <CM;+g145 >6- < PR;A7EI6< - D - GH7 / M > GH7:587EIC;]361V9 K - 2I65L< -Y/ M > 7 -=> IJ<21 > G%I >6-W/ 1n@a587:;Y< - t MZ9 + - uNI - K / M > uNI -

t - 9 +`; - 321V9XKm1eG:I69 t 1 /?- S 3 &%-f/ '21*+`3#1 / MZI6Gc7E5 uaI -f- Gc7 0 ; > ;?9:;TCJKwP^5 > Gc7U1 > 7[5 > 5l7E5m14K t = 0 321V9XI >6- GHMZI69 /=-< - @V149:5m17E5LM > < - 369 - GHG:5LM > MZI< - <C; 65l7k@VMZKLI7+`5 uaI - 3 &%- G<65 + -?> GH5LM > Gk<CI <6M;+`1V5 >6- D GHM > 7dGHI6363MZG:; - G3 - 7E5l7 - G_< - @41 > 7kKm1qKLM >70 I - IC9_< PRM > < - 3e - 9 - 9; / 9H587pKwP^;euNI2147:5LM > < -X/ M > <CI / 7:5LM > < - KL1 / '214K - IC9 -?> KwPR16G -?>6/=- < - G:MZI69 /?- G\< -[/ '61VK - I69^ E 24- ;Ru 3E _%GHMZI6GTKm1 t MV9 + - <6M >6> ; - 361V9 8 73 B

(

1

c0

∂

∂t− `h ∆

)

τ =γ − 1

βγ

1

c0

∂ pg

∂t.

8 73 \B& 1 > M47U147:5LM > pg

<C;?G:5 0Z>6- -3MZI69 e - 9 - 9 - KwP^; / 149H7k5 > Gc7U1 > 7E1 > ;e< - 3C9 - GHG:58M > 369HMB<CI6587k321V9TI >L0 ; > ;=9E147 - I691V<21V3B7E;.- /=- 7X; / 149H7X< - 369 - GHG:5LM > ;=7E1 > 7 CyKwP^MZ9:5 0 5 >C- < - KQPR; / 1V9c7]5 > Gc7U1 > 7U1 > ;< - 7 - +`3;?9:147EIC9 - τ 1VI`G - 5 > <CI2I658< - 3

K 1V9k1V58KLK - I69:G - -=> t 145LGE1 > 7TICGE1 0Z- < - KL109 - KL147E58M > KL5 > ;n149:5LGH; -ρ′g = ρ0χT (p− βτ)

<21 > GTKQPR;RuaI6147E58M > < -y/ M > <6I / 7:5LM > < - Km1 / '61VK - I69 8 73 B -C5LK@a5 -?> 7(

1

c0

∂

∂t− γ`h ∆

)

τ =γ − 1

βχT ρ0

1

c0

∂ ρ′g∂t

,8 73 9 B

MZt ρ′g 9 - 3C9:;?G -?> 7 - KQPR; / 1V9c7d5 > GH7E1 > 7U1 > ;v< - +g14G:G - @ZMVKLIJ+'5uNI - 369HMB<6IC587321V9kI >L0 ; > ;?9:147 - I69f14<21V3C7:;.- /?- 7; / 1V9H7'< - +`1VG:G - @VMZKLIJ+'5uNI - ;A7U1 > 7- /?- 7H7 -/t MZ5LG H / 5 - 707EMVI @HMZI69HG'G - K8M > e - 9 - 9- CsKwP^MZ9:5 0 5 >C- < - KwP^; / 1V9H75 > Gc7U1 > 7E1 > ;o< - 7 - +`3;?9:147EIC9 - τ ^ E 2 -6;eu 3 E 2_:3e - 9 - 9k369:MZ3MZG - <6M >6/ < - 9:;=G:MZIC<69 - KwP^;euNI2147:5LM > < -J/ M > <6I / 7E58M > < - KL1 / '214K - IC93MZI69 τ <21 > G< - IG?/ 1VGT<65bÒ;=9 -=> 7:G qKL1e@41V9:5L147E58M > < - 7 - +'3;=9E147:I69 - τ 1e3MZIC9[MZ9H5 0 5 >6- I >6- @41V9H5m147:5LM > < - 369 - G:G:58M > pg

369:Ma<6I65l7 - 321V9[I >0 ; > ;?9:147 - I69k< - @V149:5m17E5LM > < - 369 - G:GH5LM > 8 5D+`3;?<21 >6/?-o> I6K8K - B-qKL1x@V149:5m17E5LM > < - 7 - +`3;?9:147EI69 - τ 1)3MZIC9#MZ9:5 0 5 >6- I >6- @41V9:5L147E58M > < - @VMZKLI7+ - +`1VGHG:5uNI - 1/ρ′g369HMB<CI6587 - 321V9TI >0 ; > ;?9E17 - IC9< - @41V9H5m147:5LM > < - <C; 65l7 8 5 +'3;?<21 >C/?- 5 >4/2> 5 B3& P^;euNI217E5LM > < - / M > <6I / 7E58M > < - Km1 / '21VK - I69 - GH7#<CM >6/ 5 > 7 - 9H369:;A7E; - 5 / 5 / M;+,+ - I >6- ;RuNI2147E58M > < -

<65 Ì6GH5LM > <21 > G.Km1uNI - KLK - K - G -=/ M > <I+ - + 69 -Y-?> pg

8 ;eu 373 \B MZI -?> ρ′g 8 ;Ru 343 9 B - ?C3C9:5 + - Km1 / M > 7E9H5 6IC7:5LM >< PÎ >6- G:MZIC9 /?- 8 < - @41V9H5m147:5LM > < - 369 - G:G:58M > MZI< - +g1VGHG - @ZMVKLIJ+'5uNI - B[<21 > GTK - 3C9:M /?- GHG:I6Gk< - <65bÒI6GH5LM > 3

A 00 6%%(7 a5) 1 4 ! 4$((]9-?>- ` - 7-_KwP^; / 149H7#< - 369 - GHG:5LM > MZI < -

@ZMZK8IJ+ - +g14G:G:5 uNI -q-?> 7E9:1 > 7e<21 > GoKQPR;RuaI6147E58M > < -0/ M > <6I / 7:5LM > < - KL1 / '21VK - I69 >6- 9 - 369H;?G -=> 7 -'-?> 1VI / I >/ 1VGqKQPR5 > 2I -=>6/?- < PÎ >6- GHMZI69 /=- - +`1V5LGJ<6; / 9:5870KQPR;A7U147q7 ' - 9 +`Ma<7C > 1.+'5uNI - <6I)2I65L< --=> 7EMZIC7'3MZ5 > 7q<CI<6M;+`1V5 >C- < P^;=7EIC< - 3

;W> 9:;?1VKL5l7E;.-CG:5%< - GdG:MVI69 /=- Gk< - <C5 `;?9 -=> 7 - G > 147:I69 - GGHM > 7d9H;?321V9c7E5 - GT<21 > GkK - <CM;+g145 >6- -6K - I69 / M > 7E9H5 H6IC7:5LM >- GH7#3C9:5LG - -?>/ M;+'3C7 - 361V9DK - G`7 - 9 + - G ~F - q - h <61 > GDK - GD;RuaI6147E58M > GD< - : 1n@a5 - 9 H Ha7:MB - G-h< -/ M > G - 9c@4147E58M > < - Km1,+`1VGHG -b- 7f< -b/ M > <6I / 7E58M > < - Km1 / '61VK - I69 8 73 E - 73 24- 73 B 3 HB5 /?- GdG:MZI69 /?- Gy1 0 5LGHG -=> 7G:I69K - G t 9:M > 7:5g=?9 - G<6Iu<6M;+`1V5 >C- - - K8K - GkG:M > 7MCg369 -=> <69 -q-?> / M;+`3B7 - <61 > GfKQP - ?C3C9 - GHG:58M > < - G / M > <C587E58M > G1VIG? t 9:M > 7E5 =?9 - G3

cd5 > G:5 -[KwP^5 > 7 - 9:369H;=7U17E5LM > uNI65 - GH7 t 1V5l7 - < - G;RuNI2147E58M > G< - <65bÒI6GH5LM > uNI - e - 9 - 9 / ' - 9 / ' - C 9H; HG:MZIC<69 -o- GH7T5 >C- ?C1 / 7 - 37K 1V9T1V58KLK - I69HG-N5LK > P - GH7T3214G t 1V587[I6GE1 0V- <21 > G_GHM > 9E1V58G:M >C>6- + -?> 7_< - KQPR;RuNI2147E58M > < -: 1n@a5 - 9 H HN7EMB - G > 5< - KQPR;RuaI6147E58M > < -f/ M > G - 9H@4147:5LM > < - Km1I+g1VGHG - -C;euNI217E5LM > G[3MZI69c7U1 > 7 > ; /?- G:GE145L9 - GQCJKm1< - G / 9H5L3C7:5LM > <6I+'MZIC@ - + -=> 7k1 / MZICGH7E5 uNI - <21 > GTG:M >-=> G - + 6K - 3 & 1 +`;A7 '6Ma< - < - 9H;?G:MVKLIC7:5LM > IC7:5LK85LG:; -f- 7KwP^5 > 7 - 9:369H;=7E147E58M > < - G9H;?G:ICK87U17EG_M C7 -?> I6GkG:M > 7k<6M >C/ CqM CG - 9c@ - 9d< --t 1C=M >/ 9:5l7E5 uaI - 3

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

" +)@> &'567 ' "$ &%5( *+, -$6 "$ &+ / 5/+8 ! !

cd369 =?Gp1n@ZMZ589%3MZGH;XK - G.;euNI217E5LM > G.< - <C5 Ì6G:58M > 8 73 \ - 7 73 9 B - e - 9 - 99 - +`1V9uNI - uNI - K - I69 t MZ9 +bICKm147:5LM >- GH7G:5 +'5LKL1V5L9 -T- 73uNI P - KLK - GT3 - IC@ -=> 7d<6M >6/ =7E9 - ; / 9H587 - GTG:MZICGkKL1 t MZ9 + -(

1

σ

∂

∂t− ∆

)

τ(~r, t) =1

σ

∂ θg(t)

∂t3MZI69 ~r ∈ D , t ≥ 0 ,

8 73 > 1 BMZtKm1J@41V9H5m1CK - τ - Gc7kG:I6363MZGH; -e/ 1VI6G:1VK - - - 71n@ -=/

θg(t) =γ − 1

βγpg

- 7 σ = c0`h ,

<21 > GTK -o/ 14Gk<C; / 9H587k321V9_KQPR;RuNI2147E58M > 8 73 \B- - 7

θg(t) =γ − 1

βχT ρ0

ρ′g- 7 σ = γc0`h ,

<21 > GTK -o/ 14Gk<C; / 9H587k321V9_KQPR;RuNI2147E58M > 8 73 9 B3c /=- 7H7 - ;RuaI6147E58M > < - <65 Ì6GH5LM > - e - 9 - 9[1VGHG:M / 5 - Km1 8 K - G B / M > <6587:5LM > 8 G(BY1VI? t 9HM > 7E5 =?9 - G - 7hKm1 / M > <65 H

7:5LM > 5 > 587E5L1VK - G:IC58@41 > 7 - G-29 - G:3 -?/ 7E58@ - + -?> 7

τ(~r, t) = 0 , 3MZIC9 ~r ∈ S , t ≥ 0 ,8 73 > B

τ(~r, t) = 0 , 3MZIC9 ~r ∈ D , t = 0 .8 43 >V/ B

& 1v@41V9:5L16K - θg(t)- GH7hG:I6363MZGH; - I > 5 t MZ9 + - <61 > GhK - <6M;+`1V5 >C- D 3JM - 7:7 - ' Ca3MV7 '6=?G - 5 +'36KL5 uaI - <6M >6/

uNI - -CG - KLM > K -k/ 1VG / M > GH5L<6;=9:;*-CKwP^; / 1V9H7X< - 369 - G:G:58M > pgMVIDKwP^; / 1V9H7X< - +`1VGHG - @VMZKLI7+`5 uaI - ρ′g - GH7_I > 5 t MZ9 + -<21 > GvK - <6M;+`1V5 >C- - /=- Gy@41V9:5L16K - Gf;=7E1 > 7eG:I6363MZGH; - G / 1V9:1 / 7:;?9H5LG - 9y5 / 5 Km1#G:MZIC9 /?- C#KwP^MZ9:5 0 5 >C- <6I / '21.+'3

1 / MZI6Gc7E5uNI - 8 'Ca3MV7 '#=?G -v- 9H9:M > ; - B3j 1 > G K -[/ 1VG\369H;?G -?> 7 -Z<CI t 1V587 uaI - K - G <65D+ -=> G:58M > G <CIq<6M;+`1V5 >6- D GHM > 7Y3 - 7E5l7 - G < - @41 > 7YKL1dKLM >70 I - IC9< P^M > < - - KL1D369 - G:G:58M > 1 / MZI6GH7:5uNI - p - GH7yI > 5 t MZ9 + - <21 > GoK - <6M;+`1V5 >C- D 8 C / M;+'369:58Gd<21 > GoK - G / MZI / ' - GK85 +'587 - Gp7 ' - 9 +`5 uaI - G(B3 j 1 > G /=-T/ 1VG -ZG:52KL1 +g1VGHG - @VMZKLIJ+'5uNI - ρ′g - Gc7]I > 5 t MV9 + - <21 > G D 8 C / M.+`369H5LG\<61 > GK - G / MZI / ' - G[K85 +'587 - G(Bp1VKLMZ9HG-Z<OPr1V3C9B=?G]Km1y9 - Km147:5LM > ρ′g = ρ0χT (p− βτ) - τ <6MZ5l7]1VI6G:GH5 A7E9 - I > 5 t MZ9 + - <61 > GD

8 C / M;+'369H5LG\<21 > GYK - G / MZI / ' - GWKL5D+`5l7 - G B - /?- uNI65 > P - GH7]321VGYK -h/ 1VG 3cd5 > G:5:-4KQPR; / 1V9c7W< - +g14G:G - @VMZKLI7+`5 uaI -ρ′g

>6- 3 - IC7_321VG =7:9 - I > 5 t MZ9 + - <21 > G[7EMZIB7[K - <6M;+`1V5 >6- D 58K - GH7 t M >6/ 7:5LM > <6I#3MZ5 > 7X<21 > GXK - G / MZI / ' - GK85 +'587 - G3jd- +`1 > 5 =?9 - C - ?C3C9:5 + - 9Km1'G:MZK8IC7E58M >L0 ; > ;?9:1VK - <6I 369HM 6K =+ - 7 - +`3MZ9 - K 8 73 > 17- - / BGHMZI6GfKL1 t MZ9 + -< PÎ > 369:Ma<6I65l7q< -#/ M > @ZMZK8IC7E58M > < - Km1 @41V9H5m1CK - t 1n@ -=/ I >6-st M >6/ 7:5LM > < - e 9 -?-?> - e - 9 - 901VGHG:M / 5 - I >369HM 6K =+ - < - e 9 -?-=> 8 <6M > 70KL1 G:MZK8IC7E58M > - GH70Km1 9H;?3M > G - 5 +'36I6KLGH5LM >C>6- K8K - <6I G CaGH7B=+ - B uNI65 - GH70; / 9:5l7GHMZI6GkKm1 t MZ9 + - GHI658@41 > 7 -

(

1

σ

∂

∂t− ∆

)

g =1

σδ(t) 3MVI69 ~r ∈ D , t ≥ 0 ,

8 73 N 1 B

g = 0 3MZI69 ~r ∈ S , t ≥ 0 .8 73 N B

H - 9H; t ;?9:1 > 7k1VKLMV9:GOCbKQPRMVIC@a9E1 0Z- < - Mh1V9HG:Km1 - 7Z1 -0Z- 9M^ E G -C363 315 N - E 5;57-72 ]\J- G _:- e - 9 - 9h9H;?G:MVIC7K - 369:M CKg=+ - 8 43 < B[;RuNI658@41VK -?> 7<C 8 73 N B -6<61 > GkK - uNI - K%Km1bGHMZI69 /=- 5D+`3CI6KLGH5LM >6>C- KLK - δ(t) - GH7T7E9:1V<6I65l7 - GHMZI6GKL1 t MV9 + - < -y/ M > <6587:5LM > 5 > 587:5m1VK -

,+U +U )+e *P)+)+,, )#' & 4

(

1

σ

∂

∂t− ∆

)

g = 0 3MZI69 ~r ∈ D , t > 0 ,8 736<Z1 B

g = 0 3MZI69 ~r ∈ S , t ≥ 0 ,8 736< B

g = 1 3MZI69 ~r ∈ D , t = 0 .8 736< / B

& P^;euNI65l@41VK -=>6/?- < - G3C9:M 6K =+ - G 8 43 N B - 7 8 736< B - Gc7369:; / 5LGH; - -?> 1 >6>6- ? - M -?> +'M > 7E9:1 > 7/uNI PR58KLG3MVG:GB==< -?> 7Km1 + + - G:MZK8IC7E58M >0 ; > ;?9:1VK - 3

K 1V9 1V5LK8K - IC9:G-FKQP6' Ca3MV7 '6==G - G - K8M > Km1auaI - KLK - Km1 t M >6/ 7:5LM > θg(t)- GH7 I > 5 t MZ9 + - <61 > G K - <6M.+g1V5 >6- < P^;=7EIC< -

5 +'36KL5 uNI - uNI - KQPR; / 1V9c7o5 > GH7U1 > 7E1 > ;b< - +g1VGHG - @VMZKLI7+`5 uaI - MZIuKwP^; / 1V9H7f5 > Gc7U1 > 7U1 > ;b< - 369 - GHG:5LM >u- GH7yI > 5 Ht MZ9 + - <21 > G /?- <6M;+`1V5 >6- 3 [ K - Gc7\<6; +`M > 7:9:;]36KLICG'61VIC7 - 7p<21 > GpK - 321V9:1 0 9E143J' - > 343 243 2_< - KL1k9H; t ;=9 -?>C/?- ^ < _uNI - G - ICKK - < - I?C5 =+ -T/ 1VG - pg

I > 5 t MZ9 + - - - Gc7[GHI6G /=- 3C7:5 6K - < -T/ M > <CI65L9 - CoI > 9:;=G:I6Kl7U147 / MZ9:9 -?/ 7[3MZI69Km143 - 7:587 - / 1n@B5l7E; / M > G:58<6;?9H; - 5 / 5:314363 - KLM > G uaI - K - Gq@V149:5m1 6K - G pg

- 7 ρ′g >C- 9 -?> < -?> 7 > I6K8K - + -?> 7/ M;+'3C7 - <6Iu7 CB3 - < - G:MZIC9 /?-'- ? / 587U1 > 7yK - 2I658< - <21 > GyKL1 / 1n@a587E; +`1V5LG uaIOP - KLK - Go<6; / 9:5l@ -?> 7vKwP^;=7E147f7 ' - 9 H+`Ma<7C > 1.+'5uNI - <6I 6I65L< -v-?> 7:MZIC7d3MZ5 > 7d<6I <6M;+`1V5 >6- 3cd5 > G:5 - K -v/ 1VGfMZt pg

- GH7fI > 5 t MV9 + -e- GH7oK - G - I6K7E9:1V587:;f<21 > GTKL10GHI6587 - < -y/?- <6M / IJ+ -?> 7-2G:MZ5l7

θg(t) =γ − 1

βγpg

- 7 σ = c0`h ,8 7365 B

;=7E1 > 7 -?> 7 -=> <6I uNI - K - G / 1VK / I6K8GhG:M > 7kG:5D+`58Km1V589 - G[3MZI69_KwPR1VIC7:9 -o/ 1VG3j 1 > GqK - I69vMZIC@a9E1 0V- +U C +U + G+U U ^ E G _ - Mh1V9HG:Km1 - 7 Z1 -0Z- 9b<6;A@ - K8MZ363 -?> 70< - GG:MZK8IC7E58M > G_14I3C9:M 6K =+ - 8 736< B[3MZI69_<65b`O;=9 -=> 7 - G t MZ9 + - G_<CI<6M.+g1V5 >6- D 3&.- 321V9E1 0 9E1V37' - G:IC58@41 > 7y1D3MZI69 6IB7y< - 369H;?G -=> 7 - 9yKL1 +`;A7 '6Ma< - < - 9:;?GHMZKLIB7E5LM > IC7E58KL58G:; - 1V5 > G:55uaI -K - Gf9:;=G:I6Kl7U147:GoM C7 -=> I6Gy<21 > GoK -b/ 1VGyMZtuK - <6M;+`1V5 >6- D - Gc7yI >s/ CaKL5 > <69 - t - 9 +';C#G - G - ?a7E9H;+'587E;=G3 &%- G1VIC7:9 - G t MZ9 + - GW< -/ 1n@B5l7E;=GX9H;?3 - 9H7:MZ9:58; - GX<21 > GWKwPR1V9H7:5 / K - < - e - 9 - 9f^ E 2_ G:M > 7]7E9:1V587:; - G]<61 > G]KwPR1 >6>6- ? -j 3

+,/6 *+8 " +,6$ /( " %! #( 6 &&.- <6M;+`1V5 >6- < PR;A7EI6< - D - GH7h5 / 5 I >D/ CaKL5 > <69 - /2> 5< - KLM >J0 I - I69 ` -N< - 9:1 CVM > a - 7[1VIG?`32149:MZ58G]9H5 0 58< - G8 5D+`3;?<21 >6/?- < - 361V9:MZ5%5 >7/2> 5 - B31M - <6M;+`1V5 >6- 10I > @VMZKLIJ+ - V - 7kI >6- GHI69 t 1 /=- 7EMV7E1VK - A 3M[M;+`3B7 - 7 -=> Iz< - Km1#G C +';=7:9:5 -0/ CBK85 > <C9:5uNI - <CIz<6M.+g1V5 >6- < P^;=7:I6< - -.K - 369:M CKg=+ - 8 43 < BfG=PR; / 9:5l7 - -=>/ MaMZ9:<CM >6> ; - G / CaKL5 > <69H5uNI - G -

(

∂ 2

∂r2 +1

r

∂

∂r+∂ 2

∂z2 − 1

σ

∂

∂t

)

g = 0 3MVI69 r ∈ (0, a) , z ∈ (0, `) , t > 0 ,8 73 E G 1;B

g = 0 3MVI69 r = a , z = 0 MZI ` , t > 0 ,8 73 E G B

g = 1 3MVI69 r ∈ (0, a) , z ∈ (0, `) , t = 0 .8 73 E G / B

$* 7 $ 6 6( *5!/% #(/M - 3C9:M 6K =+ - 1V<J+ - 7T< - GTG:MZK8IC7E58M > GQCb@41V9:5L16K - GhG:;=321V9H; - G< - Km1 t MZ9 + -

g(r, z, t) = fr(r)fz(z)h(t) .8 43 E;E B

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

&%- 9 - 3MZ9c7T< -y/?- 7 Ca3 - < - G:MVKLIC7:5LM > <21 > GTKwP^;euNI2147:5LM > < - <65b`OI6GH5LM > 8 73 E G 1 B[<6M >6>6- 1VKLMV9:G1

fr

∂ 2fr

∂r2 +1

rfr

∂ fr

∂r+

1

fz

∂ 2fz

∂z2 =1

σh

∂ h

∂t.

8 73 E 2;BKMZI69 uNI P^I >6- t M >6/ 7E5LM > < - Km10G - I6K - @V149:5m1 6K - t 3CI65LGHG - A7E9 - 5L< -?> 7E5 uaI - C'I >6-t M >6/ 7:5LM > uNI65 >6- <6;=3 -=> <uNI - < - GT@41V9:5L16K - G_< P - GH321 /?- -8uNI - KLK - G uNI - GHMZ5 -=> 7K - G_@41VK - I69:GT< - r ∈ (0, a) - z ∈ (0, `)

- 7 t ≥ 0 -25LK - GH7> ; /?- G:GE145L9 - uNI -e/ '21uNI - + - + 69 - < - KQPR;RuNI2147E58M > 8 73 E 2;B[GHMZ587h; 0 1VK)CqI >C- + + -f/ M > GH7E1 > 7 -e> MV7E; - α 5LK@a5 -=> 7d14KLMZ9HG1

h

∂ h

∂t= −σα ,

uNI65.<6M >C>6- 3MZIC9 h(t)h(t) = e−σαt .

8 73 E B;Y> G:I65l@V1 > 7kK - + + - 9:1V5LGHM >6>6- + -=> 7T3MZI69_K - G t M >C/ 7E58M > G fr(r)

- 7 fz(z)-2KwP^;euNI2147:5LM > 8 73 E 2;B[<6M >6>C-

(

∂ 2

∂r2 +1

r

∂

∂r− αr

)

fr(r) = 08 43 E \Z1 B

- 7(

∂ 2

∂z2 − αz

)

fz(z) = 0 ,8 43 E \ B

1n@ -=/ αr + αz = α 3& 10GHMZKLIB7E5LM > Cq@41V9H5m1CK - G[G:;?361V9:; - G 8 73 E;E BX3 - IC71VK8MZ9:G A7E9 - ; / 9H587 - GHMZI6GkKm1 t MZ9 + -g(r, z, t) = ϕ(r, t)ψ(z, t) ,

MZtK - G t M >6/ 7E5LM > G ϕ(r, t)- 7 ψ(z, t) GHM > 7<C; /2> 5 - GT9 - G:3 -?/ 7E58@ - + -?> 7321V9ϕ(r, t) = fr(r)e

−σαrt - 7 ψ(z, t) = fz(z)e−σαz t .

; KLK - G_GHM > 7GHMZKLIB7E5LM > G_9 - G:3 -?/ 7:58@ - + -?> 7d< - GT369:M6Kg= + - G_G:I65l@V1 > 7:G (

∂ 2

∂r2 +1

r

∂

∂r− 1

σ

∂

∂t

)

ϕ = 0 3MZI69 r ∈ (0, a) , t > 0 ,8 73 E 9 1 B

ϕ = 0 3MVI69 r = a , t > 0 ,8 73 E 9 B

ϕ = 1 3MVI69 r ∈ (0, a) , t = 0 ,8 73 E 9 / B

- 7(

∂ 2

∂z2 − 1

σ

∂

∂t

)

ψ = 0 3MZIC9 z ∈ (0, `) , t > 0 ,8 73 E > 1 B

ψ = 0 3MZIC9 z = 0 MZI ` , t > 0 ,8 73 E > B

ψ = 1 3MZIC9 z ∈ (0, `) , t = 0 .8 73 E >V/ B

&%- GhGHMZKLIC7:5LM > G ϕ(r, t)- 7 ψ(z, t) G:M > 7_9 -?/ ' - 9 / '6; - GkG:MZICGhKm1 t MZ9 + - < - <6;=@ - KLMV363 - + -=> 7EG_G:IC9hKm1 21VG -< - G t M >6/ 7:5LM > G_3C9:MZ369 - Gk1VGHG:M / 5L; - Gk1VIG?#369:M CKg=+ - G 8 73 E 9 B - 7 8 73 E > B 58KO@a5 -?> 7

,+U +U )+e *P)+)+,, )#' & 4 7

ϕ(r, t) =+∞∑

n=1

AnJ0

(

λnr

a

)

exp−σλ2

nt

a2 ,8 73 EN B

ψ(z, t) =

+∞∑

n=0

an sinnπz

`exp

−σn2π2t

`2,

8 73 E < B

MZt'K - GY@41VK - I69HG]3C9:MZ369 - G λnGHM > 7[K - GW9E1 / 5 >C- GW< - 9E1 >J0 n < - J0(λn) = 0 3 &%- G / M - S / 5 -?> 7:G An

- 7 anG:M > 7

< - G / M > GH7E1 > 7 - G< P^5 > 7:; 0 9:147E58M > uNI65.G:M > 7k<6;=7 - 9 +`5 > ; - Gk361V9_K - G / M > <C587E58M > G_5 > 587:5m1VK - G3& 1 t M >C/ 7E58M > ϕ(r, t) <6MZ587TG:147E58G t 1V589 - CqKL1 / M > <C587E58M > 5 > 587E5L1VK -

ϕ(r, 0) = 1 =

+∞∑

n=1

AnJ0

(

λnr

a

)

,

uNI65X<CM >6>6- - 1V3C9B=?Gv369:M @ -?/ 7:5LM > G:I69JKm121VG - < - G J0(λmr/a)- 7 / M;+`3B7 - 7 -?> I < - KwP^MZ9H7 '6M 0 M > 1VKL5l7E;b< - G

;?K8;+ -?> 7EG_< -y/=- 7:7 - 61VG - 8 1 >6>6- ? - ; -2;eu 3 ; 3 \ - 7 ; 3 9 B -

An =2

λnJ1(λn).

8 73 E 5 B& 1 t M >6/ 7:5LM > ϕ(r, t) G?P^; / 9:587T<CM >6/y-?> <6; /2> 587E5l@ -

ϕ(r, t) =+∞∑

n=1

2

λn

J0(λnr/a)

J1(λn)exp

−σλ2nt

a2 .8 73 2 G B

&.- G / M - S / 5 -?> 7:G anG:M > 7b<6;=7 - 9 +'5 > ;?GJ< - Km1+ + - t 1C?M > -?> 9 - 3MZ9H7E1 > 7bKwP - ?B369 - GHG:5LM > < - ψ(z, t)<21 > GvKm1 / M > <C587E58M > 5 > 5l7E5m14K - <6I 3C9:M 6K =+ - 8 73 E > B - /?- uNI65[<6M >6>6- - 1V369 =?GJ3C9:M @ -=/ 7E58M > GHI69bKm1 21VG - < - G

sinnπz

`

- 7 / M;+'3C7 - 7 -?> I< - KwP^MZ9H7 '6M 0 M > 1VKL5l7E;d< - GT;?K8;+ -?> 7EGT< -y/?- 7:7 - 61VG - -

an =

4/(nπ) G:5 n - GH7k5 +'321V589 -0 G:5 n - Gc7k321V5L9 - .

& 1 t M >6/ 7:5LM > ψ(z, t) G?P^; / 9H587k<6M >C/

ψ(z, t) =4

π

+∞∑

n=0

1

2n+ 1sin

(2n+ 1)πz

`exp

−σ(2n+ 1)2π2t

`2.

8 73 2 E B

&.- 369HMB<CI6587_< - GkG:MZK8IC7E58M > G ϕ(r, t)- 7 ψ(z, t) <6M >6>C- KL1b9:;?3M > G - 5 +'36I6KLGH5LM >C>6- K8K - g(r, z, t)

g(r, z, t) =4

π

+∞∑

m=0

+∞∑

n=1

2

λn

1

2m+ 1

J0(λnr/a)

J1(λn)sin

(2m+ 1)πz

`

exp

[

−σt[

λ2n

a2 +(2m + 1)2π2

`2

]]

.8 73 2;2;B

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

& 1kGHMZKLIC7:5LM > τ(r, z, t) <CIJ369HM 6Kg= + - 8 73 > B - GH7 1VK8MZ9:G%<6M >6> ; - 321V9.K - 369HMB<CI6587.< -]/ M > @VMZKLIB7E5LM > 3MZ9c7U1 > 7GHI69TKm1b@41V9:5L16K - t

τ(r, z, t) = g ∗t∂ θg

∂t.

8 73 2 Bj 1 > GeK -0/ 1VGJ< - Gv3 - 7:587 - G / 1n@a587E;=G 8 / 1n@a587:;?GJ<CM > 7JK - Gy<65 + -?> G:5LM > GyG:M > 7e7E9B==GJ5 >#t ;?9H5 - IC9 - GDCKm1#36KLICG

3 - 7:587 - K8M >J0 I - I69q< P^M > < -#/ M > GH5L<6;=9:; - -Y7EMZIB7 -?> 9 - GH7U1 > 707:9B=?G0GHI63;=9:5 - I69 - G2CuKwP^;?321V58G:G - I69q< - G / MZI / ' - GK85 +'587 - G]7 ' - 9 +`5 uNI - G B -2K -d/ '21.+'3< - 369 - G:G:58M >- Gc7I > 5 t MZ9 + -- 7 >6- <6;=3 -=> < uaI - < - Km1J@41VK - I69 +'M C -?>C>6-<21 > GpK - @ZMZK8IJ+ - V < - KQPR; / 1V9c7.5 > Gc7U1 > 7E1 > ;W< - 7 - +`3;?9:147EI69 - -nK8I65 H + + -5t M >6/ 7E5LM > < - GHM > 369:MZ3C9 - 0 9E14<65 -=> 7<21 > GWK - G / MZI / ' - GXK85 +'587 - Gp7 ' - 9 +`5 uaI - G 8 @VMZ5L9Y5 >#t 9E1f;Ru 3 43 9.G MZI032149 - ? - +'36K - ^ <7-Z;Ru 3 43 > 9 _ B34cd5 > G:52G - I6K -KL1q@41VK - I69 +'M C -=>6>6- < - Km1b7 - +`3;?9:147EIC9 - <21 > GkK - <6M;+`1V5 >C- D < - @ZMZK8IJ+ - V

< τ >=1

V

∫∫∫

V

τ dV

- GH7]IC7E58K - 3 ; KLK - G?P^; / 9:587W<OPr1V3C9B=?G]KwP - ?B369 - GHG:5LM > 8 73 2 Bp< - τ - 7 / M;+'3C7 - 7 -?> I`< -k/=- uaI - θg(t)- Gc7XI > 5 t MZ9 + -

<21 > G D -< τ >=< g > ∗t

∂ θg

∂t,

GHMZ587-6<21 > GTK - <6M;+`1V5 >6- < - MVI69:5 - 9-CKwP - ?B369 - G:GH5LM > < - θg;A7U1 > 7 /?- KLK - <6M >6> ; - 361V9_KwP^;euNI2147:5LM > 8 73 5;B -

< τ >= jω < g(ω) > θg(ω) = jωγ − 1

βγpg(ω) < g(ω) > .

& 1#@41VK - I69 +'M C -=>6>6- G:3217E5m14K - -G:I69yK - @ZMZK8IJ+ - V - < - Km1#9:;=3M > G - 5 +'36I6K8G:58M >6>6- KLK - < g > <21 > GeK -<6M.+g1V5 >6- < - MZI69H5 - 9o<CM >6>6- -%< Pr14369B==G 8 73 2.2;B - KL1#369:MZ3C9:5L;A7E; 8 ; 3 B - 7eKL1#7:9E1 > G t MZ9 +`; - < -,& 1V3CKm1 /=- ^ E -2.543736< _ 8 9E1V3C3 - KL; - <61 > GkKwPR1 >6>6- ? - B-

< g >=8

π2

+∞∑

m=0

+∞∑

n=1

1

(m+ 1/2)2λ2n

1

jω +λ2

nR2 + (m+ 1/2)2π2

(1 + 2R)2

A2

V 2 σ

,

MZt R = `/(2a)- 7hMZt A

V=

2

a+

2

`<6;=G:5 0Z>6- K - 9E14363MV9H7hGHI69 t 1 /?- G:IC9[@ZMZK8IJ+ - 3 & 1J@41VK - I69+`M C -?>6>6- < -

KL1q@41V9H5m147:5LM > < - 7 - +'3;?9E17EI69 - < τ > G=PR; / 9:5l71VKLMZ9HG-6<61 > GTK - <6M.+g1V5 >6- < - MZI69H5 - 9-

< τ >=γ − 1

βγpg

+∞∑

m=0

+∞∑

n=1

8

π2

1

(m+ 1/2)2λ2n

1

1 +λ2

nR2 + (m+ 1/2)2π2

(1 + 2R)2 X2

,8 73 2 \B

MZt X - Gc7<CM >6> ;e321V9 8σ = c0`h

BX2 =

A2

V 2

c0`hjω

.

M - 7H7 - G:MVKLIC7:5LM > <6I#369:M6Kg= + - 8 736< B >6-d/ M > @ - 9 0Z- 9:1V365L< - + -?> 7XuNI - 3MZI69h< - G_3 - 7E5l7 - G[@41VK - I69:GX< - ω8 243 243 2 3r1 B /=- 7:7 -t MZ9 + - < - GHMZKLIC7:5LM > G - 9H;=@]=?K - <CM >6/ 3 - Iu3C9E147:5uNI - C`KwP^I6GE1 0V- 3 >6- 1VIB7E9 - t MZ9 + - < -GHMZKLIC7:5LM > - @V14Km16K - G:I69eI >6- 36K8I6GvKL1V9 0V- 0 1.+ + - < - t 9:;RuaI -?>6/=- -p1;=7:;`3C9:MZ3MZG:; - 361V9 e - 9 - 9 - K8K -q- Gc7369H;?G -?> 7E; - 1VI321V9:1 0 9E143J' - / 5 H < - GHG:MZICG3

,+U +U )+e *P)+)+,, )#' & 4

$*)9$ 4$(959/( % 6(9( 1 ( & 6%* 14M[M;+,+ - <21 > G'K - 32149E1 0 9:1V3J' - 369H; / ;?< -=> 7-[K - G'G:MZK8IC7E58M > Gq<6I 369:M CKg=+ - 8 73 E G BbGHM > 7g9 -?/ ' - 9 / '6; - G

G:MZICG t MV9 + - < - G:MZK8IC7E58M > G Cb@41V9:5L16K - G_G:;=321V9H; - Gg(r, z, t) = fr(r)fz(z)h(t) = ϕ(r, t)ψ(z, t) ,

MZt ϕ(r, t)- 7 ψ(z, t) GHM > 7kG:MZK8IC7E58M > G_9 - G:3 -?/ 7E58@ - + -?> 7< - GT369HM 6Kg= + - G 8 73 E 9 B - 7 8 73 E > B3& 19:;?GHMZKLIB7E5LM > < - Gq3C9:M 6K =+ - G 8 73 E 9 - 73 E > B - GH705 / 5 - ` -=/ 7:I6; - <21 > GbK - <6M;+`1V5 >6- < -& 1V36KL1 /?- - -=>

9E1.+ -?> 1 > 7 /?- GT< - IG?369HM 6Kg= + - GQCq< - Gk369:M6Kg= + - GT1VIG? / M > <65l7E5LM > Gh5 > 587E5L1VK - G > I6K8K - G 3;W> 3MZGE1 > 7 ϕL(r, t) = 1 − ϕ(r, t) -6K - 369:M CKg=+ - 8 43 E 9 BX< - @a5 -?> 7 -=>- ` - 7

(

∂ 2

∂r2 +1

r

∂

∂r− 1

σ

∂

∂t

)

ϕL = 0 3MZI69 r ∈ (0, a) , t > 0 ,8 73 2 9 1 B

ϕL = 1 3MZI69 r = a , t > 0 ,8 73 2 9 B

ϕL = 0 3MZI69 r ∈ (0, a) , t = 0 ,8 43 2 9 / B

G:MZ5l7 -B<21 > GkK - <CM;+g145 >6- < -& 1V36KL1 /?-(

∂ 2

∂r2 +1

r

∂

∂r− q2

)

ΦL = 0 3MVI69 r ∈ (0, a) ,8 43 2 > 1 B

ΦL =1

s3MVI69 r = a ,

8 73 2 > BMZt ΦL(r, s)

- GH7kKL1J7E9E1 > G t MZ9 +`; - < -& 1436Km1 /=- < - ϕL(r, t) - s - Gc7kKm1b@41V9:5L16K - < -& 1V36Km1 /?-o- 7 q2 = s/σ 3& 1qGHMZKLIC7:5LM > < -e/=- 369:M CKg=+ - G?P^; / 9:5l7ΦL =

1

s

I0(qr)

I0(qa).

8 73 2 N B& 1q@41VK - I69 +'M C -=>6>6- < ΦL >

< - ΦLGHI69TK - @ZMVKLIJ+ - V G?P^; / 9H587

< ΦL >=1

s

2

a2

∫ a

0

I0(qr)

I0(qa)r dr ,

G:MZ5l7 - -?>/t 1V5LG:1 > 7TI6GE1 0V- < - KL1q9 - KL147E58M > 8 ; 3 Bh< - KwPR1 >6>C- ? -D; -

< ΦL >=2

sqa

I1(qa)

I0(qa).

8 73 2.< B&%- 9 - 3MV9H7h< - G_<6;A@ - K8MZ363 - + -?> 7EGh1VG C +'3C7EMV7:5uNI - Gh< - G t M >6/ 7:5LM > G[< - = - G:G - K +'Ma<65 / ; - G I1(qa) - 7 I0(qa)8 ;Ru 3 ; 3 N B -2K85 +'587:;d1VI?#7E9HMZ5LG_369 - +'5 - 9:Gh7 - 9 + - G -2<21 > GTKwP - ?B369 - GHG:5LM > 369H; / ;?< -=> 7 - -<6M >6>C- 3MZI69 < ΦL >

< ΦL >≈2

sqa

[

1 − 1

2qa− 1

8q2a2 + O

(

1

(qa)3

)]

,

uNI65%G?P^; / 9:587 -=>6/ MZ9 - -

< ΦL >≈2σ1/2

a

1

s3/2− σ

a2

1

s2 − σ3/2

4a3

1

s5/2+ O

(

σ3

s3

)

.8 73 2.5 B

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

j PR1V369B==G_K - G[7E9:1 > G t MV9 +'; - G_< - & 1436Km1 /=- 5 > @ - 9HG - G 8 2*57343 B- 8 2.57373 9 B - 7 8 2.57373 > B]< - Km1J9H; t ;?9 -?>6/=- ^ E _9:1V363 - KL; - GT<21 > GkKQPr1 >6>6- ? - -2KL1J@V14K - IC9 +`M C -=>6>6- < ϕL >

< - ϕLG:IC9kK - @ZMZK8IJ+ - V <CM >6>6-

< ϕL >≈2√π

2

a(σt)1/2 − 1

4

(

2

a

)2

σt− 1

24√π

(

2

a

)3

(σt)3/2 + O(

(σt)2)

,8 73 G B

GHMZKLIC7:5LM > @41VKm1 6K - 3MZI69]< - G]7 - +'36GXG:ICS'GE1*+,+ -?> 7X3 - 7E587:G 8 -?> 9 -0 149:<g<CID321V9:1.+=A7E9 - 1/σ B 3 & 1vGHMZKLIB7E5LM >< ϕ > <6I3C9:M 6K =+ - 8 43 E 9 B - GH7k<6;=<6I65l7 - < - Km1bGHMZKLIC7:5LM > < ϕL >

32149_Km1q9 - Km147:5LM > ϕ = 1 − ϕL

< ϕ >≈ 1 − 2√π

2

a(σt)1/2 +

1

4

(

2

a

)2

σt+1

24√π

(

2

a

)3

(σt)3/2 + O(

(σt)2)

.8 73 E B

& 1 t M >6/ 7:5LM > ψ(z, t)- Gc7<C;=7 - 9 +'5 > ; - < - Km1 + + - t 1C=M > uaI - KL1 G:MZK8IC7E58M > ϕ(r, t) 3 ;W> 3MZGE1 > 7

ψL(z, t) = 1 − ψ(z, t) -2K - 369:M6Kg= + - 8 73 E > B[<6M >6>C- -2<21 > GkK - <6M;+`1V5 >C- < -& 1V36Km1 /?-

(

∂ 2

∂z2 − q2

)

ΨL = 0 3MZI69 z ∈ (0, `) ,8 73;2V1 B

ΨL =1

s3MZI69 z = 0

- 7 ` , 8 43 2 B

MZt ΨL(z, s)- Gc7kKm1J7E9:1 > G t MZ9 +'; - < -& 1V36KL1 /?- < - ψL(z, t) 3& 10GHMZKLIB7E5LM > < -y/?- 369:M6Kg= + - G=PR; / 9:5l7

ΨL =1

s

sinh qz + sinh q(`− z)

sinh q`.

8 73 B& 1q@V14K - IC9 +`M C -?>6>6- < ΨL >

< - ΨLGHI69TK - @VMZKLIJ+ - V G?P^; / 9:587

< ΨL >=2

sq`

sinh q`/2

cosh q`/2,

8 73]\BGHMZ587

< ΨL >=2

sq`

[

1 + 2

+∞∑

n=1

(−1)ne−nq`

]

.8 73 9 B

K 1V9k7E9:1 > G t MV9 +`147E58M > < - & 1V36KL1 /?- 5 > @ - 9:G - <6M >6> ; - <21 > GfK - Gd9 - Km17E5LM > G 8 2.573736< 9 B - 7 8 2.57373 9 B_< - KL19H; t ;?9 -?>6/=- ^ E _ 8 1 >6>6- ? - B-6Km1J@41VK - I69 +'M C -?>C>6- < ψL >

< - ψLG:I69TK - @ZMZK8IJ+ - V <6M >6>6-

< ψL >=4

`(σt)1/2

[

1√π

+ 2

+∞∑

n=1

(−1)nierfcn`

2√σt

]

,8 73 > B

MZt5 - 9 t / <6;=G:5 0Z>6- KwP^5 > 7E; 0 9E14K - < - KL1 t M >6/ 7:5LM > <OP - 9:9 - I69k<6; /2> 5 - 321V9

ierfc x =

∫ ∞

x

erfc t dt , 1F@ -?/ erfc x =2√π

∫ ∞

x

e−t2 dt .

& 10G:MZK8IC7E58M > < ψ > <6I369:M CKg=+ - 8 43 E > B - Gc7f1VK8MZ9:G[<6M >6> ; - 361V9

,+U +U )+e *P)+)+,, )#' & 4 U

< ψ >= 1 − 2

`(σt)1/2

[

2√π

+ 4+∞∑

n=1

(−1)nierfc n`

2

1

(σt)1/2

]

.8 73 N B

&.- 369HMB<6IC587k< - GG:MZK8IC7E58M > G < ϕ >- 7 < ψ > <6M >6>C- 1VK8MZ9:GTKL10@41VK - I69T< - Km1 t M >6/ 7:5LM > < - e 9 -?-=> -?>

+`M C -?>6>6- GHI69TK - @ZMVKLIJ+ - < g > <6I369:M6Kg= + -

< g >≈ 1 − 2√π

A

V(σt)1/2 +

8R + πR2

π(2R + 1)2

(

A

V

)2

σt+R3 − 6R2

3√π(2R + 1)3

(

A

V

)3

(σt)3/2

− 4

2R + 1

A

V(σt)1/2

+∞∑

n=1

(−1)nierfc n2R + 1

(σt)1/2A/V+ O

(

(σt)2)

.8 73;< B

& P - ?B369 - GHG:5LM > < - Km1J@41VK - I69 +'M C -=>6>6- < τ > < - τ GHI69TKm1J@ZMVKLIJ+ - V 8 ;eu 373 2 B

< τ >=< g > ∗t∂ θg

∂t,

<6M >6>C- -C<61 > G_K - <6M;+`1V5 >C- < - MZIC9:5 - 9-NKm1 t M >6/ 7:5LM > θg;A7U1 > 7 /=- KLK - 9 - 7 -=> I - <21 > G_K - 32149E1 0 9:1V3J' - 369:; / ; H

< -?> 7 8 ;Ru 373 5;B -< τ >= jω < g(ω) > θg(ω) = jω

γ − 1

βγpg(ω) < g(ω) > .

cd5 > G:5 - -?> t 145LGE1 > 7dI6GE1 0Z- -<21 > GoKQP - ?C3C9 - GHG:58M > 8 73;< B-< - Km1`7:9E1 > G t MZ9 +`; - < - & 1V3CKm1 /=- ^ E -2.543736< 9 _:-MZtKm1J@41V9H5m16K - < -& 1V36KL1 /?- s - GH7k9 - +`3CKm1 / ;o321V9 jω

8 7E9E1 > G t MZ9 +`; - < - MZI69:5 - 9 B - - 7k< - Km1b369HMZ369:58;=7:;+∞∑

n=1

(−x)n =1

1 + x,

Km1J@41VK - I69 +'M C -?>C>6- < τ > G?P^; / 9:5l7T<21 > GTK - <CM;+g145 >6- < - MZIC9:5 - 9-

< τ > ≈ γ − 1

βγpg

[

1 −X +8R + πR2

π(2R + 1)2 X2 +

3

4

√π

R3 − 6R2

3√π(2R + 1)3 X

3

+2

2R + 1

X

1 + e2(2R+1)/X+ O

(

X4)

]

,8 73;5 B

MZt X - GH7<6M >C> ;y321V9X2 =

A2

V 2

c0`hjω

.

& 1G:MVKLIC7:5LM > 8 73 E Bf3MZIC9 < ϕ > ;A7U1 > 7v5LG:GHI - < PRI > <6;A@ - KLMZ363 - + -?> 7J1VG C +'3C7EM47E5uNI -0-=> s = jω -Km1G:MZK8IC7E58M > 8 73.5 Be3MZI69 < τ >

> P - GH7q@41VKL16K - uNI - <21 > GbKm1+ - G:I69 - MZt ω > P - Gc70321VGb3 - 7:587- /=- uaIC51V3632149E1V5l7_5 / 5 <6I t 1V587_< - KL1b<6; /6> 587:5LM > <6I321V9:1.+2==7E9 - X 3

>6- 143636KL5 / 147:5LM >> IJ+';?9H5uNI - +'M > 7E9 - uNI - K - 7 - 9 + - 2

2R + 1

X

1 + e2(2R+1)/X14363MV9H7 - I >6- 3C9:; / 5 H

G:58M > 7:MZI @HMZI69HGh5 >t ;?9:5 - I69 - C 10−6 <1=GHI69hKL1v@41VK - I69[< - < g > / P - GH7h3MZIC9 /=- Km1uaIOPR58K > P - Gc7_321VGh9 - 3MZ9H7:;

<21 > GTKL1 > MZ9 + - M ;[ > E G 5]\ H 2^ 2;2_ 3

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

6 R 4 6% 8* G O2!$* 7 ' %$( * 9%/( #)(;Y> 9:;?GHIJ+';.- KL1@41VK - I69I+`M C -=>6>6- GH32147E5L1VK - < - KQPR; / 1V9c7b5 > GH7U1 > 7E1 > ;`< - 7 - +`3;?9:147EI69 - < τ > 3 - IC7

<6M >6/ =7:9 - ; / 9:587 - - -?> t M >6/ 7E5LM > < - KwP^; / 1V9H7Y5 > GH7E1 > 7U1 > ;_< - 369 - G:GH5LM > pg-ZGHMZI6GYKL1 t MZ9 + -A0 ; > ;?9:1VK - GHI658@41 > 7 -

< τ(ω) >=γ − 1

βγpg(ω)EP ,

8 73 \ G BMZt Ep

3 - IC7 A7E9 -y- ?C3C9:5 +';o< - < - IG?+g1 > 5g==9 - G

EP =+∞∑

m=0

+∞∑

n=1

8

π2

1

(m+ 1/2)2λ2n

1

1 +λ2

nR2 + (m + 1/2)2π2

1 + 2R2 X

,8 73 \ E B

8 - ?B369 - G:GH5LM >- ?C1 / 7 - 43 2 \ B -6MZI

EP ≈ 1 −X +8R + πR2

π(2R+ 1)2 X2 +

3

4

√π

R3 − 6R2

3√π(2R + 1)3 X

3

− 2

2R + 1

X

1 + e−2(2R+1)/X,

8 73 \2;B

8 - ?B369 - G:GH5LM > 73;5g@41VKm1CK - 3MZI69 X GHICS'GE1.+ + -?> 7y3 - 7E5l7B -1n@ -=/ R = `/(2a) - X2 =A2

V 2

c0`hjω

- AV

=

2

a+

2

`

- 7kMZt λn<6;?GH5 0Z>C- K - d?;=9:M'< - 9E1 >70 n < - J0(λn)

3

- * (8$:' (,:'+,6 *4670-7; &(, ! -7>/(?:'( *%&%- 9 - 3MZ9c7< - KL1J@41VK - I69 +`M C -?>6>6- < τ >

8 73 \ G BX<61 > GkKm1bKLMV57 ' - 9 +'Ma<7C > 1.+'5uNI - KL5 > ;n149:5LGH; -< ρ′ >= ρ0χT (pg − β < τ >) ,

MZt < ρ′ >- GH7kKL1 +'M C -=>6>6- G:I69TK - @ZMZK8IJ+ - V < - KQPR; / 1V9c7T5 > Gc7U1 > 7U1 > ;y< - +g1VGHG - @ZMVKLIJ+'5uNI - -6<6M >6>C-

< ρ′ >=1

c20[γ − (γ − 1)EP ] pg .

8 73 \ B& 1 +`M C -=>6>6- < - KQPR; / 1V9c7]< - +`1VGHG - @ZMZK8IJ+`5 uNI - < ρ′ >

- Gc7[9 - K85L;QCyKm1o@V149:5m17E5LM > < - @VMZKLI7+ - < - Km1 / 1n@a587:;δV < - Km1 t 1C?M > GHI658@41 > 7 - 8

Vm<6;=G:5 0Z>6- K - @ZMZK8IJ+ - +g14G:G:5 uNI - <6I 2IC5L< -b- 7 Vm0

K - @ZMZK8IJ+ - +g1VGHG:5 uaI -Gc7U147:5uNI - B

< ρ′ >

ρ0=< ρ >

ρ0− 1 =

Vm0

< Vm >− 1 =

−δ < Vm >

< Vm >= −δV

V.

& Pr1V<7+`5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - YP = 1/ZP< - KL1 / 1n@a587E;*-F9E1V3C3MZ9c7<6IJ<6;658732149:5L;A7U1VK -?> 7E9:1 > 7 8

−jω δV BGHI69TKwP^; / 149H7T< - 369 - G:GH5LM > uNI65 -?> 9:;=G:I6Kl7 - 8

pgB - - Gc7k<6M >6/ <6M >C> ; - 321V9

YP =−jω δV

pg= jω

V

ρ0c20

[γ − (γ − 1)EP ] ,8 73 \4\B

uNI65 > P - Gc7v1VIC7E9 - uNI - KwP - ?B369 - G:GH5LM > 8 243 > BdG:5 +'36K - + -?> 7 + -?> 7E58M >6> ;J<21 > GoK - 321V9E1 0 9E1V37' - 243 243 2g<6;?<658;5CKQP - ?C3MZGH;eGHI /?/ 5 > 7< - G / MV9:9 -=/ 7:5LM > G_7 ' - 9 +`5 uaI - G_369H; / M > 5LGH; - Gk321V9TKL1 > MZ9 + - M ;[ > E G 5]\ H 2 3

)+U ! C ) ( (X +U ! 6

j 1 > G /=- 7:7 -y- ?B369 - G:GH5LM > 8 43 \4\ B - EP

- Gc7T<6M >6> ;e361V9hKwP - ?B369 - GHG:5LM > 8 73 \ E B]MVI 8 73 \2;B 3 j 1 > GkK - Gh< - IG?/ 1VG -O58K - GH7o3MZG:GH5 6K - < - @V;?9H5 /2- 9 uNI - G:5pK - G - ` - 7EGf7 ' - 9 +`5 uNI - GfGHM > 7 > ; 0 KL5 0 ;?G 8`h → 0 -OGHMZ587 X → 0 B -KwPR1V<J+'587H7U1 >6/=- YP

/ MZ9H9 - GH3M > <&CqKQPr1V<7+`5l7:7U1 >6/?- 1 / MZI6Gc7E5 uaI - 14<65m16147E5 uaI - < - Km1 / 1n@a587:;.3;W> - ` - 7 -K - 7 - 9 + - EP

- Gc7o; / 9:587dG:MZICGdKm1 t MZ9 + - < PRI > <6;=@ - KLMV363 - + -=> 7 C / MVI69H77 - 9 + - 321V9k7E9E1 > G Ht MZ9 +'; - < -I& 1V3CKm1 /=- 8 73 \2;B-6K - 9H;?GHI6K87E147 EP = 1 3MZI69 `h → 0

- GH7k5 + +';?<65L147 3HB5 /=- 7 - 9 + - EP

- GH7o; / 9H587G:MZI6GdKm1 t MZ9 + - <OPRI > <C;=@ - KLMZ3C3 - + -=> 7O+`Ma<21VK 8 73 \ E B - -?> > ; 0 K85 0Z- 1 > 7kK - G- ` - 7:GT7 ' - 9 +`5 uNI - G -25LKO@a5 -=> 7

EP =8

π2

+∞∑

m=0

1

(2m+ 1)2

+∞∑

n=1

4

λ2n

= 1 ,

-?> @ - 9H7EI< - G369HMZ369H5L;=7:;?G 0 ; > ;?9:1VK - GTG:I65l@V1 > 7 - G

4

π

+∞∑

m=0

1

2m+ 1sin

(2m+ 1)π

`z = 1 ∀z ∈ (0, `)

8 73 \ 9 1 B+∞∑

n=1

2

λn

J0(λnr/a)

J1(λnr/a)= 1 ∀r ∈ (0, a) ,

8 43 \ 9 B

uNI65+bI6Kl7E5L3CKL5L; - G]321V9 - KLK - G H + + - G - 7k5 > 7:; 0 9H; - GT9 - G:3 -?/ 7E5l@ - + -?> 7G:I69 z ∈ (0, `)- 7 r ∈ (0, a) 9 - G:3 -?/ 7:5 H@ - + -=> 7k<6M >6>6-?> 7

8

π2

+∞∑

m=0

1

(2m+ 1)2 = 1- 7 +∞

∑

n=1

4

λ2n

= 1 .

;W>4/2> -2G:5 EP = 1 -6KQPr14<J+`5l7:7E1 >6/?- 1 / MVI6GH7:5uNI - < - Km1 / 1n@a587:; YP

8 43 \4\ BX@V14IC7

YP = jωV

ρ0c20

,

/?- uNI65 / MZ9H9 - G:3M > < C5 -?> CyKwPR1V<J+'587:7E1 >6/=- 1 / MVI6GH7:5uNI - < - Km1 / 1n@a587:;kGH5 K - +'MZIC@ - + -=> 7[< -k/ M.+`369 - GHG:5LM > G H<6;=7 -?> 7 - G - GH7kG:IC363MVG:;v1V<65m1 2147E5 uNI - 3

R 6() & 1g9 - Km147:5LM > < ρ′ >

ρ0= −δV

VIC7:5LK85LG:; - 361V9 e - 9 - 9 8 - 7o9 - 369:58G -v/ 5 H < - GHG:I6G Bf1V<J+ - 7

uNI - K - GY361V9:MZ58G\< - Km1 / 1n@B5l7E;_GHM > 7X361V9 t 1V5l7 - + -=> 7\9:5 0 5L< - G- /=- uaIC5 > P - GH7X321VGYK -_/ 1VG -?> 369:147E5 uNI -_-?> 9E1V58G:M >< - KL1 369:;=G -=>6/?- <6I +'5 / 9HMZ3J'6M >6- 9H; /?- 3C7 - I69 8 K -g/ '21.+'3)< - <6;=36Km1 /?- + -?> 7q<CI +'5 / 9:MZ37'6M >6- ;+ - 7:7 - I69- Gc7 - ?B369:5D+`;d321V9hK - t 1 / 7 - I69 δV B3 & P - ` - 7T< - KQPr1V<7+`5l7:7U1 >6/?- <6I+`5 / 9:MV3J'6M >6- 9:; /?- 3C7 - I69k1VI69E14587h3CI A7E9 -5 > 7E9HMB<CI6587032149 e - 9 - 9 -=> t 1V58GE1 > 7'I6G:1 0Z- < - KQPR;RuNI2147E58M > < -/ M > G - 9H@4147:5LM > < - Km1 +g1VGHG - -]GHI658@41 > 7'Km1369:;=G -=> 7E147E58M > <6M >6> ; - I > 3 - I36K8I6G_KLMZ5 > 8 7373 E -6;euNI217E5LM > G 73 \ > C 73 \ 5 B3

|gc| "( # +[ - '%) ) ! # ) - - ) + ,- . ) ! $ .; ( ) =$ ( "!% '%)

P^M - 7k< -y/?- 321V9:1Z9E143 -o- Gc7d<OP - ?C3C9:5 - 9_K -o/ 21'3< - 3C9 - GHG:58M > I > 5 t MV9 -o/ 9:;=;y<21 > GTI >6- 43 -hH7E5l7 - / 1F@a5l7E;0< - @ZMZK8I -

V- 7v< - GHI69 t 1 /?- A 32149e< - Gy@B5 69E17E5LM > Go< - 321V9HMZ5LG5^ 363"!$# &% H # > _! M - 7:7 -

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

`

r

a

z~ξ \ ~ve

onPp ! # qMBK85 > <C9 - t - 9 `; C0G - G - ?a7E9H; '587E;=GT321V9_< - G '5 / 9:MZ3CM >6- G !

/ 1F@a5l7E; - GH7[< - t MZ9 -T/ aKL5 > <69:5 uNI -T- 7 t - 9'; - CyG:M >`- ?a7E9H; '587E; z = 0 321V9]I >6- - 69E1 >6- 1 > 5'; - < P^I >'MZIC@ - -?> 7@a5 69E147:MZ5L9 - < -e/ 61`3< - <6;=36Km1 /=- -?> 7 ξ - 7k< - @a587 - G:G - ve = jω ξ - 7 CqG:M >- ?a7:9:; '587:;z = ` 361V9_I >6- - 69:1 >6- < PR1V<'587H7U1 >C/?- 1 / MZI6GH7:5uNI - Yr

ZI69 - ! # !%- 7 - 9 - 43 - 7:587 -/ 1n@a587:; <6;?GH5 >C- I >6-f/ 1n@B5l7E;f<CM > 7_K - G[<65 -=> G:58M > G[9 - Gc7 -?> 7_7:9B=?G[5 >#t ;=9:5 - I69 - G CJKL136K8I6Gd3 - 7:587 - K8M > ZI - I69k< PRM > < - λ0

/ M > G:58<6;?9H; - g145LGT<6M > 7fK - G<65 -?> GH5LM > G9 - GH7 -?> 7f7:9B=?GG:I63;?9H5 - I69 - GMCKQPR;=321V5LGHG - I69T< - G / MVI / - GKL5 `5l7 - G]@a5LG / MV7 - 9'5uNI - G δv,h

λ0 >>3√V

- 7 δv,h <<3√V ,

MZtδv ≈

√

2` ′v

k0

,- 7 δh ≈

√

2`hk0

.

%546$( *+ -7;/( '*#0:'( *#%

q I G$ ) 7/('9 G 6 Uj I t 14587'uaI - K - Gd<65 -?> GH5LM > Gd< - KL1 / 1n@a587:;bG:M > 7o7:9B=?Gd5 >#t ;=9:5 - I69 - GMCDKm1`KLM > VI - I69< P^M > < -q- 7-uNI -K -q/ 21'3s< - 369 - G:G:58M > >6- 369:;=G -?> 7 - 321VGo< - @41V9H5m147:5LM > GfG:3217E5m14K - Gf1VI @VMZ5LGH5 > 1 - < - Gv32149:MZ58G /=-/ 61`3 < - 369 - GHG:5LM > - Gc7oG:I63C3MZGH;bI > 5 t MZ9 - <21 > Gk7EMZIB7 - KL1 / 1F@a5l7E;! 149dG:IC587 - ~grad p = ~0

- 7fKL1@a5l7 - GHG - 321V9c7E5 / I6Km145L9 -d- GH7'uNI21VGH5 > I6K8K -o-=> 7EMVIC7k3MZ5 > 7T< - KL1 / 1n@a587E; /?- uaIC5 > PR5 `36K85uNI - 361VGT3MZIC914IC7U1 > 7XI > <C; 65l7[< - 321V9HMZ5 > ; VKL5 - 16K - - 7XK - G]3 6; > M= >6- GX< - @a5LG / MZGH587:;?G > Pr1436321V9:1V5LGHG -?> 7[321VG !

q I G$ ) 4$ !/(# # $ &* V 60!; K8K - 3 - IC7 A7E9 -y- ?C3C9:5'; - < PR1V369 =?GTKwP^;euNI217E5LM > ! G:MZICGkKL1 t MZ9 - 5 > 7:; Z9:1VK - GHI658@41 > 7 -

∫∫∫

V

[

∂ ρ′

∂t+ div ρ0~v

]

dV = 0 ,

GHMZ587∫∫∫

V

∂ ρ′

∂tdV + ρ0

∫∫

A

~v. d ~A = 0 . ! \ >

- 9H7E1V5 >6- Gv361V9:MZ58Go< - Km1 / 1n@a587:;0;=7U1 > 7v1 > 5'; - Go<OPRI > `MVIC@ - -=> 7y@a5 C9E147:MZ5L9 - < - <6;?36KL1 /?- -=> 7~ξ 5 / 5 -?> z = 0 '5 / 9HMZ36M >6- ; - 7H7 - IC9 - 7k< PR1VIC7:9 - Gk;=7E1 > 7 / 1V9:1 / 7:;?9:58G:; - Gk321V9_K - I69T5'3;=<21 >6/=- 369 - G:GH5LM > @a587 - G:G - Z 5 / 5 -?> z = ` '5 / 9HMZ36M >6- 9H; /=- 3C7 - I69 /=- 7H7 - ;RuNI2147E58M > 3C9 -?> <#Km1 t MZ9 -

GHI658@41 > 7 - <61 > GTK - <6Mg1V5 >6- t 9H;euNI -?> 7:5 - K ∫∫∫

V

jω ρ′ dV + jω ρ0

∫∫

A

~ξ. d ~A+ ρ0

∫∫

A

p

ZdA = 0 .

! \ N

)+U ! C ) ( (X +U !

j I t 1V587h< - KwP^I > 5 t MV9'587E;k<6I / 21'3< - 369 - G:G:58M > 58KO@a5 -?> 7∫∫

A

p

ZdA = p

∫∫

A

1

ZdA =

pA

Z,

! \

MZt 1/Z <6;=G:5 >6- KL1q@41VK - I69 'M -?>6>C- < - KwPR1V<'587H7U1 >6/=- < - 321V9:MV5 - 7Km1q9 - Km17E5LM > ! \ N < - @a5 -?> 71VK8MZ9:G 1

ρ0

∫∫∫

V

ρ′ dV + δV +pA

jω Z= 0 ,

! \

MZt δV =

∫∫

A

~ξ. d ~A <C;?G:5 >6- KL1`@41V9:5L147E58M > < - @ZMZK8I - < - Km1 / 1n@a587E;JKL5L; - 1VI? @a5 69:147E58M > G< - Km132149:MZ5 `5 / 9:MV36M >6- ; - 7:7 - I69 -=> z = 0 !

q I #9$ J7%( 6 '2* # 6( 6 4&3* 109 - KL147E58M > K85 > ;?1V9:58G:; -

ρ′ = ρ0χT (p− βτ)

9 - 3MZ9H7:; - <21 > GkKm1b9 - KL147E58M > < - Km1q9 - Km17E5LM > ! \ 3 - 9 - 7T< P^;?KL5 `5 >6- 9[Km1J@41V9:5L16K - ρ′ - 7k<6M >6>C-[

1 +A

jωZχTV

]

p− β

V

∫∫∫

V

τ dV =δV

χTV,

!% ;RuaI6147E58M > uaIC5 `M > 7:9 -_-?> 32149H7E5 / I6K85 - 9 uNI - K -h/ 61`30< - 369 - GHG:5LM >'>C- <C;?3 -?> <'321VGY< - τ(~r, t) `1V5LG< - G:1J@41VK - I69 'M -?>6>C- G:I69[7:MZIC7_K - @VMZKLI -

V < - Km1 / 1F@a5l7E; 1VKLMZ9HG - uNI - τ @41V9H5 - < - t 1C?M >5 `3MZ9H7E1 > 7 - <61 > GTK - G / MZI / - GK85'587 - GX7 - 9 `5 uNI - G !q I #9$ ) 4$ ]$ ) * V ) 1* (

MVI6GG:1 t MZ9 - ! - K8K - G=PR; / 9:5l7k<21 > GTK - <CM&g145 >6- t 9H;euNI -=> 7:5 - K (

jω

c0+ `h ∆

)

τ =γ − 1

βγ

jω

c0p ,

! # MZI -?>C/ MZ9 -

τ =c0`hjω

∆τ +γ − 1

βγp .

! MZt `h <6;=G:5 >6- I >6- KLM > ZI - I69 / 1V9E1 / 7E;=9:58GH7E5 uNI - < - <65 `I6GH5LM > 7 - 9'5uNI - ! M&'3C7 - 7 -=> I#< - KQPRI > 5 Ht MZ9 `5l7E;f<CI / 21 `3< - 369 - G:GH5LM > p /=- 7H7 - < - 9 > 5g==9 - 9 - Km147:5LM > <CM >6>6-

∫∫∫

V

τ dV =c0`hjω

∫∫

A

~grad τ. d ~A +γ − 1

βγV p .

!

( +,/6 *+ 1G:MVKLIC7:5LM > < - KwP^;euNI217E5LM > ! # uNI65WG:147E58G t 1V5l7 CKL1 / M > <C587E58M > τ = 0 G:I69eK - G t 9:M > 7E5 =?9 - G 3 - IB7

=7:9 - ; / 9:587 - K - t 1 / 7 - I69 ejω t - GH7kM&'5LG

τ =γ − 1

βγp[

1 − e−j khu]

, ! \

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

MZt u - Gc7dKL1'3MZGH587E58M > <6I 3MZ5 > 7 / M > GH5L<6;=9:;J< - KL1 / 1n@B5l7E; 9 - 3;?9H; - 361V99:1V363MZ9H7<C0Km1 > MZ9 g1VK - G:MV9H7U1 > 7 -C0Km1bGHI69 t 1 /=- < - Km1q321V9HMZ5 u = 0 G:IC9kKL1b321V9HMZ5 ! - 7H7 - t MV9 - < - G:MVKLIC7:5LM > - Gc7 / MZ9:9 -?/ 7 - GHMZI6Gg9H;?G - 9H@ - < P^I >6- 321V9H7 uNI - KQPR;=321V58G:G - I69'< - G / MZI / - GK85'587 - Gh7 - 9'5uNI - GdG:MV587T7E9 =?Gd3 - 7:587 - 32149d9E14363MV9H7d1VIG?<65 -=> G:58M > Gk< - KL1 / 1F@a5l7E; / P - Gc7 H C H <6589 - uNI - KL1t M >C/ 7E58M > e−jkhu <6; / 9HMZ5LGHG - 7E9 =?GT@B5l7 - 3MZI69TI > 3MZ5 > 7'uNI65.G=PR; / 1V9c7 - < - KL1'361V9:MZ5 - 7k< Pr14IC7E9 - 32149H73uNI -GHMZ587 t 1V5l7_K -o/ CMZ5b?

kh =−1 + j√

2

√

k0/`h , !

3MZI69 uNI -g/=- 7:7 - t M >C/ 7E58M > 9 - 3C9:;?G -?> 7 - C5 -?> I >6- M > < - < - <65 `I6G:58M > 7 - 9'5uNI - uaIC5X3; > =A7E9 - <21 > GvKL1/ 1F@a5l7E; Cq32149H7E589_< - GT32149:MZ58G ! 1V9TGHI6587 - GHI69TKm1qG:I69 t 1 /?- u = 0

∂ τ

∂u= −γ − 1

βγp

1 + j√2

√

ω

c0`h- 7∫∫

A

~grad τ. d ~A =

∫∫

A

∂ τ

∂udA = −1 + j√

2

√

ω

c0`hAγ − 1

βγp ,

/=- uNI65%<6M >6>6- 2> 1VK - -?> 7T3MZI69_KwP - ?B369 - GHG:5LM > ! ∫∫∫

V

τ dV =

[

V − 1 − j√2A

√

c0`hω

]

γ − 1

βγp .

! > %- 9 - 3MZ9H7[< -d/=- 9:;=G:I6Kl7U147[<21 > GhKL1v9 - Km17E5LM > ! % <CM >6>6- KQP - ?B369 - G:G:58M > G:IC58@41 > 7 - 3MZIC9[KwPR1'36KL5l7EI6< - < -KL103C9 - GHG:58M > <61 > GTKm1 / 1F@a5l7E;

p = − γδV

χTV

1

1 +γA/V

jω χTZ+

1 − j√2

(γ − 1)A

V

√

c0`hω

. ! N

MZ9HG / MZI / - G_KL5'587 - G KwPR1V<'587H7U1 >6/=- < - Gh361V9:MZ58G - GH7 > I6K8K - <21 > GXK -d/ 1VGh369H;?G -?> 7_GE14I t 3MZIC9XKQP - ?B7:9:; H'587:; z = ` uaIC5X1#3MZI69y1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - Yr

! 1D@41VK - I69 `M -?>6>6- 1/Z < - KQPr14<`5l7:7E1 >6/?- < -32149:MZ5 - GH71VKLMV9:Gh<6M >6> ; - 321V9

A

Z=

∫∫

A

1

ZdA =

∫ 2π

0

∫ a

0

Yr

πa2 r dr dθ = Yr ,

/=- uNI65%<6M >6>6- 3MVI69_KwP - ?B369 - G:GH5LM > ! N < - Km1b369 - G:GH5LM > 1 / MZI6GH7:5uNI -

p = − γδV

χTV

1

1 +γ

jω χTVYr +

1 − j√2

(γ − 1)A

V

√

c0`hω

. !

%- 6I65L< -y-=>#t - 9 `;y<21 > GKL1 / 1n@B5l7E; - Gc7d< - KQPr1V589 uaIC5%3 - IC7 A7E9 -e/ M > GH5L<6;=9:; / M& - I > 1]dy32149 t 1V5l7 /=- uNI65p5 `3CKL5uNI - ρ0c20 = γ/χT ≈ γP0

! PR1V<'587H7U1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9:1 > G t - 9H7 YT =Sve + Yep(0)

p(`)<6IG aGc7>= - `5 / 9:MV36M >6- G Hi/ MZI63CK - IC9 ;eu ! !$# G=PR; / 9:5l7k5 / 5

YT = − jω δV

p+ Ye = jω

V

γP0

[

1 + (γ − 1)A

V

1 − j√2

√

c0`hω

]

+ Ye + Yr . !

+)# +U d+U ) 7

- 7:7 -D- ?B369 - GHG:5LM > ! < - KwPR1V<'587H7U1 >C/?- 1 / MVI6GH7:5uNI - < - 7E9:1 > G t - 9H7 YT

- GH7 /=- KLK - 369HMZ3MZG:; - 321V9k58;=7 - 7 %-?/ MZKLK85 >6- 7^ &% _B1V5 > G:5 uNI - 361V9[1VKLKL1&^ _a3MZI69pK - G;=7E1VKLM >6> 1 - G -=> 21VGHG - G t 9:;euNI -=>6/?- G ! - 7:7 -- ?B369 - G:GH5LM >)- GH7b1VI6G:GH5[; 1VK - C KQPr1V< `5l7:7U1 >6/?- YP

7E9HMZIC@V; - 32149 e - 9 - 9 ;Ru ! ! \4\ ;eu ! ]\ < - KwPR1V9H7:5 / K -< - e - 9 - 9T^$# _ EP

;A7U1 > 7T<6M >6> ;y321V9hKL1bG:MZK8IC7E58M > C / MZI69H7_7 - 9 - 361V9h7E9:1 > G t MV9'; - < - 1V3CKm1 /=- ;Ru ! ! \ ;eu ! !#< - e - 9 - 9 CbKwP^MZ9:<C9 - # -?> X - ?B369 - G:GH5LM > uNI65 > P - GH7_3214G /?- KLK - 9 - 7 -?> I - <61 > GhKL1 > MZ9 - ;[6> # % \ H ^ _ !

1DGHMZKLIB7E5LM > ! 3MZI69fK -t 1 / 7 - IC9 −jω δV/p - GH7o58< -?> 7E5 uNI - CgKm1`G:MZK8IC7E58M > < - e - 9 - 9 ! \4\ 3MVI69 −jω δV/pg 6MV9'5LGbKQPr1V< `5l7:7U1 >6/?- YruNI65 > P - GH7'3214G`369H5LG -#-?> / M&'3C7 - <21 > G0K - 'MB<6==K - < -

e - 9 - 9 -?> ; / 9:58@41 > 7 1/√

j = (1 − j )/√

2 !

|g < $ # - A! "! ) #? ! ) j 1 > GoK -J/ 14GoMZtuK -v/ 21'3u< - 369 - G:GH5LM >u> P - Gc7y36KLICGfI > 5 t MZ9 - <21 > GoKm1 / 1n@B5l7E; K - Gf<65 -?> G:5LM > Gd< -

Km1 / 1F@a5l7E; >6- G:M > 7v36KLI6Gy3 - 7:587 - Gy< - @41 > 7vKm1DKLM > ZI - I69y< PRM > < - I > 14IC7E9 - 'Ma<6=?K - 3 - IC7 =7E9 - 3C9:MZ3MZG:;3MVI69o9 - 369:;=G -?> 7 - 9JK -q/ 21'3x1 / MZI6Gc7E5 uaI - <61 > GeKm1 / 1n@a587:; Km1#369:MV321 147:5LM >u- GH7 /?- KLK - < PRM > < - Ge36KL1 >6- GG:I65l@41 > 7JKQPr1? - < - KL1 / 1n@a587:; / BK85 > <C9:5uNI -'- 7JKwPR1'MZ9H7:5LG:G - -?> 7v< - GvM > < - Gb<)1VI? - ` - 7EGv@a58GuNI - I? - 77 - 9'5uNI - G - Gc7q369H5LG -=>)/ M&'3C7 - <21 > GJK -g> M& 69 - < P^M > < -D- 7q321V9vKwP^5 > 7E9HMB<CI / 7:5LM > < P^5'3;=<21 >6/=- Gb< -321V9HMZ5%1VIG? - ?a7E9H; '587E;=G ^ H #!_! - 'Ma<6=?K -f- Gc7 -?> 321V9c7E5 / I6KL5 - 9 /=- KLIC5%369:MZ3MZGH;e361V9h1VK8Km1& ^ _ !j 1 > G]I > GHMZI / 5< -/ Km1V9c7E;T<6I'<65LG / MZI69HG /?- 9H7E1V5 >6- GX< - GX1V36369HM9?B5 g147:5LM > G t 1V587 - GW<21 > G /=- 32149E1Z9:1V3 -3 - IC@ -=> 7h321V9:1 L7:9 - I > 3 - Ig9:1V365L< - G ! ;Y>`/ M&'36KL; -=> 7W< - KL1o9:;?GHMZKLIB7E5LM > 3C9:MZ3MZG:; - <61 > G /?- 321V9E1 Z9E1V3 - I >6- 1V363C9:M / - 36KLI6G_<C;=7U145LKLK8; - <CI - 369:M6Kg= -f- GH7<6M >6> ; - <21 > GTKQPr1 >C>6- ? - e !

4 67()*+ * /* %5 >)/?- 9H7E1V5 >)> M 69 - < P a3MV7 6==G - G`GH5'36KL5 2/ 147:9:5 /?- Ge3 - IC@ -?> 7 =7E9 - 9 - 7 -=> I - G'3MZIC9bG:5 `36K85 2- 9yK - G

;euNI2147:5LM > G[< - KQPr1 / MZI6Gc7E5 uaI - ! # ! - 7 ! ^ ! N _! ; KLK - GXGHM > 7_369H;?G -=> 7E; - Gk1VI t I69 - 7QC - GHI69 -< - K - I69_I6GE1 - !

.- G BGc7>= -k/ M > G:5L<C;?9:; - GH7 /?- K8I652<6; / 9:5l7W321V9WKm1 VI69 - !$# <6;6IC7X<6I`361V9E1V9E1V3 - ! I >6-T/ 1n@a587E;/ aKL5 > <69H5uNI - < - 9E1 ZM > a - 7y< - K8M > ZI - I69 ` - Gc7 - ? / 5l7E; - CGHM >s- ?a7E9H; '587:; z = 0 32149fI > `5 / 9HMZ36M >C-; - 7H7 - IC9d< PR1V<'587H7U1 >6/=- C / 589 / IC587MZIC@ - 9H7 Ye

I > '5 / 9:MZ3 6M >6- 9:; /?- 3C7 - I69o< PR1V<'587:7E1 >6/=- 1 / MZI6Gc7E5uNI -C / 5L9 / I65l7_MZIC@ - 9c7 Yr

- Gc73CKm1 / ; C0KQP - ?a7E9:; `5l7E; z = ` ! 1 - C9E1 >6- < - KQPR; - 7:7 - I69 - Gc7GHI6363MZG:; - 1n@ZMV5L9#I > 9E1 ZM > ; Z1VK C /=- K8I65o< - Km1 / 1n@a587:;! M >

`MVIC@ - -=> 7 - GH7'14G:G:5 `58KL;C /?- KLI65[< PRI > 365LGc7EM > 36KL1 > - KLK -g/ 9H; - I > <C; 65l7 Sve

uaI61VG:5hI > 5 t MZ9 - G:I69GE1 GHI69 t 1 /=- ! P^; / 149H7q5 > GH7E1 > 7U1 > ;#< - 369 - G:G:58M > p ;=7E1 > 7 uNI21VGH5 I > 5 t MZ9 - G:I69qI >6- G -?/ 7E58M > <6I)7EI - 5LK - GH7 / M > G:58<6;?9H; / M& - 5 > <6;=3 -=> <21 > 7'< - Km1 / MBMZ9H<6M >6> ; - r - 7'3 - IC7 A7E9 -> MV7E; p(z) ! 1V90G:IC587 - KL1/ M&'3MZGE1 > 7 - 9E14<65m1VK - <CI V9E1V<65 -?> 7h< - 369 - G:GH5LM >- Gc7 > ; ZKL5 - 1CK - 321V9h9:1V363MZ9H7OCbGE1 / M`3MZGE1 > 7 - 1?B5m14K -- 7 321V9TG:IC587 - KL1 / M&'3MVGE1 > 7 - 9:1V<65L1VK - < - KL1J@a587 - G:G - 32149H7E5 / I6KL1V5L9 - 3 - IC7 A7E9 -y> ; ZKL5 Z; - vr << vz

! 1V9p1V5LK8K - I69:G -?> 9E1V58G:M > < - KQPR5 `3MZ9c7U1 >6/=- < - G - ` - 7:G K85L;?GCkKm1h@a5LG / MZG:5l7E;]< -[/ 5LG:1V5LK8K - -?> 7 Km1_@41V9:5L147E58M >< - Km1q@B5l7 - G:G - 321V9c7E5 / I6Km145L9 - v = vz

G:I65l@41 > 7dKL10@41V9H5m16K - r - GH7k7E9 =?GG:I63;?9H5 - IC9 - C'GE10@V149:5m17E5LM > G:I65l@V1 > 7Km1b<6589 -=/ 7E58M > z ∣

∣

∣

∣

∂

∂zv(r, z)

∣

∣

∣

∣

<<

∣

∣

∣

∣

∂

∂rv(r, z)

∣

∣

∣

∣

. ! > %

5 > 14K - -?> 7G - I6K - KL1 / M&'3MZGE1 > 7 - 1?B5m1VK - < - KQPR;RuNI2147E58M > < -5: 1F@a5 - 9 H 7EMB - G - GH7DC - ?C3C9:5 - 9 - K8K -

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

369 -?> <Km1 t MZ9 - 1436369:M / 6; - GHI658@41 > 7 - (

1

c0

∂

∂t− ` ′

v1

r

∂

∂rr∂

∂r

)

vz(r, z) = − 1

ρ0c0

∂

∂zp(z) , r ∈ (0, a) , z ∈ (0, `) .

! > #F1 c /=- 7H7 - ;euNI217E5LM > ! > #F1 G:M > 7T14G:G:M / 58; - G_K - G[< - IG? / M > <6587:5LM > G]G:IC58@41 > 7 - G_G:I69_KL1 / M`3MZGE1 > 7 - z < - KL1@a587 - GHG -

vz(0, z)9 - GH7 - 2> 5 -y- 7 vz(a, z) = 0 ∀z . ! > #

1'GHMZKLIC7:5LM > 3MZI69 vz< -e/?- 369:M6Kg= - ! > #F1 G?P^; / 9H587 -?> 9:; Z5 - 21V9 `M > 5uNI - -=> M& - 7H7U1 > 7kK -

t 1 / 7 - I69 ej ω t vz(r, z) =

j

ωρ0

∂

∂zp(z)

[

1 − J0(kvr)

J0(kva)

]

, ! >

MZtK -o> M& 69 - < PRM > < - < - <65 `I6GH5LM > kv

- GH7k<6; 6> 5.321V9kv =

1 − j√2

√

k0/` ′v .

1q@41VK - I69 'M -?>C>6- < -v/=- 7:7 - @a5l7 - GHG - G:I69TI >C- G -?/ 7:5LM > <6I7EI - 'MB<6==K - M > < - 36Km1 >C- G?P - ?B369H5 -1VK8MZ9:G / M& - GHI6587

< vz >a=2

a2

∫ a

0

vz(r, z)r dr =j

k0ρ0c0

∂ p(z)

∂z

[

1 − 2

kva

J1(kva)

J0(kva)

]

. ! >

PR;RuaI6147E58M > < -y/ M > G - 9c@V17E5LM > < - KL1 g1VGHG - ! 369 -=> < KL1 t MZ9 - G:5 `36K85 ; -

jωγ

c20(p(z) − βτ(r, z)) + ρ0

∂

∂zvz(r, z) = 0 ,

! > \ - 7G:10@41VK - I69'M -=>6>6- G:I69TI >C- G -?/ 7:5LM > <6I7EI - G?P^; / 9:5l7

∂

∂z< vz >a +

jω

ρ0

γ

c20(p(z) − β < τ >a) = 0 .

! > ;Y> 2> <6I t 1V587 uNI - Km1u@41V9:5L147E58M > < - 7 - `3;?9:147EI69 - τ G=Pr1 >C> ICK --?> r = a C KwP^5 > Gc7U1V9q< - KL1u@a5l7 - GHG -32149H7E5 / I6KL1V5L9 - ! > % G:1g@V149:5m17E5LM > GHI658@41 > 7oKL1'@41V9:5L16K - r - Gc7f7E9 =?GoG:IC3;=9:5 - I69 - CgGE1g@41V9H5m147:5LM > G:I65l@41 > 7KL1J@V149:5m1 6K - z !1cd5 > G:5 KQPR;RuaI6147E58M > ! 3MZI69_KL1 / M > <6I / 7:5LM > < - KL1 / 21VK - I69_369 -?> <Km1 t MZ9 - 1V36369HM / 6; -GHI658@41 > 7 -

(

1

c0

∂

∂t− `h

1

r

∂

∂rr∂

∂r

)

τ(r, z) =γ − 1

βγ

jω

c0p(z) , r ∈ (0, a) , z ∈ (0, `) .

! > > 10G:MZK8IC7E58M > < -y/?- 7:7 - ;RuaI6147E58M > uNI65.GE17E5LG t 1V587h1VIG? / M > <65l7E58M > G

τ(0, z) 9 - Gc7 - 2> 5 -e- 7 τ(a, z) = 0 ∀z , ! > N G=PR; / 9:5l7

τ(r, z) =γ − 1

βγp(z)

[

1 − J0(khr)

J0(kha)

]

, ! >

MZtK -o> M& 69 - < PRM > < - kh

- Gc7k<6; 2> 5%321V9kh =

1 − j√2

√

k0/`h ;

+)# +U d+U )

GE1b@41VK - I69'M -?>6>C- GHI69TI >6- G -=/ 7E58M > G?P^; / 9H587

< τ >a=γ − 1

βγp(z)

[

1 − 2

kha

J1(kha)

J0(kha)

]

, ! >

%- 9 - 3MZ9H7< -J/=- 7H7 - ;RuNI2147E58M > ! > <21 > GKwP^;euNI2147:5LM > ! > / M > <6I65l7TC'I >6- G -?/ M > < - 9 - KL147E58M >-?> 7E9 -369 - G:GH5LM >- 7T@a587 - GHG - ∂

∂z< vz >a= − jω

ρ0c20

[

1 + (γ − 1)2

kha

J1(kha)

J0(kha)

]

p(z) , ! N %

uNI65 1n@ -=/ Km1s369 - `5 =?9 - ;eu ! ! > 'M > 7E9 - uNI - K - 3C9:M 6K = - 3 - IB7 A7E9 - 7E9:1V587:;32149g1 > 1VKLM&V5 - 1n@ -=/Km1g76;?MZ9H5 - < - GyKL5 >6- GMC / M > GH7E1 > 7 - Gv9H;?321V9c7E5 - G ! ;W> - ` - 7 K - Go;euNI2147:5LM > G ! > - 7 ! N % -?> > MV7U1 > 7w(z) = S < vz >a

K - <6;6587f1 / MVI6GH7:5uNI - MZt S = πa2 <C;?G:5 >6- KL1`G -=/ 7:5LM > <69HMZ587 - <6I / aKL5 > <69 - G:M > 7< - Km1 t MZ9 -

∂ p

∂z+ Zvw = 0

- 7 ∂ w

∂z+ Yhp = 0 ,

! N # MZtKwP^5'3;=<21 >6/=-f-?> G:;=9:5 - Zv

- 7kKwPR1V<'587:7E1 >6/=-d-?> 321V9:1VKLK =?K - Yh32149kI > 587:;o< - KLM > VI - I69_G=PR; / 9H58@ -=> 7

Zv =1

S

j k0ρ0c01 −Kv

- 7 Yh = Sj k0

ρ0c0[1 + (γ − 1)Kh] ,

! N 1n@ -?/

Kh,v =2

kh,v

J1(kh,va)

J0(kh,va).

! N

@7( (8: +,6#*46$ -7( / : /* $-$ # 1h3C9:MZ321 147E58M > < - GM > < - G G:I63C3MZGH; - Gp36KL1 >6- G <21 > GK -\/ aK85 > <69 -\- Gc79:; V5 - 32149%KwP^;euNI2147:5LM > GHI658@41 > 7 - M B7 -?> I -y-=> ;=KL5'5 > 1 > 7hKm1J@41V9H5m1CK - w <21 > GTK - GT;RuaI6147E58M > ! N #

(

∂ 2

∂z2 + k2z

)

p(z) = 0 , z ∈ (0, `) ! N \

MZtk2

z = −ZvYh = k20

1 + (γ − 1)Kh

1 −Kv

, 1F@ -?/ Re kz > 0- 7 Im kz < 0 .

! N pMZIC9g7EMZICG#K - G'7 a3 - G< - / MZIC36K - I69 - 7D7:MZIC7 - Km1 Z1 - < - t 9:;RuaI -?>6/=- G / M > G:5L<C;?9:; - |kh,va| > 8 1V5 > G:5OK -f/ aK85 > <69 - 3 - IC7 A7E9 -o/ M > GH5L<6;=9:;oG:M - I > 7EI - 4KL1V9 - |kh,va| >> 1 ! > <6;A@ - K8MZ363 - -?> 71VG '3C7EM47E5uNI - C KQPRMV9:<69 - # < - GbK8M > ZI - I69:G / 1V9:1 / 7:;?9:58GH7:5uNI - G ` ′

v

- 7 `h < - G t M >6/ 7E58M > GJ< - - GHG - K/ M > 7 -?> I - Gk<21 > G Kh,v<6M >C>6- 1VKLMZ9HG_3MVI69 kz

k2z ≈ k2

0

[

1 +1 − j√

2

2

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

, ! N >

G:MZ5l7kz ≈ k0

[

1 +1 − j√

2

1

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

. ! N4N

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

1 / M > <65l7E58M > 1VIG?#KL5'587 - GXG:I69_KL1bG:I69 t 1 /?- <6I `5 / 9HMZ36M >C- ; - 7H7 - IC9kG=PR; / 9H587 -?>/t 1V5LG:1 > 7TI6GE1 - < -KL103C9 - '5g==9 - 9 - Km17E5LM > ! N #

− 1

Zv

∂ p

∂z= Sve − Yp p

3MVI69 z = 0 , ! N

MZt Yp

- GH7kI >6- 1V<'587H7U1 >C/?- 9 - 369:;=G -=> 7E1 > 7dK - G - ` - 7:GT< -y/ MZI / - GdK85'587 - GX7 - 9 `5 uNI - GT1VI@ZMV5LG:5 > 1 - < -KL10361V9:MZ5 @ZMV5L9 ^ ! !$#!_ - 7d1 >6>6- ? -

Yp =S

ρ0c0

1 + j√2

√

k0(γ − 1)√

`h . ! N

;Y>z = ` KQPRM > < -[- GH7\9H;=2; / 65 - G:I69\I > 36Km1 > < P^5'3;=<21 >6/=- Yr+Yp /?- uNI65a<6M >6>C- 3MVI69 KL1 / M > <6587:5LM >1VI?KL5 `5l7 - GXG:I69TK - 9:; /=- 3C7 - I69 < Pr14369B==GKL1b369 - `5 =?9 - ;RuNI2147E58M > ! N #

1

Zv

∂ p

∂z= −(Yr + Yp) p

3MZI69 z = ` . ! %

10GHMZKLIB7E5LM > < - KwP^;euNI2147:5LM > < - 369HMZ321Z147E58M > ! N \ 3MZI69_KL1q369 - G:G:58M >- Gc7d< - KL1 t MZ9 -p(z) = B(e−j kzz +R` ej kzz) ,

! # MZtK - G / M > GH7U1 > 7 - G B - 7 R`

G:M > 7k<6;=7 - 9 `5 > ; - Gk361V9_K - G / M > <C587E58M > GT1VI? t 9:M > 7:5g=?9 - G ! 1 / M > <65l7E58M > 1VIG? t 9HM > 7E5 =?9 - G ! &% -=> z = ` <6M >6>6- 1VK8MZ9:G

R` = e−2j kz`Yi − (Yr + Yp)

Yi + (Yr + Yp),

! MZt Yi 1V3C3 - KL; - KQPr1V< `5l7:7U1 >6/?- 587E;=9E147:58@ - <6I / aKL5 > <69 - G?P^; / 9:587

Yi =

√

Yh

Zv=

j kz

Zv=

S

ρ0c0

√

[1 + (γ − 1)Kh](1 −Kv) , !

GHMZ587 -?> 369 -=> 1 > 7K - <6;A@ - KLMZ363 - -?> 7d1VG '3C7EM47E5uNI - < - G t M >6/ 7:5LM > G_< - - GHG - KCqKwP^MZ9H<69 - K - 36KLI6G 214G

Yi ≈S

ρ0c0

[

1 +1 − j√

2

2

a√

k0

(−√

` ′v + (γ − 1)

√

`h )

]

, ! ]\

;Y> 2> KL1 / M > <65l7E58M > 1VIG? t 9:M > 7E5 =?9 - G ! N -=> z = 0 <6M >6>6-

B =Sve

Yi(1 − R`) + Yp(1 +R`).

! - * ( $:' (,:'+,6 *4670-7; &( !

1 uNI21 > 7:587:;eIB7E5LK - <21 > GkKL1 `;A7 6Ma< - < P^;=7E1VKLM >C> 1 - 32149_9:; / 5L369HM / 587:; - GH7kKQPr1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5 uaI -< - 7E9:1 > G t - 9H7 YT uNI6MV7E5 -?> 7<6Iv6IG?e< - @a587 - G:G -[-?>e/ MVI69H7 H / 589 / I65871 / MZI6Gc7E5uNI - 369HMB<CI6587321V9.K - `5 / 9:MV36M >6-IC7:5LK85LG:; -=> ; - 7:7 - I69y321V9yKm1#369 - GHG:5LM > 1 / MZI6GH7:5uNI - 1Z58G:GE1 > 7eGHI69vKL1 - 69E1 >6- IC7E58KL5LGH; -q-=> 9:; /?- 3C7 - I69 #&! !

YT =Sve + Yep(0)

p(`).

! >

U , o +e @o U

; KLK - G?P^; / 9H587 -?>/t 1V5LG:1 > 7TI6GE1 - < - GT;euNI2147:5LM > G ! # ! - 7 !

YT = j sin kz`

[

Yi +(Ye + Yp)(Yr + Yp)

Yi

]

+ cos kz` (2Yp + Ye + Yr) . ! N

;W> 214G:G - G st 9:;euNI -=>6/?- G KwP - ?B369 - G:GH5LM > ! N < - @a5 -=> 7YT ≈ j kz` Yi + 2Yp + Ye + Yr ,

! & G:MZ5l7 -?>/t 1V5LG:1 > 7TI6GE1 - < - G - ?B369 - G:GH5LM > G ! ]\ ! N - 7 ! N]N 1n@ -?/ S = πa2

YT ≈ jωV

ρ0c20

[

1 + (γ − 1)

(

2

a+

2

`

)

√

c0`hjω

]

+ Ye + Yr ,

G:MZ5l7 YT ≈ jω

V

γP0[1 + (γ − 1)X] + Ye + Yr ,

! 1n@ -?/

X =A

V

1 − j√2

√

`hk0

,

K - 9:1V363MZ9H7TGHI69 t 1 /?- @VMZKLI - < - KL1 / 1n@a587E;o@41VIC7 AV

=2

a+

2

`!

- 7:7 -g- ?C3C9 - GHG:58M > < - KwPR1V<'587H7U1 >6/=- YT

- Gc7 65 -?> Km1 - CKQPRMZ9H<69 - # -?>X uaI - KwP - ?B369 - GHG:5LM > ! \4\ ! \ 7E9HMZIC@V; - <21 > GTI > < - Gk< - IG? `Ma<6;=K - G[< - e - 9 - 9 - 7T7E9HMZIC@V; - ; 1VK - -?> 7_<21 > GK - `Ma<6==K -

43 - 7E5l7 -o/ 1F@a5l7E; 9 - KL147E58M > ! !

|g ! - ' ) ) % . $ ) ! - ( ` '_) )[ - ! -W MZIB7 - GXK - G ';=76MB< - G[< -/ 1VK / I6K < - KQPr1V< `5l7:7U1 >6/?- 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7[IB7E5LK85LGH; - G C /?- HMVI69XM > 7

;=7:;k3C9:;?G -?> 7E; - Gh<61 > G /?-k/ 21V365l7E9 - ! pMZI69Y7 - 9'5 >C- 9 I >6- @V;?9H5 2/ 147E58M >0> I `;=9:5uNI - < - GX369HMZ3MZGY7 -?> I6GXG:I69/?- Gk<C5 `;?9 -=> 7EGT9:;=G:I6Kl7U147:G - GH7k369H;?G -=> 7E; - 5 / 5!

+ -5/?:&@7( 9-7 7 & *+ 6 $* + .- 'MB<#=?K - 3C9:MZ3MZG:;]321V9 Z1V9c@a5LG ^$# _ - 79 - 369:58G.321V9Km1 > MZ9 - ;[0> # % \ H ;eu ! !$# !$# G=Pr1V3C36I65 -G:I69vK - 9E1V58G:M >C>6- -?> 7v< - e - 9 - 9v369H;?G -?> 7E;#<21 > GJK - 32149E1Z9:1V3 - ! ! %-g/ 14GJ9 - 7 -=> I)361V9 Z149H@a5LG - GH7/?- KLI65pMZtuKwP^; / 1V9H7o5 > GH7E1 > 7U1 > ;b< - `1VG:G - @ZMZK8I'5uNI - ρ′g 1VGHG:5'5LK8;ZCDI > 7 - 9 - < - G:MZI69 /?-q- GH7yI > 5 t MZ9 -

<21 > GTKL1 / 1n@a587E; /Rt ! ! !$# ! - 3 -?> <21 > 7 / M& - 5 > <65uNI6;D<61 > GqK - 321V9E1 Z9E1V3 - ! !$# K - G -=/ M > < - C9 - < - KQPR;RuNI2147E58M > < -/ M > <6I / 7:5LM > < - Km1 / 61VK - I69 7 - KLK - uNI P - KLK -k- GH7 t MZ9 bI6KL; - 321V9 e - 9 - 9 ;eu ! ! \ - 7 ! >6- 9 - 369:;=G -?> 7 - 321VGKwPR1 / 7:5LM > < P^I >6- G:MVI69 /=- <21 > GoK - <6M&`1V5 >6- `1V5LGKwP^;=7E147 - <6Iu2IC5L< - <61 > GfKL1 / 1n@a587:;! p1V9o1V58KLK - I69:G <21 > GvI >6-D/ 1n@B5l7E;`<6M > 7bK - GJ<C5 -?> GH5LM > GeG:M > 7J3 - 7E5l7 - Gv321V9v9E1V363MZ9c7 C KL1K8M > ZI - I69e< PRM > < - 43 - 7E5l7 -/ 1n@a587:; /?- uNI65 - Gc7kK -f/ 1VGT5 / 5 K -o/ 21'3< - 369 - G:GH5LM > 1 / MVI6GH7:5uNI -o- Gc7kI > 5 t MV9 - ^ > ! ! !_ /?- uNI65- Gc7h5 >C/ M&'32147:5 6K - 1n@ -?/ KwP a3MV7 #=?G - < PRI >D/ 21'3g< P^; / 1V9H7[5 > Gc7U1 > 7E1 > ;d< - `1VG:G - @VMZKLI'5uNI - I > 5 t MZ9 - / 1V9_K -o/ 21 `3< - 7 - `3;?9:147EI69 -o> P - GH7kK8I65 H - 321VGTI > 5 t MZ9 - !cd5 > G:5 K - 9H;?G:ICK87U17T3MZIC9_KwPR1V<'587:7E1 >6/=- 1 / MZI6GH7:5uNI - < - 7E9:1 > G t - 9H73uNI65%< - @a9E1V5l7 =7E9 - IC7E58KL5LGH; - GH7 /?- KLI65<6;=7 - 9 `5 > ;3MZI69]KwP a3MV76=?G - pg

I > 5 t MZ9 - ;eu ! ! \4\ - 7 > M > 321VG /=- KLIC53C9:MZ3MZG:;f1 / 7:I - K8K - -?> 7]321V9Km1 > MV9 - ;eu ! !$# ! # !

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

: MV7:M > G -?> 2> uNI - KwP - ?B369 - G:GH5LM > < - KwPR1V<'587H7U1 >C/?- < - 7E9:1 > G t - 9H7 YT;A7U16K85 - 361V9 e - 9 - 9 ;Ru ! ! \4\ - GH7k;euNI65l@V14K -?> 7 - C /=- KLK - ;=7E16KL5 - <21 > GkKL1 / 1V<C9 - < - KwP^;=7:I6< - <6I / 21'31 / MZI6Gc7E5 uaI - <61 > GTI >6- 43 - 7:587 -/ 1F@a5l7E; ;Ru ! ! GHMZI6Gk9:;?G - 9c@ - < - 369 -=> <69 -e-?>/ M`3C7 - KwPR1V<'587H7U1 >6/=- < - G '5 / 9:MZ3 6M >6- G !

+ ( &(,*+ -7 #%6$/ () + 6 (,5:0-7* %! & + -5/%- GuNI217E9 - `Ma<6==K - Ge369H;?G -=> 7E;?Gq3C9:; / ;?< - -?> 7q3 - IB@ -?> 7 A7E9 - 69H5g==@ - -?> 7 / 1V9E1 / 7E;=9:5LGH;?G / M& -

GHI6587 qK - <6;=@ - K8MZ363 - -?> 7 'Ma<21VK ;eu ! ! \4\ ! \# ;A7U1CKL5 32149 e - 9 - 9 ^$# _ - 7k369H; / M > 5LGH;e361V9_K -y> MZ9 - ;[ N 1 > 7:;?9:5 - I69 - C'KL1 > MZ9 - ;[6> # % \ H ^ # _ qK - G -=/ M > < 'Ma<6=?K - <6587TC / MZI69c7o7 - 9 - ;A7U1CKL5321V9 e - 9 - 9 ;Ru !& ! \]\ ! \ 3MVI69fKwP a3MV76=?G - pg

I > 5 t MZ9 - qK - 'MB<6==K - 58G:GHI <CI / 1VK / I6K <CI / 61`3 1 / MZI6Gc7E5uNI -q-=> 43 - 7E5l7 - / 1F@a5l7E; ;Ru ! ! 369HMZ3MZG:;q32149T5L;A7 - 7 %-?/ MZKLK85 >6- 7k1V5 > GH5KuNI - 361V9 h1VKLKL1& - 7k;RuaIC58@41VK -=> 7 C'KQPRMV9:<69 - K - 36KLICG 214Gk< - Gd<C;=@ - KLMZ3 H3 - -?> 7:G14G `3B7EMV7:5uNI - G C /=- K8I65 < - e - 9 - 9T3MZI69 pg

I > 5 t MZ9 - qK - `Ma<6==K - ;A7U1CKL5632149 e - 9 - 9 ;eu ! !$# ! # 3MZIC9\KwP B3MV76=?G - ρ′g I > 5 t MZ9 - 369:; / M > 58G:;_32149Z149H@a5LGD^ # _ - 7d1 / 7EI - KLK - -?> 7 > MV9`1VKL58G:;^ _ !%- G_9:;=G:I6Kl7U147:G_M C7 -?> I6Gk3MZIC9 / 21 / I > < -o/=- G `Ma<6==K - G[G:M > 7 / M`32149:;?G > I';?9H5uNI - -=> 7 ! ;Y> 76;?M H

9H5 - K - Gv7E9HMZ5LGv369 - `5 - 9HG `Ma<6==K - G / 5l7E;?GJG:M > 7q;RuaIC58@41VK -=> 7EG / 1V9b9 - 3MVGE1 > 7qGHI69bK - G - G a3M47 6==G - G !j 1 > GbKm1369:147E5 uaI - 5LKY3 - IC7 =7E9 - 5 > 7:;?9 - G:G:1 > 7q< -`/ M&'321V9 - 9 > I `;=9:5uNI - -=> 7 /?- GJ7E9HMZ5LG 'MB<#=?K - GJ1 2>< - <6;A7 - 9 `5 >6- 9TK - uNI - K%< - GT7E9:MV5LGT3 - IC7 =7:9 - K - 36K8I6Gk1V<21V3C7:;y3MZIC9kK - GT;=7E1VKLM >6> 1 - G ! %- Gk9:;=G:I6Kl7U147:GT369:; HG -?> 7E;=G'3MZ9c7 -?> 7JG:I69bK - @VMZKLI - 1 / MZI6Gc7E5 uaI - < - 7:9E1 > G t - 9c7 VT

<6I)G aGH7B= - '5 / 9HMZ36M >6- G Ho/ MVI636K - I69 369HMZ3MZ9H7:5LM >6>C- K7CqKwPR1V<'587H7U1 >C/?- < - 7:9E1 > G t - 9c7 YT

VT =γRPR

jωYT ,

MZt γR- 7 PR

GHM > 7d9 - GH3 -=/ 7E5l@ - -=> 7dK - 9E14363MV9H7< - G / 1V361 / 5l7E;?G / 14KLMZ9H5 uNI - G Cp- 7 Cv

`1VG:GH5uNI - G < -KQPr1V589 - 7kKL1b369 - G:G:58M > GH7U17E5uNI - <21 > GTK - G / M > <65l7E58M > G_GH7E1 > <2149:<< - 9:; t ;=9 -=>6/?- !pMZIC9f< - G / M& `Ma<6587:;?Gk< P - ?B36KLMV587U17E5LM > < - Go9:;?GHI6K87E147EG K - @ZMVKLI - < - 7:9E1 > G t - 9c7 VT

- GH7 > MZ9`1VKL58G:;32149DGE1x@41VK - I69g1VIG? 21VG:G - G t 9H;euNI -?>C/?- G VTbf

-=> KwPR16G -?>6/=- < P - ` - 7EGD7 - 9 `5 uaI - GD<6; 2> 5 - / M& - KL1GHM& - <6Is@ZMZK8I - Z;?M&';=7:9:5 uaI - < - Km1 / 1n@a587:; - 7e< - Ge@VMZKLI - Go;RuNI658@41VK -?> 7EGe1VIG? 21VG:G - G t 9:;RuaI -?>6/=- G< - G `5 / 9:MV36M >6- G VTbf = V +Vbfe +Vbfr MVt'K - G\@VMZKLI - G ;euNI658@41VK -?> 7EGY< - G `5 / 9HMZ36M >C- GYG:M > 7WK85L;=GCK - I69HG14<`5l7:7E1 >6/?- GT1 / MVI6GH7:5uNI - Gk9 - G:3 -?/ 7E58@ - G Ye,r

-?> 21VGHG - G t 9:;RuaI -?>6/=- Gd361V9 Vbfe,r = γRPR/jω Ybfe,r ! ! !R1 ! .- G Z9:1V365 uNI - G< - G ZIC9 - G ! ! ! \9 - 369H;?G -=> 7 -?> 7Y<CM >6/ K - 9E1V3C3MZ9c7 VT

VTbf

-?> t M >6/ 7E5LM >< - KL1 t 9H;euNI -=>6/?- !

%- GY<65 -=> G:58M > G < - G / MZIC36K - I69:G G / 6; g1VG -=> 1 >6>6- ? - c - 7WK - G / 149E1 / 7E;?9H5LGc7E5uNI - GY< - G '5 / 9:MZ3CM >6- GIC7:5LK85LG:; - G<21 > G /=- 7H7 - 1V3C36KL5 / 147:5LM >> I';?9:5 uNI - G:M > 7f<6M >6> ; - Go<21 > GfK - G7E16K - 1VIG? !$# - 7 ! ! %- <C;=@ -@HK8MZ363 - -?> 7 `Ma<21VK ;Ru ! ! \# - GH7 / 14K / ICKL;y1n@ -=/ # %%? # %&%J7 - 9 - G !

[ Ko1V36361V9E1 87 -?> 369 - '5 - 9gKL5 - I uNI - KL1 <65 `;?9 -?>6/=-s-?> 7E9 - K - G#9H;?G:ICK87U17EGM C7 -?> I6G3MZIC9#K - 'MB<#=?K -> MV9`1VKL58G:; - 7 3MZIC9pK - Gp1VIC7E9 - G`Ma<6=?K - G / 21 `3b< - 369 - G:GH5LM > I > 5 t MV9 - > P - GH7 321VG > ; ZK85 - 16K - / 149 - KLK -3 - IC7 147H7 - 5 > <69 - % %&T< -?> 9 - 1V9:<e< - Km1T369:; / 58G:5LM > 9 -?/ - 9 / 6; - 3MVI69pK - G.;=7E1VKLM >6> 1 - G.< - '5 / 9:MZ3CM >6- G 1 / 7:I - K8K - -?> 7JKm19 - 369HMB<CI / 7:5 658KL587:;0G:I69bKL1 <6;A7 - 9 `5 > 147:5LM > < P - S / 1 / 587:; -?> 369 - G:GH5LM > <61 > GbK - G 61VG:G - Gt 9H;euNI -?>C/?- G - GH7 < - KQPRMV9:<69 - < - % % #q % % _< ^ \ _ - 7 KQPR5 >6/?- 9H7:587EIC< -W- ?B3;=9:5 -=> 7E1VK -W- GH7\;A@41VKLI6; - C -?> @a5L9HM >% % q< !

+ 6

P - ?B369 - GHG:5LM > C / MZI69c7v7 - 9 - ;=7U1 6KL5 - 361V9 e - 9 - 9e3MZIC9eKQP a3MV7 6==G - pgI > 5 t MZ9 - ! ;Ru !

! \ ! \4\ - GH7f; 1VK - C0KwP^MZ9H<69 - # -?> X C'KQP - ?B369 - G:G:58M > MC7 -=> I -J-?>/ 1VK / I6KL1 > 7TK -y/ 21'3 < - 369 - GHG:5LM >-?> 43 - 7:587 - / 1n@a587:; ! ;eu ! ! C /?-?/ 5Y369B==G-uNI - e - 9 - 9 > PR5 > 7:9:Ma<6I65l7d3214GoK - Gf1V<'587:7E1 >6/=- G Ye- 7 Yr! 1 / M > 7E9H5 6IC7:5LM > < - G]7 - 9 - Gh< P^MZ9H<69 - G:I63;?9H5 - IC9OC # - GH7_1VG:G - df9H;?<6I65l7 -/ 1V9 <21 > GhKm1 1 - < -

t 9:;RuaI -?>6/=- GTIC7E58KL58G:; - Z; > ;?9E14K - -?> 7_3MZI69[K - G[;=7U14KLM >6> 1 - G G:IC3;=9:5 - I69 - C% d - KLK - 9 - GH7 - 5 >#t ;?9H5 - I69 -C % % #< @41VK - I69 `1?B5`1VK - M C7 -?> I - 3MZI69'< - G '5 / 9HMZ36M >6- G ! P - ?B369 - G:G:58M > C KwP^MZ9:<69 - #-?>X

- GH7y<6M >6/ G:IBS`G:1 > 7 - 3MZI69 - ?B369:5 - 9fK - G - ` - 7:Gd7 - 9'5uNI - Gf<21 > G /=- 7H7 - 1 - < -t 9:;euNI -=>6/?- G9 -=/ M& g1 > <6; - 321V9fKL1 > MZ9 - ! W MZIC7 -t MZ5LG 3MZI69 t 1V589 - < - G - G:IC9 - G CD< - G t 9:;RuaI -?>6/=- Ge36KLI6G21VG:G - G 5LKG - 9E1V5l7v3C9:; t ;=9E1CK - 3MZI69eM C7 -?> 589vI >C- 36K8I6G V9E1 > < - 369H; / 5LG:58M > < P^IC7E58KL58G - 9yK - <C;=@ - KLMZ3C3 - -=> 7 C / MZI69H77 - 9 -f/ M&'36K - 7hMVI - K - <6;=@ - K8MZ363 - -?> 7 `Ma<21VK uNI65 / M > @ - 9 - 36K8I6Gh9E14365L< - -=> 7 -?> 7E9 =?G 21VGHG - Gt 9:;RuaI -?>6/=- G -?> 7 -?> 1 > 7 / M&`3B7 - < - Km1b9 - `1V9uNI - <6M >6> ; -e-?>2> <6I321V9:1Z9E143 - ! ! \ !

MZI63CK - IC9 MZI636K - I69 MZIC36K - I69_< -/ aK85 > <69H5uNI - # / aK85 > <69H5uNI - # V9E1 > <#@VMZKLI - #j 5m12==7:9 -o- ?B7:9:; `5l7E;?G # > # > M > ZI - I69 - ?a7E9H; '587E;=G % % # j 5m12==7:9 -o/=-?> 7E9 - # > # \ M > ZI - I69 /?-?> 7:9 - \ N #

"! p !$# q j 5 -=> G:58M > G_< - G / MZI636K - I69HG !

# # # Y9:M t M > < - IC9 t 9HM > 7U1VK - % \ % \ # N > # NYWMZK8I - 61VG:G - G t 9:;eu ! 3 N #0\ N N # # > 9:;RuaI -?>6/=- < - 9:;=G:M > 1 >6/?- d &%&%&% %%% %&% %&% 1 / 7 - I69T< - uNI21VK8587:; < H # N N H # N N H % \ H % \

"! p !q 1V9:1 / 7:;?9:58GH7:5uNI - GT< - G `5 / 9:MV36M >6- G !

6 I & /( '92 7* % 70 27a 4(% )$ D4$)% * (

102

103

104

−0.2

0

0.2

0.4

0.6

0.8Gerber modal, p

g unif.

Gerber CT, pg unif.

Petite cavitéGerber CT, ρ

g’ unif. (norme)

!

onPp ! qYWMZK8I - 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 > MV9`1VKL58G:; VT/VTbf-=> t M >6/ 7E58M > < - Km1 t 9:;RuaI -?>6/=- G - K8M ><65b`O;=9 -=> 7:G `Ma<6=?K - G_3 aG:5 uNI - G3MZIC9_< - G `5 / 9HMZ36M >C- G # - 7TI >/ MZI636K - I69 / aK85 > <69H5uNI - !

102

103

−0.1

0

0.1

0.2

0.3

0.4Gerber modal, p

g unif.

Gerber CT, pg unif.


g’ unif. (norme)

!

onPp ! qYWMZK8I - 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 > MV9`1VKL58G:; VT/VTbf-=> t M >6/ 7E58M > < - Km1 t 9:;RuaI -?>6/=- G - K8M ><65b`O;=9 -=> 7:G `Ma<6=?K - G_3 aG:5 uNI - G3MZIC9_< - G `5 / 9HMZ36M >C- G # # - 7kI >/ MZI63CK - IC9 / aKL5 > <69:5 uNI - !

+

102

103

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Gerber modal, p

g unif.

Gerber CT, pg unif.


g’ unif. (norme)

!

onPp ! \q YWMZKLI - 1 / MZI6GH7:5uNI - < - 7:9E1 > G t - 9c7 > MZ9 g14KL5LGH; VT/VTbf

-=> t M >6/ 7E5LM > < - Km1 t 9:;euNI -=>6/?- G - KLM ><65 `;?9 -?> 7EG `Ma<6==K - Gk3 aG:5 uaI - Gy3MVI69< - G'5 / 9:MZ3CM >6- G # # 1n@ -?/ I > / MZI636K - I69<6587 4< - Z9E1 > <@ZMZK8I - !

* (),'%$& ( " (z!"%%%$&

*D # % W* % * % * %%

1 > MZ9 - ;[ > # % \ H ^ _CG:I69\KQPR;A7U1VK8M >6> 1 - 32149\9:; / 5L3C9:M / 587:;[< - G `5 / 9HMZ36M >C- G\;A7U1VK8M > G\< - Km1M H9E147:MZ5L9 - 3C9:MZ3MZG - I > 'MB<6==K - 1 / MVI6GH7:5uNI - <6I#G aGH7>= - < - < - IG? '5 / 9:MZ3CM >6- G ; - 7:7 - I69 - 7_9H; /?- 3C7 - I69 / MZIC36KL;=GX32149[I >6-/ 1n@a587:; / aK85 > <69H5uNI - /t ! ! ! - 'MB<#=?K - uNI659 - 3MZG - G:I69XKm1y369:MZ361 147:5LM > < P^M > < - G36Km1 >6- Gy<21 > GeKL1 / 1n@a587E; > ; ZK85 - K - G 'MB< - GoG:I63;?9H5 - I69:G 'MB< - Gy9E1V<C5m1VIG? uNI65Y3MZI69:9:1V5 -=> 7J1n@VMZ5L9yI >6-5 > 6I -?>6/=-g> M > > ; VKL5 - 16K - G:I69JK - Ge9:;=G:I6Kl7U147:GJ< P^;=7E1VKLM >6> 1 - ! P - ?B58GH7 -=>6/?- < -g/=- G 'MB< - G - GH7J<6I - 14It 1V587 uNI - K -/ 21'3#< - <6;=36Km1 /=- -?> 7_< - KL1 - C9E1 >6- <6I `5 / 9HMZ36M >C- ; - 7H7 - I69 > P - Gc7T321VG_I > 5 t MZ9 - /t ! 5 >#t 9:1 - 7iuNI - Km1o321V9c7E5 - 'M 658K - < - KL1 - 69E1 >6- < - G `5 / 9HMZ36M >C- G > PR1e3214G - ?C1 / 7 - -?> 7hK - -<65m1 =A7E9 - uaI - Km1 / 1n@B5l7E; !$c5 > GH5 KL1q369 - G:G:58M > G:I69kKm1 - 69:1 >6- <6I9:; /?- 3B7 - IC9 > P - Gc7d3214GkI > 5 t MZ9 -o- 7 < - 36K8I6G /?-o> P - Gc7361VG - ?C1 / 7 - -?> 7TK -f/ 21'3< - 369 - G:GH5LM > 'M -?> uNI65 - GH7 - G:I69H;o321V9 /=- K8I65 Hi/ 5 / 1V9hG:1 - 69:1 >6-o> PR1q321VG > M > 36KLICG_I > 'MZIC@ - -?> 7k< - 365LGc7EM > 36KL1 > !

1VG bI6GHG -?> ^ __369HMZ3MVG - < - 369 -?> <69 --=> / M&'3C7 - K - G 'MB< - G09E1V<65L1VIG?x321V90I > 7 - 9 -#/ MZ9H9 -?/ 7E5 tCD1V363CKL5uNI - 9< -t 1C?M > bI6Kl7E5L3CKL5 / 147E5l@ - CgKQPr14<`5l7:7E1 >6/?- 1 / MZI6GH7:5uNI - < - 7E9:1 > G t - 9H7d<6;=7 - 9'5 > ; - G - KLM > K -`Ma<6==K - 3 aG:5uNI -J> MZ9`1VK85LG:; ! %-J/ 14K / ICKp< -v/?- 7:7 -J/ MZ9H9 -?/ 7E58M > 9 - 3MZG - GHI69dI > `Ma<6==K - g1476; `147:5uNI -9:5 ZMZI69 - IG?z1VIz<6;?361V9H7 `1V5LGe1V< - 7 2> 1VK - -=> 7 /?- 9H7E1V5 >6- G a3MV7 6==G - G uaIC5X9 - Kg=A@ -=> 7b36KLIC7BAV7e< - KQPR5 >GH7EI65l7E58M > Km1 > 17EI69 - <65LGHG:5832147E5l@ - <6Iu2I65L< -b> P - GH7y361VGf369H5LG -J-?>s/ M&'3C7 -b/ 1V9 - KLK - GHMZ9H7y< - GoM -=/ 7:5 t Gf< -KwPR1VIC7 - I69 C /=- 7 -=> <69HMZ587 ! ;Y> - ` - 7 58K]3C9 -?> < 65 -=>x-=> / M&'3C7 - < - t 1C=M > 1 > 1VK N7:5uNI - K - t 1V5l7 uaI - Km1369 - G:GH5LM > G:I69hK - 9:; /=- 3C7 - I69 > P - GH7_361VG[I > 5 t MZ9 - C / 1VI6G - <6ID369HM K > M > I > 5 t MV9 - <CI / 21'3g< - <6;?3CKm1 H/?- -=> 7y< - Km1 - C9E1 >6- < - KwP^; - 7H7 - I69`1V5LG >6- 7E5 -?> 7 / M`3C7 - uNI - < - t 1C=M >s- '365L9H5uNI - < - KwP - ` - 7<6I / 21'3< - <C;?36KL1 /?- -=> 7 > M > I > 5 t MZ9 - <6I<C5m1V3C9E1& - 9H; /?- 3C7 - I69kG:I69TK -o/ 61`3< - 369 - G:GH5LM > ! P - GH7T1V5 > G:5uNI - <21 > G /?-f/ 61V36587:9 - K - 'MB<#=?K -k- Gc7_9 - 369:58G[< -3t 1C?M > 1 > 1VK N7E5uNI - G:I69hKL1 214G - <OPRI >C-G:MZK8IC7E58M > `Ma<21VK - <6Iu3C9:M 6K = - `1476; `147E5 uNI - uNI65 9 - 3MZG - G:I69fKL1 t MZ9 bI6KL147E58M > 5 > 7E; V9E1VK -v- 7-uNI65 < -G:I69 / 9HM L7d369 -=> < -=>u/ M&'3C7 - Km1 > 17EI69 - <65LGHG:5832147E5l@ - <CIs6I65L< - /?- uNI65 ;=@a587 - < PR1F@VMZ5L9TCDK - 7:9E1V5l7 - 9G:MZICGKm1 t MZ9 - < - 7 - 9 - G / MZ9H9 -?/ 7E5 t G1q3MZGH7 - 9H5LMZ9H5 !<=~ ' - A!/ ! ;( ) ! - y) #%$ ;& '( )

.- G aGH7B= - ;=7:I6<65L; 9 - 369H;?G -=> 7E;uG:IC9DKm1 ZI69 - \!$# - GH7 / M > Gc7E587:I6; < - < - I?'5 / 9HMZ36M >6- G KQPRI >; - 7H7 - IC9X< - 9E1 ZM > ae KwPR1VIC7:9 - 9:; /?- 3B7 - IC9h< - 9:1 ZM > ar / MZI636K8;?G[361V9XI >6-d/ MZKLM >C>6- < - 2I658< -k/ M > 7 -=> I -<21 > GXI >D/ BK85 > <C9 - CvG -=/ 7:5LM >D/ 5L9 / I6KL1V5L9 - < - 9:1 VM > a G:I63;?9H5 - I69h1VI?g9:1 ZM > G ae

- 7 ar

- 7]7E9 =?Gh3C9:M / - < -> N

) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

/=- IG? H / 5! %- G - 69E1 >C- G[< - KQPR; - 7:7 - I69 - 7h<CID9:; /?- 3B7 - IC9hG:M > 7[1 > 5 `; - G]9 - GH3 -=/ 7:58@ - -=> 7_< - G]@a587 - G:G - Gve 5'3MVG:; - - 7 vR

<6I - 1VI / 21'3D< - 3C9 - GHG:58M > 9 - C?I#GHI69hKm1 - 69:1 >6- ! .- 369:M6Kg= -k- Gc7_G:I63C3MZGH;CoG ';=7:9:5 - 1?B5m1VK - > ;n1 > 'MZ5 > G\3MZI69\< - GW9E1V58G:M > G < P - ?B3MZGH; KQPr1 > ZK - θ 1V36321V9:1 L7\1VI0<6; CIC7W<6I / 21V365l7E9 -`1V5LG[Km1qG ';=7E9H5 - 9 - GH7 - GHI6363MZG:; - <61 > G /?- 9H7U145 >6- GkGH5'36KL5 2/ 17E5LM > G !

aae ar

ve vR

`

z

V

onPp \ ! # q j 5LGH3MZGH587:5 t 3MZI69TKwP^;=7E1VKLM >C> 1 -f-=> 3C9 - GHG:58M > G / 6; g10< - 369:5 >6/ 583 - !

8 467()*+ +$-7( (,/ ' : +7-7**+ (86 &+*! & 6$ 9/ ( &+,* /( %! #(8/

%- Gy369:5 >6/ 58321VK - Gd@41V9H5m16K - Gf<C; / 9H58@41 > 7yKwP^;=7E147o7 - 9'Ma< > 1'5uNI - <6Is2I658< - 145L9 G:M > 7eK - G - GuNI - 369:; / ;=< - -=> 7 ! # !j 1 > GWKL1fG:I65l7 - K - 369HM 6Kg= -_- GH7 - ?B369H5';QCo321V9H7:5L9\< - GY7E9HMZ5LGY;RuaI6147E58M > G t M > <61 -=> 7U1VK - GY< - KwPR1 / MZI6G H7:5uNI -J-=> 2I658< - 7 - 9 `MF@a5LG"uNI - IG? 14KL5 > ;n1VGk5 5L5 5L585 C`GE1n@VMZ5L9KwP^;euNI2147:5LM > < - : 1F@a5 - 9 H 7EMB - G /=- KLK - < -/ M > G - 9H@4147:5LM > < - Km1 g1VGHG -0- 7 /=- K8K - < -0/ M > <6I / 7E58M > < - Km1 / 214K - IC9 ! j 1 > G /?- Gy;RuaI6147E58M > G I >s/?- 9c7U1V5 >> M 69 - < P a3MV7 #=?G - G0G:5'36K85 2/ 147E9H5 /?- Gy3 - IC@ -=> 7 A7E9 - 9 - 7 -?> I - G ! ; K8K - GvG:M > 7q3C9:;?G -?> 7E; - G'1VI t IC9 - 7 C - GHI69 - < - K - I69TI6G:1 - ! ;Y> 369 - '5 - 9TK85 - I KwP^; / 149H75 > Gc7U1 > 7U1 > ;v< - 369 - G:GH5LM > ;=7E1 > 7 uNI21VGH5pI > 5 t MZ9 - G:IC9I >C- G -=/ 7E58M > <6I / BK85 > <C9 - 5LK - Gc7 / M > G:58<6;?9H; / M& - <6;=3 -=> <21 > 7y3 - Iu< - G / MaMZ9H<6M >6> ; - G r - 7 θ !'cd5 > G:5 K - G / M&`3MZG:1 > 7 - Gq< - Km1@a5l7 - GHG - ~v G - KLM > r - 7 θ G:M > 7q3 - 7:587 - Gb< - @41 > 70GE1 / M`3MZGE1 > 7 - 1?B5m14K - vz

KL1> M47U147:5LM > vz

> P - Gc7_IC7:5LKL58G:; - 5 / 5uaI - 3MZIC9[< - GPuaI - GH7:5LM > Gh< - GH5'36KL5 2/ 147:5LM > G]ICK87E;=9:5 - I69 - G ! p1V9_1V58KLK - I69HG -=> 9E1V58G:M > < - KwP^5'3MV9H7U1 >6/?- < - G - ` - 7:GqKL5L;=GZCKm1@a5LG / MZGH587:;g< -D/ 5LG:1V5LK8K - -?> 7 Km1@41V9H5m147:5LM > G:IC58@41 > 7bKL1@41V9H5m16K - r < - KL10@a587 - GHG - 32149H7E5 / I6KL1V5L9 - vz

- Gc7d7:9B=?GG:I63;?9H5 - I69 - C`GE10@41V9:5L147E58M > G:I65l@V1 > 7Km10<6589 -?/ 7E58M > z /=- uNI65%5 `36K85uNI --?> 321V9H7:5 / I6KL5 - 9 uNI - K - t 1 / 7 - I69TKL58;MCqKm1J@a5LG / MZGH587E;y< - @VMZKLI - 3 - IC7 =7:9 -y> ; ZKL5 Z;!5 ! 1 / M&'3MZGE1 > 7 - 1?B5m14K - < - KwP^;euNI217E5LM > < -+: 1F@a5 - 9 H 7EMB - GJ369 -=> <)Km1 t MV9 - 1V3C369:M / 6; - GHI658@41 > 7 - ;RuaI6147E58M > < - 1 -=>GH MZ5LG - I658KLK - /Rt ;Ru ! ! > #y1

(

1

c0

∂

∂t− ` ′

v1

r

∂

∂rr∂

∂r

)

vz(r, θ, z) = − 1

ρ0c0

∂

∂zp(r, θ, z) .

\!$# c /?- 7:7 - ;euNI217E5LM > \!$# G:M > 71VGHG:M / 5L; - GTK - GT< - IG? / M > <65l7E58M > GhGHI658@41 > 7 - GTG:I69TKm1 / M&`3MZG:1 > 7 - z < -Km1J@a587 - GHG -

vz(0, θ, z)9 - Gc7 - 2> 5 -o- 7 vz(a, θ, z) = 0 ∀z, θ . \!

5L5 ! 1`G -=/ M > < - ;euNI2147:5LM > 369:58G -v-?> / M&`3B7 -e- GH7 /?- K8K - < -J/ M > <6I / 7E58M > < - Km1 / 21VK - I69 ! j I t 1V587'uNI - KL1@41V9:5L147E58M > < - 7 - `3;?9:147EI69 - τ G=Pr1 >C> ICK -f-=> r = a

- 7h< - Km1eG ';=7E9H5 - <CI#G aGH7B= - KL1e7 - `3;?9:147EIC9 -- Gc7k7:9B==G_3 - I<6;=3 -=> <21 > 7 - < - KQPr1 > ZK - θ - 7kGE1q<6;=9:5l@Z; - G - KLM > z - GH7 > ; ZKL5 - 1CK - < - @41 > 7kGE1q<6;?9H58@V; -

12 @+ $+ ) +

G - K8M > r <21 > G'K - G / MVI / - G`KL5 `5l7 - Ge7 - 9'5uNI - G !p1V9'G:I65l7 - KQPR;RuNI2147E58M > < -/ M > <6I / 7:5LM > < - Km1/ 21VK - I69D^ ;eu ! ! > _%369 -?> <Km1 t MV9 - 1V36369HM / 6; - G:I65l@V1 > 7 -

(

1

c0

∂

∂t− `h

1

r

∂

∂rr∂

∂r

)

τ(r, θ, z) =γ − 1

βγ

jω

c0p(r, θ, z) ,

\! C0KL1uNI - K8K - GHM > 71VG:GHM / 5L; - GK - G / M > <65l7E5LM > G

τ (0, θ, z) 9 - GH7 - 2> 5 -o- 7 τ(a, θ, z) = 0 ∀z, θ . \ ! \ 5L585! - GH7 - C - ?B369H5 - 9oKL1g7E9:MV5LG:5 = - ;RuaI6147E58M > t M > <21 -?> 7E1VK -b/ M > GH5L<6;=9:; - /?- K8K - < -'/ M > G - 9H@4147:5LM > < -

Km1 `1VG:G - jω ρ′ + ρ0div ~v = 0 .

;W>/t 1V58GE1 > 7TI6G:1 - < - Km1b9 - Km17E5LM > < PR;A7U147ρ′ =

γ

c20(p− βτ) ,

\ ! 3MZIC9_;?KL5 `5 >6- 9XKm1b@41V9:5L16K - ρ′ KQPR;RuNI2147E58M > < -y/ M > G - 9c@4147E58M > < - Km1 g14G:G - 3 - IB7d14KLMZ9HG =7E9 - ; / 9H587 -

γ

ρ0c20

∂

∂t(p− βτ ) +

1

r

∂

∂r(rvr) +

1

r

∂

∂θvθ +

∂

∂zvz = 0 .

\ ! > .- GyG:MZK8IC7E58M > Gd< - Gy;euNI2147:5LM > G \!$# C \! > 3 - 9 - 7:7 -?> 7 -?> GHI6587 - < P - ?B369:5 - 9oKwP^;euNI2147:5LM > < - 3C9:M H

321 17E5LM > <6; / 9H58@41 > 7#Km1 369:MV321 147:5LM > < P^I >6- M > < - uNI21VGH5d36KL1 >6- <61 > GDKL1 / 1n@a587:; -?> 9 - '36Km1 Cn1 > 7`K - GuNI21 > 7E5l7E;?G vz τ - 7 p 32149XK - I69HGX@41VK - I69:G 'M -?>6>C- GhGHI69[I >C- G -=/ 7:5LM > <CI / aKL5 > <69 - /?- uaIC5 / M > <6I65l7OCeI >369:M6Kg= - 14IG?KL5 `5l7 - G[<6;=3 -=> <21 > 7I > 5uNI - -=> 7< - KL1 / MaMZ9:<CM >6> ; - 1?C5L1VK - !

, 4 67()*+9-7 $ &+ (,()#*+ P^; / 1V9H7W5 > GH7E1 > 7U1 > ;_< - 369 - G:GH5LM > p ;=7E1 > 7 / M > G:5L<C;?9:; / M& - 7:9B==GY3 - I`<6;=3 -=> <21 > 7]< - G / MaMZ9:<6M >6> ; - G

r, θ <21 > GdK - G / MZI / - GoKL5 `5l7 - Gh@a58G / M47 - 9'5uNI - GdG:I69K - G321V9HMZ5LGkKm147:;?9:1VK - G M > < - uNI214G:5 3CKm1 >6- - 7fKL1/ M&'3MZGE1 > 7 - vz< - Km1s@a587 - G:G - 321V9H7:5 / I6Km1V589 - ;=7E1 > 7 t MZ9H7 - -?> 7D<C;?3 -?> <21 > 7 - < - r <21 > G#K - G / MZI / - G

KL5 `5l7 - Gh@a58GuNI - ICG - G K - G -=/ M > < - 69 - < - KQPR;RuNI2147E58M > \!$# - GH7dG:I6363MZGH;b5 > <C;?3 -?> <21 > 7d< - r /?- uNI65/ M > <6I6587MC; / 9H5L9 - KL1gG:MVKLIC7:5LM > vz<6Is369:M6Kg= - \!$# \! -?> 9:; V5 - 21V9 `M > 5uNI - G:MZICGoKm1 t MZ9 -

1V36369HM / C; - K - t 1 / 7 - I69 ejω t - GH7kM&'5LG /t ! ! \! #

vz(r, θ, z) ≈j

k0ρ0c0

∂

∂zp(r, θ, z)

[

1 − J0(kvr)

J0(kva)

]

, \! N

MZtK -o> M 69 - <OPRM > < - < - <C5 `I6G:58M > kv

- Gc7k<6; 2> 5361V9

kv =1 − j√

2

√

k0/` ′v .

j 1 > GvKm1#G:I65l7 - KQP - ` - 7J<C5LG:GH5L3217E5 t <x1VIG? / MVI / - GJKL5'587 - Gd@a58GuNI - ICG - Gb1VI > 5l@ - 1VIx< - GJ32149:MZ58GeKL1 H7E;=9E1VK - Gb< - KL1 / 1n@a587E; - GH7'369H5LG -=> / M&`3B7 -D-=> 369 -?> 1 > 7'KL1 @V14K - IC9 `M -?>6>6- GHI658@41 > 7 r <6I 369HM Kh<CI/ 21'3< - @a587 - G:G - 5 `3MZG:;o361V9K - G / MZI / - GKL5'587 - G[@a58GuNI - ICG - G 1V< - 7H7U1 > 7k< - Km10G:MZ9c7 - KwP B3MV76=?G -uNI -J/=- 7 - ` - 7f<658G:GH5L32147:5 t KL58;v14IG? / MZI / - GfK85'587 - Gh@a5LGuNI - I6G - G -?> r = a 1 ` -?/ 7 - < - Km1 - `1 > 5 =?9 -

7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

KQPRM > < - 36Km1 >C- ! \ - 7'KwP^M > < - uNI21VGH5T36Km1 >C- !cd5 > G:5 Km1 @41VK - I69 'M -?>C>6- 9 - 7 -?> I - 5 / 5_3MZ9c7 - G:I690KL1G - I6K - t M >C/ 7E58M > [1 − J0(kvr)/J0(kva)]

uNI65%@41V9H5 - 7E9 =?Gk9E1V3658< - -?> 7<21 > GkKm1 / MZI / - KL5'587 - @a5LG"uaI - I6G - K - t 1 / 7 - I69 ∂

∂zp(r, θ, z) t M >6/ 7:5LM > < - r - 7 θ ;A7U1 > 7kG:MZ9c7E5 < - KQPR5 > 7E; V9E1VK - G:I69 r - 7 θ !

- 7H7 - @41VK - I69`M -?>6>6- 321V9c7E5 / I6K85g=?9 - > MV7:; - vz G=P - ?B369:5 - 1VK8MZ9:G / M& - GHI6587

vz ≈ j

k0ρ0c0

∂

∂zp(r, θ, z)

1

πa2

∫ 2π

0

∫ a

0

[

1 − J0(kvr)

J0(kva)

]

r dr dθ

=j

k0ρ0c0

∂

∂zp(r, θ, z)

[

1 − 2

kva

J1(kva)

J0(kva)

]

. \ !

: MV7EM > GT1VI321VG:G:1 - 3MZIC9_K - G - GHMZ5 > G_< - Km1bGHI6587 - uNI - K - G / M&'3MZGE1 > 7 - G vr

- 7 vθG - KLM > r - 7 θ < - KL1@a587 - GHG - 321V9c7E5 / I6Km145L9 - uNI65 G:M > 7 / M >#t M > <6I - Gy1n@ -=/ K - I69@V14K - IC9`M -?>6>6- 1VIuG -?> Gy5 > <65uNI6;J369:; / ;?< - H

-?> 7 / 1V9 > M > 1` -=/ 7E; - Gk5 / 5 321V9_KL1J@B58G / MZG:5l7E; 3 - IC@ -?> 7 A7E9 -y- ?C3C9:5'; - G -?> 369 - `5 =?9 - 1V36369HM9?B5 g147:5LM >32149

vr(r, θ, z) =j

k0ρ0c0

∂

∂rp(r, θ, z) ,

\!

vθ(r, θ, z) =j

k0ρ0c0

1

r

∂

∂θp(r, θ, z) .

\! # % jd- Km1 - t 1C=M > uaI - 3MZI69YMC7 -=> 5L9\KQP - ?B369 - G:G:58M > < - vz

\! N KL1fG:MZK8IC7E58M > τ < - KwP^;euNI2147:5LM > \! / M&`3B7 - 7 -=> I< - G / M > <65l7E5LM > G \ ! \ - GH7k<6M >C> ; - 321V9 ! \!$#

τ(r, θ, z) =γ − 1

βγp(r, θ, z)

[

1 − J0(khr)

J0(kha)

]

, \!$#&#

MZtK -o> M& 69 - < PRM > < - kh

- Gc7k<6; 2> 5%321V9

kh =1 − j√

2

√

k0/`h ,

- 7XG:1y@41VK - I69 'M -=>6>6- G:I69WI >6- G -?/ 7:5LM > G?P^; / 9:5l7 -?> 1V< - 7H7U1 > 7WK - G - GW1V36369HM9?B5 g147:5LM > G uNI -T/?- K8K - G9 - 7 -=> I - G369:; / ;?< - -=> 7kGHI69 vz ;Ru !1\!

τ(r, θ, z) =γ − 1

βγp(r, θ, z)

[

1 − 2

kha

J1(kha)

J0(kha)

]

. \!$#

cd5 > G:5 KL10@41VK - I69 `M -?>6>6- < - KQPR;RuaI6147E58M > \! > 1VI G -?> Go< - Gd;RuaI6147E58M > G \! - 7 \ ! # / M&'3C7 -7 -?> I0< - G\;RuNI2147E58M > G \! \! \! # % - 7 \ ! # / M > <CI6587 <65L9 -?/ 7 - -?> 7CfKQPR;RuNI2147E58M > < - 369:MV321 147:5LM >GHI658@41 > 7 -

[

k20

1 + (γ − 1)Kh

1 −Kv+

1

1 −Kv

(

1

r

∂

∂rr∂

∂r+

1

r2

∂ 2

∂θ2

)

+∂ 2

∂z2

]

p(r, θ, z) = 0 , \!$#

MZtK - G t 1 / 7 - I69HGTKL5L;=GQCqKm1b@a5LG / MZG:5l7E; - 7 CqKL1 / M > <6I / 7E58M > 7 - 9'5uNI - Kv- 7 Kh

G:M > 7k<6M >6> ;?Gk321V9

Kh,v =2

kh,va

J1(kh,va)

J0(kh,va).

\!$#0\

) + + 7 U

PR; / 1V9c7`< - 369 - GHG:5LM > p <6;=3 -=> <21 > 7`3 - I< - G / MBMZ9H<6M >6> ; - G r, θ G - G'<6;?9H58@Z; - G`321V9H7:5 - KLK - GbG - KLM > /?- G/ MaMZ9:<CM >6> ; - Gf<21 > GfKwP^;euNI217E5LM > \ ! # G:M > 7f< - Gd7 - 9 - Gd3 - 7E5l7EG'uaIC5\3 - 9 - 7H7 -?> 7d< PR; / 9H5L9 - 1 −Kv ≈ 1-?>/t 1 / 7 - I69 !6cd5 > G:5%KQPR;RuNI2147E58M > \ ! # 3 - IC7 =7E9 - ; / 9H587 - G:MZI6GkKm1 t MV9 - G:5'36K85 ; -[

1

r

∂

∂rr∂

∂r+

1

r2

∂ 2

∂θ2 +∂ 2

∂z2 + k2

]

p(r, θ, z) = 0 , \! #

1n@ -?/k2 = k2

0

1 + (γ − 1)Kh

1 −Kv

≈ k20

[

1 +1 − j√

2

2

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

. \!$# >

cd5 > G:5 K - 'MZIC@ - -?> 7_<6Ig6I65L< - <21 > GXKm1 / 1n@a587E; - Gc7_<6; / 9:5l7h321V9]KwP^;euNI217E5LM > < - 3C9:MZ321 147E58M > \!$# <6M > 7qK - GqG:MVKLIC7:5LM > Gv<6MZ5l@ -?> 70G:147E58G t 1V589 - K - G / M > <65l7E58M > Gq1VI? t 9HM > 7E5g==9 - G -?> z = 0- 7 z = ` 369 -=> 1 > 7-?> / M&'3C7 - KL1 t MV9 - <CI / 21'3 < - @a5l7 - GHG - < - G - 69E1 >6- G0< - `5 / 9HMZ36M >C- G 9:1V5LGHM > <OPA7E9 - < - G

MZ3;?9E17 - IC9:G -?> r - 7 θ < - KwP^;euNI2147:5LM > \!$# 1V5 > GH5uNI - K - G - ` - 7:G_< - G / MZI / - GkK85'587 - GW@B58G / MV7 - 9'5uNI - GG:I69 /=- GT321V9HMZ5LG !

< "( # +[ - `%) !/ ! ) + - . )6$ ( . ) -%._ $ '( ) ) 35$ ) ) !

1 / MZK8M >6>6- < - 2I658< -/ M > 7 -=> I - <21 > G]Km1 / 1n@a587:; - GH7XK - G:5g= - < PRI >g/ 21'3#1 / MZI6Gc7E5uNI - 21V9 `M > 5uNI -/ 9H;?;0321V9oKL1g@B5 69E17E5LM > < - Km1 - 69:1 >6- <6I '5 / 9:MZ3CM >6- ; - 7H7 - I69oG:5l7EI6; -'-=> z = 0 !7\!$# ! %- G

321V9HMZ5LGXG:587:I6; - G -?> z = 0- 7 z = ` 369:;=G -=> 7 -?> 7k9 - G:3 -?/ 7:58@ - -?> 7TK - G_1V<'587H7U1 >6/=- G Ye

- 7 YruNI65 G:M > 7hK - G

1V<'587H7U1 >6/=- GX1 / MVI6GH7:5uNI - G[< - G '5 / 9:MZ3 6M >6- GC / 5L9 / I65l7[MZIB@ - 9c7 ! .- G - ` - 7:Gh< - <65LGHG:5L36147E58M > @a58GuNI - I? - 77 - 9'5uNI - GXGHM > 7[369H5LG -?>D/ M&'3C7 - 321V9]K -T/ 6MZ5.?'< P^I >6-'t M >6/ 7:5LM > < - e 9 -=-?> uaIC5G:147E58G t 1V5l7WG:I69 /?- GX321V9HMZ5LGz = 0

- 7 z = ` CqI >6-o/ M > <6587:5LM > 1V<21V3C7:; - / M > <65l7E58M > '5b?a7 - <61 > G_Km1auaI - KLK - Km1 t M >6/ 7:5LM >#- 7TGE1q<6;=9:58@V; -G:M > 7kKL58; - G_G:I69TK - G_321V9HMZ5LG_32149_KwP^5'3;=<21 >6/=- @a5LG / MV7 - 9'5uNI - < - 32149:MZ5 ! 1q369HMZ321Z147E58M > < - GTM > < - Gk<21 > GTK -o/ aKL5 > <69 -o- GH7k9:; Z5 - 361V9TKwP^;euNI2147:5LM > \! #

(∆ + k2)p(~r) = 0 , ∀~r ∈ D . \!$# N

c /?- 7:7 - ;euNI217E5LM > < - 3C9:MZ321 147E58M > G:M > 71VGHG:M / 58; - GTK - G / M > <6587:5LM > GT1VIG? t 9HM > 7E5g==9 - GkGHI658@41 > 7 - G

ρ0c0vz = ρ0c0ve − ζz0p ,3MZI69 r ∈ (0, ae) , θ ∈ (0, 2π) , z = 0 ,

\! # Z1 ρ0c0vz = −ρ0c0vR + ζz`p ,

3MVI69 r ∈ (0, ar) , θ ∈ (0, 2π) , z = ` , \! #

ρ0c0vz = −ζz0p ,3MVI69 r ∈ (ae, a) , θ ∈ (0, 2π) , z = 0 ,

\! # / ρ0c0vz = ζz`p ,

3MZI69 r ∈ (ar, a) , θ ∈ (0, 2π) , z = ` , \! # V<

MZt ζz0

- 7 ζz`G:M > 7]K - GW1V<'587:7E1 >6/=- GY1 / MZI6Gc7E5uNI - GWG:3; / 5 uNI - G]< - G]361V9:MZ58G -?> z = 0

- 7 z = ` 7E9E14<6I65LG:1 > 7K - G - ` - 7EGk< - G / MZI / - GK85'587 - GX7 - 9 `MF@a5LG"uNI - I6G - GG:I69TK -o/ 61`31 / MZI6GH7:5uNI - <21 > GkKL1 / 1n@a587:; 1 >6>6- ? - - 7

vr = 0 3MZI69 r = a , θ ∈ (0, 2π) , z ∈ (0, `) , \!$# - / 1V9%K - G - ` - 7:G.< - G / MZI / - GKL5 `5l7 - G7 - 9'MF@B58GuNI - I6G - G Km17E;?9:1VK - G G:M > 7p369H5LG - G -?>e/ M&'3C7 - <61 > G.K -Y> M 69 -

< P^M > < -y/ M`36K - ? - k !.- GT1V<'587H7U1 >6/=- G ζz0

- 7 ζz`9 - GH7E1 > 7T3 - 7E5l7 - G / M`3C7 - 7 -=> I< - GT;euNI2147:5LM > G \! \!$#0\ /=- G / M > <65 H7E58M > G_1VI? t 9:M > 7:5g=?9 - G \ ! # G?P^; / 9H58@ -?> 7 1 >6>6- ? -

∂ p

∂n= −∂ p

∂z=

j k0ρ0c01 −Kv

vz ≈j k0ρ0c01 −Kv

ve − j k0ζz0 p \!$# 1

7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

3MZI69 r ∈ (0, ae) θ ∈ (0, 2π) z = 0 ∂ p

∂n=∂ p

∂z= − j k0ρ0c0

1 −Kv

vz ≈ j k0ρ0c01 −Kv

vR − j k0ζz` p \!$#

3MZI69 r ∈ (0, ar) θ ∈ (0, 2π) z = ` ∂ p

∂n= −∂ p

∂z= −j k0ζz0 p

\!$# / 3MZI69 r ∈ (ae, a) θ ∈ (0, 2π) z = 0

∂ p

∂n=∂ p

∂z= −j k0ζz` p ,

\!$# < 3MZI69 r ∈ (ar, a) θ ∈ (0, 2π) z = `

∂ p

∂n=∂ p

∂r= 0 ,

\!$# - 3MZI69 r = a θ ∈ (0, 2π) z ∈ (0, `) !

pMZIC9]I >6- M > < - 21V9 `M > 5uNI - K -T/ 21'3`< - 3C9 - GHG:58M > 1 / MZI6Gc7E5 uaI - p(~r) <21 > G]I > <6M&`1V5 >C- D K85'587E;32149_I >6- G:I69 t 1 /?-t - 9 `; - S 3 - IC7 =7:9 - ; / 9:5l7TG:MZI6GTKL1 t MV9 - 5 > 7E; Z9E14K - GHI658@41 > 7 - ^ > ! !_ K - t 1 / 7 - IC9ej ωt - GH7M&`58G - 7kK - @ -=/ 7 - I69 ~r - Gc79 - 369H;?G -?> 7E;v5 / 5O321V9_K - G_7E9:MV5LG / MaMZ9H<6M >6> ; - G r θ - 7 z

p(~r) =

∫∫∫

DG(~r, ~r0)f(~r0) dD0 +

∫∫

S

[

G(~r, ~r0)∂

∂n0

p(~r0) − p(~r0)∂

∂n0

G(~r, ~r0)

]

dS0 . \!%

1 t M >6/ 7E58M > f(~r0)9 - 369:;=G -=> 7 - KwPR1 / 7:5LM > < - G[G:MVI69 /=- Gh9H;?321V9c7E5 - G CvKwP^5 > 7:;?9:5 - I69W<6Ig<6M&`1V5 >6- D <61 > GK -h/ 1VGY3C9:;?G -?> 7 f(~r0) = 0 ! 1 t M >6/ 7:5LM > G(~r, ~r0)

- GH7WI >6-Xt M >6/ 7E5LM > < - e 9 -?-=> GE147:5LG t 1V5LG:1 > 7CfKwP^;euNI217E5LM >(∆ + k2)G(~r, ~r0) = −δ(~r, ~r0) <61 > G D \ ! #

MZt δ(~r, ~r0) - Gc7bKL1<658GH7:9:5 CIC7E58M > < -Dj 589E1 / - 7 C< - G / M > <C587E58M > Ge1VIG?zK85'587 - Go13C9:5LMV9:5 uNI - K / M > uaI - GGHI69KL1qG:I69 t 1 /?- S ! 10<6;?9H58@Z; - ∂

∂n0

9 - 369H;?G -?> 7 - Km1q<6;?9H58@V; - G:I65l@V1 > 7kKL1 > MZ9`1VK - K8M / 1VK - C'KL10GHI69 t 1 /?- SMZ9H5 -=> 7:; - @ - 9:GkKwP - ?a7E;=9:5 - I69_<6I<6M&`1V5 >6- ! - 7H7 - 9 - KL147E58M > Z; > ;=9E1VK - 58G:GHI - <6I 7 6;=MZ9B= - < - e 9 -?-?> ^ > ! ! _k143636KL5 uNI6; - CzKL1 / 1F@a5l7E; < -/ MZI636KL1 - 3 - 9 - 7W<OP - ?C3C9:5 - 9 1VG:G - dTG:5 `3CK - -?> 7 V9>X /=- 14I / CMZ5b?0<OPRI >C- t M >6/ 7:5LM > < - e 9 -?-=> 1V<21V3C7:; - KL103C9 - GHG:58M > 1 / MZI6Gc7E5 uaI -e-=> 7:MZIC7T3MZ5 > 7T< - KL1 / 1n@a587E; !;Y> t 1V5LG:1 > 7[ICGE1 - < - G / M > <C587E58M > G]1VIG? t 9:M > 7:5g==9 - G \ ! # 1?q - 3MZI69[K -/ 61`3`< - 369 - GHG:5LM > K -/ CMZ5b?< P^I >6-t M >6/ 7E58M > < - e 9 -?-=> G:147E58G t 1V58GE1 > 7TG:IC9kKQP -=> G - 6K - < - Gk321V9HMZ5LGh< - KL1 / 1n@a587E;y14IG? / M > <6587:5LM > G

∂

∂nG(~r, ~r0) = −j k0ζz0G(~r, ~r0) ,

3MZI69 r ∈ (0, a) , θ ∈ (0, 2π) , z = 0 , \! V1

∂

∂nG(~r, ~r0) = −j k0ζz`G(~r, ~r0) ,

3MZIC9 r ∈ (0, a) , θ ∈ (0, 2π) , z = ` , \!

∂

∂nG(~r, ~r0) = 0 , 3MVI69 r = a , θ ∈ (0, 2π) , z ∈ (0, `) ,

\ ! / 3 - 9 - 7<OP - ?C3C9:5 - 9TKQPR5 > 7:; Z9:1VK - < - G:I69 t 1 /?- 1V3C321V9E145LG:G:1 > 7k<21 > GkKm1q9 - Km147:5LM > \ ! % uNI65.9 - 369:;=G -=> 7 - KL1369 - GHG:5LM > 1 / MZI6Gc7E5uNI -y-=> 7:MZIC7k3MV5 > 7_< - Km1 / 1F@a5l7E; < - KL1 g1 > 5g==9 - GHI658@41 > 7 -

p(~r) =j k0ρ0c01 −Kv

[∫∫

Se

ve(~r0)G(~r, ~r0) dS0 +

∫∫

Sr

vR(~r0)G(~r, ~r0) dS0

]

, \!

12 R) @+ $+ @+ K) + X) +) 7 6

Se

- 7 Sr;=7E1 > 7dK - GTG:I69 t 1 /?- G9 - GH3 -=/ 7E5l@ - Gf< - Km1 - 69:1 >6- < - KwP^; - 7H7 - IC9 - 7<6I9H; /=- 3C7 - I69 K - G - ` - 7:G< - G / MZI / - G7 - 9'M @a58GuNI - ICG - GY;A7U1 > 7p3MV9H7E;=Gp361V9pK -it 1 / 7 - I69 Kv

uNI65a5 > 7 - 9H@a5 -?> 7 <61 > GKwP^5'3;=<21 >6/=- <CI`58KL5 - I - 7[361V9XKL1 t M >6/ 7:5LM > < - e 9 -=-?>D- KLK -hH - <61 > G]K -k> M& 69 - <OPRM > < - k 3MZIC9XK - G - ` - 7EGXKm147:;?9:1VIG?- 7[<21 > GXK - G[1V<'587H7U1 >6/=- G]G:3; / 5 uNI - G ζz0

- 7 ζz`1VI > 5l@ - 1VID< - G - 69:1 >6- GX< - G `5 / 9HMZ36M >C- G - 7hK - I69

3;=9:5836;?9H5 - !4c > MV7 - 9WKm1oG ';=7:9:5 - ; - 7:7 - I69 H 9:; /?- 3C7 - I69 <21 > G]KwP - ?B369 - G:GH5LM > <6I / 21'3`< - 369 - GHG:5LM >1 / MVI6GH7:5uNI - /=-e/ 61`3<6;=3 -=> < < -e/=- GT@a587 - G:G - G3uaI - KuNI P -?> G:MZ5l7_K - I69 / 14I6G - !<c| #/$ ) 0="! ) ? "!/+ - A! ) 35$ ) ) ! #%$ "& '( ) eD

)T $ #/$ #/$ ) 1 t M >6/ 7E58M > < - e 9 -?-=>g- GH7XG:MZK8IC7E58M > <6ID369HM 6Kg= - <C; / 9H587X321V9]KwP^;euNI2147:5LM > \! # - 7h< - G / M > <C587E58M > G1VIG? t 9:M > 7E5 =?9 - G \! V1 /

(∆~w +∂ 2

∂z2 + k2)G(~w, z; ~w0, z0) = −δ(~w − ~w0)δ(z − z0)<61 > G D ,

\! \ 1 ∂

∂n0G(~w, z; ~w0, z0) = − ∂

∂z0G(~w, z; ~w0, z0) = −j k0ζz0G(~w, z; ~w0, z0)

3MZIC9 z = 0 , \! \

∂

∂n0G(~w, z; ~w0, z0) =

∂

∂z0G(~w, z; ~w0, z0) = −j k0ζz`G(~w, z; ~w0, z0)

3MZI69 z = ` , \! \ /

∂

∂n0G(~w, z; ~w0, z0) =

∂

∂r0G(~w, z; ~w0, z0) = 0 3MVI69 r = a ,

\ ! \V< 1n@ -?/ ∆~w =

1

r

∂

∂r

(

r∂

∂r

)

+1

r2

∂ 2

∂θ2!

1DGHMZKLIB7E5LM > < -q/?- 369HM 6Kg= -J- GH7y9 -?/ - 9 / 6; - GHMZI6GyKm1 t MV9 - < P^I > <C;=@ - KLMZ3C3 - -=> 7^ > !$#&! \ / _G:I69TKL1 21VG - < - G t M >6/ 7:5LM > G_3C9:MZ369 - GTMZ9c7 6M > MZ9'; - G ψt

µν(r, θ)<6I / aKL5 > <69 -

G(~r, ~r0) =∑

µ,ν=0,1...

t=c,s

gµν(z, z0)ψtµν(r0, θ0)ψ

tµν(r, θ) .

\!

.- G t M >6/ 7:5LM > G_369:MZ3C9 - GTGHM > 7dGHMZKLIC7:5LM > Gh<6I369:M CKg= - 1VIG?g@41VK - I69:GT369HMZ369 - G(∆~w + k2

wµν)ψtµν(r, θ) = 0 ∀r ∈ (0, a)

- 7 ∀θ ∈ (0, 2π) , \! > 1

∂

∂nψt

µν(r, θ) = 0-?>

r = a- 7 ∀θ ∈ (0, 2π) ;

\! > - K8K - G[G?P^; / 9H58@ -?> 7 t = c, s 9 - GH3 -=/ 7:58@ - -=> 7d3MZI69 cosµθ

- 7 sinµθ ψc

µν(r, θ) =1

aµbµν

cos µθJµ(kwµνr) , \! N 1

ψsµν(r, θ) =

1

aµbµν

sin µθJµ(kwµνr) , \! N

- 7_K - G[@41VK - I69HGh369HMZ369 - GT1VG:GHM / 5L; - G kwµνG:M > 7_5 `3MZGH; - G[321V9_Km1 / M > <C587E58M > 1VI? t 9HM > 7E5g==9 - G_<6I#369:M6Kg= -

\! > / M > <6587:5LM > G_< -D:- I g1 >C> kwµν = γµν/a ,

\!

7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

MZt γµν

ν = 0, 1, 2, 3... 9 - 3C9:;?G -?> 7 - K - ν + 1 5g= - d?;=9:M`< - KL1'<6;=9:5l@Z; - 369 - `5 =?9 - < - Km1 t M >C/ 7E58M > < -

- G:G - K Jµ GHMZ587 J ′

µ(γµν) = 0 ! - 3C9:M 6K = - 369H;?G -=> 7 - <CM >6/ I >6- <6; Z; > ;=9 - G /?-=>6/?- < PRMV9:<69 - !%- G / M - S / 5 -?> 7EGk< -y> MZ9 g14KL5LG:147E58M > aµ

- 7 bµνG:M > 7k<6M >C> ;?Gk321V9

a2µ =

∫ 2π

0

cos2 µθ dθ =

∫ 2π

0

sin2 µθ dθ = (1 + δµ0)π , \!

- 7b2µν =

∫ a

0

J2µ(kwµνr)r dr ,

GHMZ587 -?> IC7E58KL58GE1 > 7_KL1b9 - KL147E58M > ^ ! \! _%36I658G_K - G_9 - KL147E58M > G ^ ! \ N !$# - 7 _%< - KL109H; t ;=9 -?>C/?- ^$# _

b200 =a2

2

b2µν =a2

2

(

1 − µ2

γ2µν

)

J2µ(γµν)

3MZI69 (µ, ν) 6= (0, 0) .

\!&%

%- <6;=@ - KLMV363 - -=> 7 \! < - KL1 t M >6/ 7:5LM > < - e 9 -=-?> GE147:5LG t 1V587 65 -=> 145 > GH57CqKm1 / M > <6587:5LM >g-?>/t 9:M >GH7:5g=?9 - G \! \Z< ! - Gc7 - C - ?C3C9:5 - 9JK - G / M - S / 5 -?> 7EGq<CI)<6;=@ - KLMV363 - -=> 7 gµν(z, z0)

< - g1 > 5g=?9 - C /=-uNI - KwP - ?B369 - GHG:5LM > < - G G:147E58G t 1VGHG - CDKQPR;RuaI6147E58M > < - GoM > < - G \! \ 1 - 7e1VI? / M > <65l7E5LM > G -=> t 9HM > 7E5 =?9 - G \! \ - 7 / ! [ K @B5 -?> 7T7EMZIB7 / 1VK / I6K t 14587 1 >6>6- ? -D[ [ ! #

gµν(z, z0) = −cos(kzµνz + ϕ0µν) cos(kzµν(z0 − `) − ϕ`µν)

kzµν sin(kzµν`+ ϕ0µν + ϕ`µν),

\! #F1 GH5 z < z0 - 7k321V9

gµν(z, z0) = −cos(kzµνz0 + ϕ0µν) cos(kzµν(z − `) − ϕ`µν)


\! # GH5 z > z0 MVt ϕ0m ϕ`m

G:M > 7k<6M >6> ;?Gk321V9 ;eu ! [ ! N

tanϕ0,`µν = − j k0ζz0,`µν

kzµν.

\!

<# < ;( # +[ - `%) !/ = + , - .%- / 21'3< - 369 - G:GH5LM > 1 / MZI6Gc7E5 uaI - <21 > GgKm1 / 1n@a587:; - GH7#M B7 -?> I -=> 9 - 3MZ9c7U1 > 7`Km1 t M >6/ 7:5LM > < -e 9 -=-?>s- ?B369H5'; - <21 > GoK - 32149E1Z9:1V3 - 369:; / ;?< -?> 7 \! <21 > GoKwP^;euNI2147:5LM > \! ! M&`3B7 - 7 -?> Iz< -

KL1 t MZ9 - < -/?- 7:7 - t M >6/ 7E58M > < - e 9 -=-?> Km1s369 - G:GH5LM > p(r, θ, z) 1V3C321V9E1 L7'1VK8MZ9:G0G:MVI6GgKL1 t MZ9 - <OPRI ><6;A@ - KLMZ363 - -?> 7G:I69TK - G t M >6/ 7E58M > G_369HMZ369 - G ψt

µν(r, θ)

p(r, θ, z) =j k0ρ0c01 −Kv

∑

µ,ν=0,1...

t=c,s

[

gµν(z, 0) < ve|ψtµν >e +gµν(z, `) < vR|ψt

µν >r

]

ψtµν(r, θ) ,

\ !

MZt Km1 > MV7E147E58M > < ve|ψtµν >e

<6;?GH5 >6- KwP^5 > 7:; Z9:1VK - GHI69 Se<6I 369HMB<CI6587 ve(r, θ)ψ

tµν(r, θ)

- 7MZt <vR|ψt

µν >r<6;=G:5 >6- KwP^5 > 7E; Z9E1VK - GHI69 Sr

<CI369HMB<CI6587 vR(r, θ)ψtµν(r, θ)

!

) + 7

1qGHMZKLIC7:5LM > 3C9 -?> <<6M >6/ Km1 t MZ9 -

p(r, θ, z) =∑

µ,ν,t

ptµν(z)ψ

tµν(r, θ) ,

\!]\ 1n@ -?/

ptµν(z) =

j k0ρ0c01 −Kv

[


µν >r

]

. \!

j 1 > G`Km1uG:I65l7 - < - KQP - ?B3MZGH; K -/ 21'3< - @a587 - GHG - ve

/ 21 `3 @B5 69E17EMZ589 - G:MVI69 /=- - GH7gG:I6363MZGH;5 > <C;?3 -?> <21 > 7e< - KwPR1 > VK - θ - 7J361V9yG:I65l7 - / M&`3B7 - 7 -=> Iz< - Km1#G ';=7E9H5 - < - 9:;A@ZMZK8IC7E58M > < - KL1 / 1n@a587E;- 7g< - /?- KLK - <6I `5 / 9:MV36M >6- 9:; /?- 3C7 - I69 K - / 21 `3< - 369 - G:GH5LM >- GH7#K8I65 H - 5 > <6;=3 -=> <21 > 7D< - θ - 7 / M > G:; / IC7:58@ - -=> 7#Km1 @a587 - G:G - vR; 1VK - -?> 7 ! -=/ 5 - Gc7 / MZ9:9HM MV9:; -?> / 1VK / I6KL1 > 7`KQPR5 > 7E; V9E1VK - ve,R<6;?GH5 > 1 > 7T5 > <65b`O;=9 - -?> 7 ve

MZI vR

< ve,R|ψtµν >e,r=

∫ ae,r

0

r0 dr0ve,R(r0)

∫ 2π

0

dθ0ψtµν(r0, θ0) ;

\! > < PR1V369B==GoKwP - ?B369 - G:GH5LM > \! N < - G ψt

µν KwP^5 > 7E; Z9E14K - G:I69 θ0 - Gc7 > I6K8K - 3MZI69 µ 6= 0 - 7y< - G:IC9 / 9HM L7 - KLK -- Gc7; 1VK - -?> 7 > I6KLK - 3MVI69 t = s µ = 0 1 >6>6- ? - [ [ ! !cd5 > G:5 KL10GHMZKLIB7E5LM > \!]\ < - @a5 -=> 7

p(r, z) =∑

ν

pc0ν(z)ψ

c0ν(r) .

\! N jd- `1 > 5g==9 - C / 1VK / I6K - 9vK - G pc

0ν(z)<6M > 7JKwP - ?B369 - G:GH5LM >)- GH70<6M >6> ; - 321V9bKQPR;RuaI6147E58M > \! 5LK - Gc7> ; /=- GHGE1V589 - < P - ?B369H5 - 9vKm1@a5l7 - GHG - vRuNI65X<6;?3 -?> < - KLK - - <6I / 61`3 1 / MZICGH7E5 uNI - p(r, z) ! pMVI69/?- t 1V5L9 - 321V9 - ?a7 -=> G:58M > < - KL19 - Km17E5LM > SrvR = −Yrp(`)

9 - KL5L1 > 7JKQPr1V< `5l7:7U1 >6/?- Yr<6I '5 / 9:MZ3 6M >6-

9:; /?- 3B7 - IC9_1VIG?`369 - G:GH5LM > G - 7X@B5l7 - G:G - G_1 / MZI6GH7:5uNI - GhKLMZ9HGuNI - K -d/ 61`3g1 / MVI6GH7:5uNI -f- GH7h5 > <6;=3 -=> <21 > 7_< -r Km1b369 - G:GH5LM >- Gc7k; / 9H587 -

SrvR(r) = −Yr(r)p(r, `) , \!

MZtKwPR1V<'587H7U1 >6/=- Yr(r) 9 - 369:;=G -=> 7E1 > 7dKQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - KLM / 1VK85LGH; -ot M >6/ 7:5LM > < - r <6I9:; /?- 3 H7 - I69 - Gc7J; / 9H587 - <21 > GvKL1GHI6587 - GHMZI6GvKm1 t MZ9 - G:I65l@41 > 7 - -=> t M >6/ 7E58M > < - G:1@41VK - I69 `M -?>6>6- G:I69eKm1G:I69 t 1 /?- <6I9:; /?- 3C7 - I69 < Yr >r

Yr(r) =< Yr >r +δYr(r) , \!

δYr(r);=7E1 > 7k7 - KKuNI - < δYr >r= 0 !

cd5 > G:5 K - t 1 / 7 - I69T<6;?3 -?> <61 > 7d< - vR< - Gk9 - Km147:5LM > G \! - 7 \! N G=PR; / 9H587

−Sr < vR|ψc0ν >r=< Yrp(`)|ψc

0ν >r

\! \ % - 7 321V9T9 - 3MV9H7T< - KwP - ?B369 - GHG:5LM > \! N 3MVI69 z = `

−Sr < vR|ψc0ν >r=

∑

ν′

pc0ν′(`) < (< Yr >r +δYr)ψ

c0ν′ |ψc

0ν >r , \! \#

G:MZ5l7 < Yr >r;=7E1 > 7T5 > <C;?3 -?> <21 > 7< - r


∑

ν′

pc0ν′(`)

[

< Yr >r< ψc0ν′ |ψc

0ν >r + < δYrψc0ν′ |ψc

0ν >r

]

. \! \

7 ) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

j I t 1V587 uNI - < δYrψc0ν′ |ψc

0ν >r3 - IC7 =7:9 - ; / 9:587T5 / 5%G -?> GH5 6K - -?> 7k; 1VK)C < δYrψ

c00|ψc

00 >r

K - 7 - 9 -M&'5LGW<6M >6> 1 > 7XI >6-d/ M > 7E9H5 6IC7:5LM > G:IC363MVG:; -> ; VKL5 - 16K - C p - 7 ψc

00;=7U1 > 7[5 > <C;?3 -?> <21 > 7X< - r 5LK23 - IB7

A7E9 - 9 - '36Km1 / ;o321V9 < δYr >r< ψc00|ψc

00 >r= 0 < P^MZt−Sr < vR|ψc

0ν >r=∑

ν′

pc0ν′(`) < Yr >r< ψc

0ν′ |ψc0ν >r .

\! \ %- 9 - 3MZ9H7]< -/?- 7:7 -- ?B369 - G:GH5LM > <21 > GXKwP - ?B369 - G:GH5LM > \! N < - Km1o369 - G:GH5LM > / M`3C7 - 7 -?> ID< - Km1y9 - Km147:5LM > \! / M > <6IC587fC -?> ; / 9:5l@V1 > 7 < Yr >r= Yr

p(r, z) =j k0ρ0c01 −Kv

∑

ν

[

g0ν(z, 0) < ve|ψc0ν >e

− Yr

Srg0ν(z, `)

∑

ν′

pc0ν′(`) < ψc

0ν′ |ψc0ν >r

]

ψc0ν(r) ,

\ ! \4\

MZt K - G'7 - 9 - G < ψc0ν′ |ψc

0ν >r<6;=G:5 >6-?> 7DK - G05 > 7:; Z9:1VK - G'GHI69'Km1sG:I69 t 1 /?- Sr

< - Gg3C9:Ma<6I6587:G ψc0νψ

c0ν′ <6M >6> ; - G -?> 1 >6>6- ? - [ 361V9JK - G - ?B369 - G:GH5LM > G [ ! ##F1 ! %- Gv7 - 9 - G - ?B369H5`1 > 7b< - G / MZI636KL1 - G -?> 7E9 -

<65b`O;=9 -=> 7:G `Ma< - G ν 6= ν ′ GHM > 7o7E9B==Gy3 - 7E587:Gf< - @V1 > 7eK - Gy1VIC7:9 - G - 7e3 - IC@ -=> 7v<CM >6/ =7:9 -q> ; ZKL5 Z;?G K - GMZ9H<69 - GT< - Z9E1 > < - I69TGHM > 7<CM >6> ;=G<21 > GTK - 321V9:1Z9:1V3 -M[ ! ! 1'369 - GHG:5LM > 1 / MVI6GH7:5uNI - p(r, z) - GH7f<CM >6/ 9 - 7 -?> I - GHMZI6GdKL1 t MV9 - -?> t 1 / 7:MZ9:58GE1 > 7KwP^5 > @ - 9:G - < - Km1GHI69 t 1 /=- S = πa2

p(r, z) =j k0

1 −Kv

ρ0c0S

∑

ν

[g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p

c0ν(`)]ψ

c0ν(r) ,

\ ! \

MZt Yr0ν = Yra2

a2r

< ψc0ν |ψc

0ν >r!

1 / M&'321V9:1V5LGHM > < - G - ?B369 - G:G:58M > G \ ! N / M&'3C7 - 7 -?> I< - \! - 7 \! \ 3 - 9 - 7_< - <6;A7 - 9 `5 >6- 9K - G / M - S / 5 -?> 7EG pc

0ν(z)

pc0ν(z) =

j k0

1 −Kv

ρ0c0S


c0ν(`)] ,

\ ! \ > GHMZ587 3MVI69 z = `

pc0ν(`) =

j k0

1 −Kv

ρ0c0S

g0ν(`, 0)S < ve|ψc0ν >e

1 +j k0

1 −Kv

ρ0c0S

Yr0νg0ν(`, `)

, \! \ N

/=- uNI65%<6M >6>6- 3MVI69_KwP - ?B369 - G:GH5LM > < - pc0ν(z)

pc0ν(z) =

j k0

1 −Kv

ρ0c0S

[

g0ν(z, 0)S < ve|ψc0ν >e

− Yr0νg0ν(z, `)

j k0ρ0c01 −Kv

g0ν(`, 0) < ve|ψc0ν >e

1 +j k0

1 −Kv

ρ0c0S

Yr0νg0ν(`, `)

]

; \! \

7C7

K - 3C9:Ma<6I6587TG / 1VKL1V5L9 - < ve|ψc0ν >

- GH7 / 14K / ICKL;f<21 > GKQPr1 >6>6- ? -M[ [ ! ! 1z369 - G:G:58M > 1 / MZI6Gc7E5uNI --?> 7:MZIC7D3MZ5 > 7`<6I / aKL5 > <69 - ;=7U1 > 7D1V5 > G:5 -?> 7E5 =?9 - -=> 7D<6; 2> 5 - 321V9'K - G

;euNI2147:5LM > G \! N - 7 \! \ 5LK29 - GH7 - 1VK8MZ9:G Ce<6;=7 - 9'5 >C- 9WKwP - ?B369 - G:GH5LM > < - KQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - < -7E9:1 > G t - 9H7 YT

<CIG a7B= - !

< ( -0- !/+ ) +[ 0- '/) ) - $ !/ )6$ -.-y/ 21 `3< - 369 - G:G:58M > - 7kK -y/ 21'3< - @a5l7 - GHG -v> PR;A7U1 > 7321VGTI > 5 t MZ9 - G_<61 > GkKm1 / 1n@a587:; KwPR1V<'587 H

7U1 >C/?- 1 / MZI6Gc7E5 uaI - < - 7:9E1 > G t - 9c7 9:1V363MZ9H7[<6ID<6;6587[< - KQPR; - 7:7 - I69 -=>#/ MZIC9H7 Hi/ 589 / IC587[1 / MZICGH7E5 uNI - G:IC9[Km1369 - G:GH5LM >-?> z = ` - Gc7I >6- t M >6/ 7:5LM > < - r !

PR1V<'587H7U1 >C/?- 1 / MZICGH7E5 uNI - < - 7:9E1 > G t - 9c7 > MZ9 - ;["> # % > \ H ^ _ - GH7q<C; 2> 5 - 5 / 5 -=> 9 - `36KL1 HCn1 > 70K - G uNI21 > 7:587E;=G uaIC5h<6;?3 -?> < -?> 7`< - Km1 @41V9H5m16K - r 321V9bK - I69b@41VK - I69 'M -=>6>6- G:I69qKL1 G:IC9 t 1 /=- <CI7E9:1 > G:<CI / 7 - I69T5'36K85uNI6; 321V9 - ?a7 -?> GH5LM > < - Km1q9 - Km147:5LM > ! # C0GE1n@ZMV5L9

YT =Se < ve(r) >e + < Ye(r)p(r, 0) >e

< p(r, `) >r

, \! \

Ye;=7U1 > 7TKQPr14<`5l7:7E1 >6/?- 1 / MZICGH7E5 uNI - KLM / 1VK85LG:; - t M >6/ 7:5LM > < - r < - KwP^; - 7H7 - IC9 ! ;Y> ; / 9H58@41 > 7

Ye(r) =< Ye >e +δYe(r) , \!%

- 7p(r, 0) =< p(0) >e +δp(r, 0) ,

\! # K - G -?/ M > <7 - 9 - <6I > I';?9E17 - IC9k< - @a5 -=> 7

< Ye(r)p(r, 0) >e = < (< Ye >e +δYe(r))(< p(0) >e +δp(r, 0)) >e

≈ < Ye >e< p(0) >e + < Ye >e< δp(0) >e + < p(0) >e< δYe >e ,

G:MZ5l7< Ye(r)p(r, 0) >e≈< Ye >e< p(0) >e .

\! 1V9TG:IC587 - -?> 3MZGE1 > 7

Ve =< ve >e=2

a2e

∫ ae

0

ve(r0)r0 dr0 , \!

- 7Ye =< Ye >e=

2

a2e

∫ ae

0

Ye(r0)r0 dr0 , \! \

5LK@B5 -?> 7YT =

SeVe + Ye < p(0) >e

< p(`) >r,

\! G:MZ5l7 -?> 9 - 3MZ9c7U1 > 7kKwP - ?B369 - G:GH5LM > < - Km1b3C9 - GHG:58M > p(r, z) GHMZI6GkGE1 t MZ9 - \! N

YT =

SeVe + Ye

∑

ν

pc0ν(0) < ψc

0ν >e

∑

ν

pc0ν(`) < ψc

0ν >r

. \! >

76) A ) 1% 14$a)& 4 ) 0 )* )( ) 7 * 6! 0)(+* $( * #9$

'9/"( 1* & %( * 1 * 9

[ K%9 - GH7 - 14KLMZ9HG C - ?B36KL5 / 5l7 - 9_K - G / M - S / 5 -=> 7:G pc0ν(`)

- 7 pc0ν(0) !

%-o/ M - S / 5 -?> 7 pc0ν(`)

- GH7<6M >6> ;e361V9_KwP - ?B369 - G:GH5LM > \ ! \ N

pc0ν(`) =

S < ve|ψc0ν >e

1 −Kv

j k0

S

ρ0c0

1

g0ν(0, `)+ Yr0ν

g0ν(`, `)

g0ν(0, `)

, \! N

- 7K -o/ M - S / 5 -?> 7 pc0ν(0)

- Gc7k<6M >6> ; -=> t M >C/ 7E58M > < - pc0ν(`)

361V9_Km1b9 - Km17E5LM > \! \ >

pc0ν(0) =

j k0

1 −Kv

ρ0c0S

g0ν(0, `)

[

S < ve|ψc0ν >e

g0ν(0, 0)

g0ν(0, `)− Yr0νp

c0ν(`)

]

. \!

%- <6;=7E1V5LKO<6I / 1VK / I6K < - G / M - S / 5 -=> 7EG pc0ν(`)

- 7 pc0ν(0)

- Gc7<CM >6> ;y<21 > GkKQPr1 >6>6- ? - [ [ ! > ! .- 9H;?GHI6K87E147M B7 -?> I3MZIC9_K -o/ M - S / 5 -?> 7 pc

0ν(`)369 -=> <Km1 t MZ9 - 1 >6>6- ? -D[ 9 - Km147:5LM > G [ ! - 7 [ ! %

pc0ν(`) =

SeVeS

a0b0νV0ν

YT00ν + Yr0ν

[

Yp0ν

Yi0νj sin kz0ν`+ cos kz0ν`

] , \!

MZt V0ν- GH7fI > / M - S / 5 -?> 7d<6;?3 -?> <61 > 7o<6I 369HM K< - @a587 - GHG - < - KwP^; - 7H7 - IC9 / 1VK / I6K8;v<61 > GfKQPr1 >C>6- ? -Z[ [ ! 9 - Km147:5LM > G [ ! #F1 / MVtKQPr14<`5l7:7E1 >6/?- YT00ν

9 - KL147E58M >+[ !% G?P^; / 9:587

YT00ν =

[

Yi0ν +Y 2

p0ν

Yi0ν

]

j sin kz0ν`+ 2Yp0ν cos kz0ν` , \! > %

- 7MVtK - t 1 / 7 - I69 Yp0ν- GH7<6M >C> ;y321V9 [ !

Yp0ν =S

ρ0c0ζz0ν

- 7 Yi0ν =S

ρ0c0

kz0ν

k0(1 −Kv) .

\! > # p1V9k1V5LK8K - IC9:G K - G / M - S / 5 -?> 7EG pc

0ν(0) G=PR; / 9:5l@ -=> 7 1 >C>6- ? - [ 9 - Km147:5LM >+[ ! #

pc0ν(0) = pc

0ν(`)

[

j sin kz0ν`Yr0ν + Yp0ν

Yi0ν+ cos kz0ν`

]

. \! >

PR1V<'587:7E1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9E1 > G t - 9H7kG?P^; / 9:587T<CM >6/ 2> 1VK - -?> 7TG:MVI6GKL1 t MZ9 -

YT =

1 + Ye

∑

ν

SV0ν

a0b0ν< ψc

0ν >e

cos kz0ν`+Yp0ν + Yr0ν

Yi0νj sin kz0ν`

YT00ν + Yr0ν

(

Yp0ν

Yi0ν

j sin kz0ν`+ cos kz0ν`

)

∑

ν

SV0ν

a0b0ν

< ψc0ν >r

1

YT00ν + Yr0ν

(

Yp0ν

Yi0νj sin kz0ν`+ cos kz0ν`

)

. \! >

j 1 > GJK -'/ 1VGb361V9H7:5 / ICKL5 - 9yMZt K - GJ9:1 VM > G ae ar

- 7 a G:M > 7J; 14IG? KwPR1V<'587H7U1 >C/?- / MV9:9:5 Z; - Yr0ν \! \ - Gc7; 1VK - C Yr!

7

5C< - 3CKLI6G 5LK > P q1f369HMZ321Z147E58M > uNI - < P^M > < - GY36KL1 >6- G <21 > GYKL1 / 1n@a587:; ν = 0 - 7WGH5CK - G - C9E1 >6- G< - G '5 / 9:MZ3CM >6- GYM > 7]I > 'MZIC@ - -=> 7XI > 5 t MZ9 - 1VK8MZ9:G < ψc

00 >e,r=1

a0b00

- 7 V00 = 1 /?- uNI65 <6M >6>C-3MVI69 YT

YT = j sin kz00`

[

Yi00 +(Ye + Yp00)(Yr + Yp00)

Yi00

]

+ cos kz00`(Ye + Yr + 2Yp00) , \! > \

MZtsK -0> M& 69 - < P^M > < - kz00- GH7v<6M >6> ;'361V9yKm1#9 - Km147:5LM > [ ! \ / M&'3C7 - 7 -?> I < - G - ?B369 - G:GH5LM > G \ ! 1n@ -?/ γ00 = 0 - 7 \!$# >

k2z00 = k2 − k2

w00 = k2 = k20

[

1 +1 − j√

2

1√

k0

2

a(√

` ′v + (γ − 1)

√

`h )

]

, \! >

KwPR1V<'587H7U1 >6/=- < - 32149:MZ5 Yp00361V9_Km1b9 - Km17E5LM > [ !$#

Yp00 =S

ρ0c0ζz00 =

S

ρ0c0

1 + j√2

√

k0(γ − 1)√

`h , \! > >

- 7TKQPr1V< `5l7:7U1 >6/?- 5l7E;=9E147:58@ - Yi00321V9 ;eu !1\! > #

Yi00 =S

ρ0c0

kz00

k0(1 −Kv) ≈

S

ρ0c0

[

1 +1 − j√

2

1

a√

k0

(−√

` ′v + (γ − 1)

√

`h )

]

. \! > N

> @Z;=9:5 2- 1V5 > GH5uNI - KwP - ?B369 - G:GH5LM > \! > \ - GH75L< -=> 7E5uNI - C_KwP - ?B369 - GHG:5LM > < - KwPR1V<'587H7U1 >C/?- 1 / MZI6Gc7E5uNI -< - 7:9E1 > G t - 9c7 YT

<6I - G aGH7>= -d/ 1VK / I6K8; -k-?>D>6-d/ M > GH5L<6;=9E1 > 7 uaI - Km1y369:MV321 147:5LM > < P^M > < - Gh36KL1 >6- G 32149 - ? - '36K - KQP - ?B369 - G:G:58M > 369:MZ3MZGH; - 321V9h1VK8Km1 ^_ !

< : . # - - -%._ $ '/) c 2> < - '5 - I? uNI21 > 7:5 2- 9#KwP^5 > 2I -?>6/=- < - Km1 369HMZ321Z147E58M > < P^M > < - G9E1V<65L1VK - GgGHI69Km1x@41VK - I69D< -

KwPR1V<'587H7U1 >6/=- 1 / MZI6Gc7E5uNI - < - 7:9E1 > G t - 9c7 uNI - K uNI - Gk9:;=G:I6Kl7U147:GkGHM > 7k369:;=G -?> 7:;?Gd<61 > G /?- 321V9:1Z9:1V3 - ! - I6K8GYK - G 'Ma< - G % % - 7 % # c G:M > 7 Co369 -=> <69 -T-=>`/ M&`3B7 - <21 > G]K -_/ 1VK / I6K ! ;Y>'- ` - 7]Km1 / M > 7E9:5 6I H

7E58M > < - G `Ma< - G]G:IC3;=9:5 - I69:G CyKm1o@41VK - I69W< - KwPR1V<'587:7E1 >6/=- 1 / MZICGH7E5 uNI - < - 7E9:1 > G t - 9H7 YT

;Ru !/\! > - GH75 >#t ;=9:5 - I69 - C # % −3 < <CM >6/0> ; VKL5 - 16K - 5 / 5 K - Gf7 - 9 - G - ?C3C9:5`1 > 7e< - G / MZI636KL1 - G -=> 7:9 - <65 `;?9 -?> 7EG`Ma< - G_3MZIB@V1 > 7 A7E9 -e> ; ZK85Z;=G_<21 > GTK -o/ 1VK / I6K /Rt ! [ ! !.- G ZI69 - G\! - 75\ ! 9 - 369H;?G -=> 7 -?> 7`Km1 <C5 `;?9 -=>6/?- < - G 'MB<6ICK - Gv< - G'1V< `5l7:7U1 >6/?- Gb1 / MZI6GH7:5uNI - G< - 7:9E1 > G t - 9c7 YT

/ 1VK / I6K8; - GvG - KLM > I > `Ma<6=?K -`>C- 369 -=> 1 > 7 -=>)/ M&'3C7 - uNI - Km1369:MV321 147:5LM > < P^M > < - G36Km1 >6- G ;eu !\! > \ - 7yG - K8M > K - 'Ma<6=?K - <6;=@ - K8MZ363;q<21 > G /?-0/ 21436587:9 - ;Ru !\! > 1n@ -=/ uNI217E9 -0/ 1F@a5l7E;?G< - KLM > ZI - I69:Gf<C5 `;?9 -=> 7 - G K - G / 1n@a587E;=Gy< PR;A7U1VK8M >6> 1 - <658G:3M > 5 CK - Gd1VI ): ; 3MZI69d< - G '5 / 9HMZ36M >6- G K - GJ9:1 VM > Gq< - G - 69:1 >6- G ae

- 7 ar;A7U1 > 7q; 1VIG? - 7b@41VKL1 > 7 ae = ar = 0, 96 a ! j 1 > GJKm1 ZI69 - \! Km1 - 69E1 >C- < - KwP^; - 7:7 - I69 - GH7eG:IC363MVG:; - 1n@ZMV5L9oI > `MVIC@ - -=> 7 > M > I > 5 t MZ9 - <6M >6> ;

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J DeSUH J RJ Q b J \ +0POHNF K FID J Q HUC KGK FIHpS b \ OFI\GHNCVDeSSNHUC _ J HNQUCZDqS RJ EPaOHNF K FID J M J SNS J \GH QN\GH R CS J DmQNPFVD OPH^O\GPoS FI\m_ J H^S CV\ > FVHND J Q b \,EPaOHNF K FID JH^MO JK S J \GH-C[S J DGQ^PFID OPaHNO\GPaSfFI\ _ J HpS J Q^S. J Q^\GHNM JJ D9\mSNP R PaQUCVDeS R C S J O DmP u \ J b J R C S J DGQNPaFID0PDGQ^MHNM J i0//'#'1

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C4EPaQ J J D 1[\m_ H J b J O J SNS J 0MS F b J b J S J Q^S%CLOFID b \GPaS b J Q%HNMQN\ R SYCZSNQ50M b PaF OH J Q D J QUCZSNPQStnCVP *QNCVDeS K CVQ2CV\ SUF R MHNCVDGO J Q K H J QNOHNPoS J Q K CZH R CLDmFIH6 J K FI\GH RJ Q QNP& K CVHNC SUH J Q b M&S J H60PD J H 9 QNC _IFIPaHRaJ Q b J \ ]_IF R \ J Q dRJ Q b J \ HNMQNPQpSYCVDmO J QWCVOFI\mQ^SUP u \ J Q J S RaJ Q b J \ PD b \GOSUCVDGO J QWCVOFI\GQpSUP u \ J QWH J O GJ H *O M J Q7 g J Q^S K FV\GH u \GFIP\GD J 0F b M R PaQUCZSNPFID KmR \GQ OF' KGR S J b J Rhg FIH J P RaRJ CZH^SUP32 OP JRaRJTJ S b J Q J QTM R M- J DeSNQOFIDGQ^SNPaSN\mSUP t`Q2C M&SUM b M&_ JR F KGK M J98-

: .2/ / ; ;2169:c; / ;21 3 2/ <>4 ;21 < . 6 1306;21 / ;21 ;24!-1 / ; #"j. : ;%3 ; <[:$3 E% 3; & ;

% CVDGQ RaJ > \ S b J _VMH^P2 J H R C[_VC R P b PaSUM b J Q.0F b RJ Q b g P K M b CVDGO J b g J DeSNHNM J b J Q SN\ >J Q J S b JWR C t J DeS J O CZH6IMQ>FI\DGFID K CVH[\GD_IF R \ J K H^F K FVQNMQ b CVDmQ R C H^MRt`MH J DmO J 8 i \GD J MSU\ b JEJ K MHNP J DeSUC RJJ QpS J DGM J CZ\ )( (' R C 0F b M R PQUCXSUPFVD b \Q Q^S J \ SUP R PQ^M K FI\GHWO J S^S J M&SU\ b J J Q^S K H^MQ J DeSUM J b CVDGQ O JK CVHNCIHUC KGJ # J SNS J _ZC R P b CXSUPFVD J K MH^P J DeSYC RaJ J QpS J<J OSN\GM J Q JR FVD\mD J EM&S F b J QNPEP R CVPaH J.9R CEM&S F b J b J H^MOP K HNF OPoSUM K FI\GH R g M&SYC R FIDGD C JTJ D K H J QNQ^PFID b J Q 0POHNF K FID J Q J SNS J EM&S F b J J S RJ b PQ K FIQ^PaSUP t EPaQ J D 1[\m_ H J CV\ )( K FI\GH J Q^\GH J HEO J Q P K M b CVDGO J Q QNFVDqSb MOHNPoSUQ b CVDmQ RJ K CZHUCIHNC KGJ # 3/'5 J Q K CVHUCIHUC KGJ QQ^\GPa_ZCVDeSUQ # *) J S # K HNMQ J DeS J DeS b J \ EF b RJ Q K Q^P u \ J Q K FIQNQ^P >GRaJ Q K FI\GH%O J b PaQ K FIQNPoSUP t +-,/.0,1 2436587:9;<=;:<?>@<BADC:EF<G;<BAIH>JK3B;:LNM:O6<BAI;QPR<BMS5DEF3B<G;<BAI3BTU3>@<BMS5VAW;:<

TPR9XEF<BHUTUTU<YLZED5VHU[O6HU<BTUTU< J Q Q^S J MSN\ b PM J Q^S b MOHNPaS K CVH RaJ Q^O M-EC b J(R C 2I\mH J # 6 R J Q^SfP b J DeSUP u \ J O JR \mPm\mSNP R PaQNM b CVDGQ

RJ OC b H J b J[R g M&SYC R FIDGD C J b J EPaOH^F K FID J Q K CVH R C 0MS F b J b J HNMOP K HNF OPaSNM # b J \ EPaOHNF K FID J Q \mDM- J S^S J \mH nJ&\ O CVOPaSUM Mpe

C b 0PaS^SYCVDmO J CVOFI\mQ^SUP u \ J Ye _ PaS J QNQ J b JR C0 J > HUCVD J ve

J S \GDHNMO JK S J \mHnJ&\ OCVOPoSUM Mpr C b 0PaSNSUCVDGO J CVOFV\GQ^SNP u \ J Yr

_ PaS J QNQ J b J J > HUCVD J vR QNFVDqS OFI\ KGR MQ CVOFV\GQ^SNP u \ J8* J DeS K CVH9\mD J O C _ PaSNM b J OFI\ KGR C J J SNS J O C _ PaSUM J Q^S OFIDGQ^SNPaSN\GM J b J b J \ K FIHpSUPaFIDGQ b J O R PD b H J

b J J b P C SUH J a J Q J > HNCVD J Q b J Q EPaOH^F K FID J Q FIDeS b J Q HUC IFIDmQ b P MH J DeSUQ b J O JR \GP b \O R PaD b H J J QM R M J DqSNQ OCVHUCVO&SUMHNPQ J H2QNFVDqSPDmQNMH^MQ R C QNM K CVHUCXSUPFVD b J QO R PD b H J Q-$ J Q M R M- J DeSUQ !u \ CXSUH JSU\ >dJ Q b g C b EPoSNSUCVDGO J PD b Pa_ P b \ JRaRJ Yt

FI\ \GD J t J DeS J CVDmD \ R CVPaH J EPaDGO J b g C b 0PaSNSUCVDGO J Yf

QNFVDqSWH JK HNM *Q J DqSNMQ K CVH R g C b EPoSNSUCVDGO J Q K MOP2 u \ J y1

QN\ KGK FVQNM J R F OC R PaQNM J QN\GH R C OPH^OFIDt`MH J DGO J, r = a b CVDGQ RJ

KGR CZDGQ b J QNM K CVHUCXSUPFVD z = `11 J Q J K H J QNQNPaFIDGQ b J Q C b 0PaS^SYCVDmO J Q Yt

J S Yf

QNFVDqSWO C R O\ R M J Q b CVDGQ RJ QK CVHNCIHUC KGJ Q * ) J S * b J[R C HNMt`MH J DGO J 8

r

aae ar

ve vR

]``1

z

^`_ baUced#fgcehgijbk iblmlUibn#a

^`_ baUced#fgcehgia j oibd#lUibn#a

p N k _qlml H h oi Lmd j b_*rDstngiy1ibh

z = `1ibl

r = a

"!/Y[Z # 6 q % PaQ K FIQNPoSUP t \mSNP R PQNM K FI\mH R C J Q^\GH J b J QLP K M b CVDmO J Q b g J DeSUH^M J b J Q M R M- J DeSUQ b J0Rhg FIH J P RRaJCVH^SNP2 OP JRRJ

J QTS J O DGP u \ J Q b J J QN\GH J QNFIDeS RaJ Q J Q u \ J O JRRaJ Q\mSUP R PQ^M J Q K FI\GH Rhg MSYC R FIDGDGC J[J D K H J Q^QNPFVDK CVH R C EM&S F b J b J H^MOP K H^F OPaSUM i) C u \ CVDeSNPaSUM J QN\GH^M J R g C b EPoSNSYCZDGO J M RJ O&SUH^P u \ J b J SUHNCVDGQSt J H^SYE = 1/ZE

b \ Q QpS J J Q^SfOF' K CVHNM J b J Ql_ZC RJ \GHNQ b J YE

b MS J H60PDmM J Q b g C K H Q R CWH JR CZSUPaFID //

YE =1

MpeMprYT ,

!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5

RaJ Q J&\ OCVOPoSUMQ b J Q EPaOHNF K FID J QTM&SYCVDeS OFIDGDq\ J Q J S Rhg C b 0PaS^SYCVDmO J CVOFV\GQ^SNP u \ J b J SUHUCZDGQ"t J H^S YT

MSUCVDeSb M&S J H60PDGM J CVD C R SNP u \ J J DqS b g C K H Q0\mD 0F b RaJ K QNP u \ J b \ Q Q^S J OFIDGQ^SNPaSN\GM K CVH RaJ Q0POHNF *K FID J Q RJ OFV\ KGRJ \GH J S R g C b EPoSNSYCZDGO J y1

%TJ \ EF b M R PQNCZSUPaFIDGQ b J O J Q QpS J H JK H J D CVDeS J D IHNCVD b J K CZH^SUP J RJ Q EF b RJ QTMSYC >GR PaQ b CVDGQ R C

H^MRt`MH J DGO J 98- QNFIDeS K HNMQ J DeSNM J Q b CZDGQ O J K CVHNCIHNC KGJ 5C K H J 0P H J EF b M R PQNCZSUPaFID QN\ KmK FIQ J u \ J R CK H^F K CICZSUPaFID b CVDGQ R C[O C _ PaSUM J QpS(O JRRaJ b g FID b J Q KGR CVD J Q b FIDeS R g C0FIHpSUPQ^Q J J DqS b CV\ J8 J SUQ_ PaQ u \ J \ J SES GJ HEP u \ J Q J Q^S K HNPaQ J D OF' K S J b CVDGQ RJ DGF' > H J b g FID b J b CVDGQ R g P K M b CVDGO J PoSUMHUCZSNPa_ J J S K CZHR g PaDqSNHNF b \GO&SUPFVD b g \mD J P K M b CVDGO J b J[K CVHNFVP CV\ J SUHNMEPoSUMQ-CEQ J OFID b J 0F b M R PaQUCZSNPFIDPaDqSNHNF b \GPoS RJ Q EF b J QHNC b P CZ\ nJ SWM&_ J DeSN\ JRaRJ J DeS M R POF < b CV\ b CZDGQR C]O C _ PaSNM b J OFI\ KmR C J RaRJ H JK FIQ J QN\GH R C t`FVH6 \ R CZSUPaFID PDeSUMIHUC RJ b J Q K HNF >GR J QLCV\ R P0PaS J Q RJ QJ8 J SUQ b J b PQ^QNP K CZSUPaFID9_ PQ u \ J \ J S%S GJ HEP u \ J Q2MSUCVDeS K HNPaQ J D OF' K S J b CZDGQ RJ DGF' > H J b g FID b J J D]O Ju \GP"OFIDGO J HND J O J \ QN\GH RaJ Q K CZHNFIPaQ R CXSUMHNC RJ Q r = a

J S K CZH RJ O FIP& b g \GD J t`FIDGO&SUPFVD b J @ H JJ D u \GPQNCZSUPaQ"tnCVPoSlQN\GH RaJ Q K CVH^FIPQ J D z = 0 J S ` \GD J OFID b PaSUPaFID 0P& S J b CVDGQ R C u \ JRRJR C t`FIDGO&SUPaFID J SQUC b MHNPa_VM JQ^FIDeS R PaM J Q K CZH R g P K M b CZDGO J _ PQ^OFVS GJ HEP u \ J b J>K CVHNFVP t`FIDGO&SUPFVD b J @[H JJ D O C R O\ R M J CV\ K CVHUCVHUC KmJ6 1+-, . , 249;BTU<LZAV3 AVC E TUL JE89ZJLNL 5VHU9ZM ;KP 9ZM:;<BA J TULZM<BA! !# !7-')(*!#"' 5! !05(8+'-$!3

J Q HNC IFIDGQ b J Q"EPaOHNF K FID J QQ^FIDeS(Q^\ KGK FIQNMQ%POP MeCV\ O JR \GP b JTR C[O C _ PaSNMC K H^F K CeCXSUPFVD b J QFID b J Q Q^\ KGK FIQNM J Q KGR CVD J Q b CZDGQ2O CVO\mD b J Q O R PD b H J Q J Q^S HNMIP J K CVH Rhg M u \ CZSNPFID 6

(

∂ 2

∂z2 + k2z

)

p(z) = 0 , 0 ≤ z ≤ ` # / F RaJ DGF' > H J b g FID b J kz

u \GP K H J D b J D OF' K S JWRhg PDG\ J DGO J b J Q J<J SUQ%_ PQ u \ J \ J S(S GJ HEP u \ J Q b CZDGQRaJ Q2OFI\mO GJ Q R PEPoS J Q R CZSNMHNC RJ Q J QpS b FIDGDGM K CVH R g M u \GCZSUPaFID I

k2z = k2

0

1 + (γ − 1)Kh

1 −Kv

≈ k20

[

1 +1 − j√

2

2

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

. # *)'C QNF R \mSUPaFID K FV\GH R C K H J Q^QNPFVD CVOFI\GQpSUP u \ J[J Q^S MOHNPaS J

p(z) = A(e−jkzz +Bejkzz) , ∀z ∈ (0, `1) , # C

p(z) = C(e−jkzz +Dejkzz) , ∀z ∈ (`1, `) , # >

RaJ Q OFVDGQ^SUCVDeS J Q A B C D MSUCVDeS b MS J H60PDmM J Q K CVH RJ QWOFID b PaSNPFIDGQ MOH^PH J CZ\ PaDqS J HStnCVO J Q z = 0 z = `1

J S z = ` J Q[OFVD b PaSNPFIDmQ[CV\ t`HNFVDqSNP H J Q J D z = 0 J S z = `

Q^FIDeS RaJ Q J Q u \ J b CVDGQ RJK CZHUCIHNC KGJ 6 )P C OFID b PaSNPFID CV\ st`HNFIDeSNP H J Q J D z = ` J Q^S b FIDmDGM J[K CVH R g M u \GCZSUPaFID

1

Zv

∂

∂zp = −(Yr + Yp) p ,

K FI\GH z = ` , # 6 C _ J O M u )'

Zv =1

S

j k0ρ0c01 −Kv

# #'

[ !"#$&% &# %#%#$ ' (#%#$ *)

J S M!+ 8 Yp =

S

ρ0c0

1 + j√2

√

k0(γ − 1)√

`h , #-,

O J +q\GP b FVDGD J

D = e−2j kz`Yi − (Yr + Yp)

Yi + (Yr + Yp), #

F Rhg C b 0PaS^SYCVDmO J PoSUMHNCZSUPo_ J Yi_ZCV\mS M.+ J S 6

Yi =j kz

Zv≈ S

ρ0c0

[

1 +1 − j√

2

2

a√

k0

(−√

` ′v + (γ − 1)

√

`h )

]

. #

PP % CVDmQ RaJ KGR CZD z = `1 R C OFIDeSUPaDq\GPaSNM b J Q K H J QNQ^PFIDGQ b FVDGD J

C = Ae−j kz`1 +Bej kz`1

e−jkz`1 +Dejkz`1. # 8

F' K S J S J Dq\ b \ b M > PoS σ1

ρ0c0y1p(`1)

J DeSUHNCVDeS b CVDGQ Rhg M R M J DeS CZOFI\GQpSUP/+ \ J5 O CVHNCVOSNMHNPaQ J H lR COFVDqSNPDq\GPoSUM b J Q b M > PoSUQ b FIDGD J

− 1

Zv

A∂

∂z(e−jkzz +Bej kzz) = − 1

Zv

C∂

∂z(e−j kzz +Dejkzz) + Y1A(e−jkzz +Bej kzz) ,

K FI\GH z = `1 ,

FY1 =

σ1

ρ0c0y1J QpS Rhg C b 0PaS^SYCVDmO J CVOFV\GQ^SNP$+q\ J b JLRhg M R M J DeS OCVHUCVO&SUMHNPQ J H D10nCVPaQUCVDeSW\mQUC J

b JER C]H JR CZSUPaFID # 8 J DeSUH JERJ QLOFVDGQ^SUCVDeS J Q C J S A O J SNS J H JR CZSNPFID b FIDmD J R C]OFVDGQ^SUCVDeS J B J D0`FIDGO&SUPaFID b J D OFIDGDq\ M!+ $#

B = −e−2j kz`1Y1 +De2j kz`1(Y1 − 2Yi)

(Y1 + 2Yi) +De2jkz`1Y1

. #3/

PPaP C OFID b PaSUPaFID J D20`HNFVDqSNP H J Q J D z = 0 b FIDGDGM J[K CVH M.+$ ' #

− 1

Zv

∂ p

∂z= Sve − Yp p ,

K FI\mH z = 0 , # /'/ KdJ H6 J S b g J K HNP J H RaJ b M > PaS Sve

#

Sve = A[Yi(1 − B) + Yp(1 +B)] . # / ) J S^S J J K H J Q^QNPaFID b \ b M > PoS Sve

KJ H6 J S b g J K HNP J H Rhg C b 0PaSNSUCVDGO J CVOFI\GQpSUP$+q\ J b J SUHUCZDGQ30 J H^S YTb \0Q Q^S J 0POHNF K FID J Q * OFI\ KGRaJ \GH eJ S b FIDmOTCV\mQNQNP Q^FID C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SUHNCVDGQ30 J H^S YEb g C K H Q R C H JR CXSUPFVD /'3/ 1

!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5

$ /((8!7"$ =(8*+ (8!#"'-1 8(C H JR CZSUPaFID /'/ b FIDmD J Rhg J K H J QNQ^PFID b \ K HNF b \GPoS b J Q J\ O CVOPaSNMQ b J Q b J \ 0POHNF K FID J Q J D

0`FIDmOSUPaFID b JR g C b EPoSNSUCVDGO J CVOFI\mQ^SUP/+q\ J b J SNHUCVDGQ 0 J HpS YT O C R O\ R M J K CVH^SNPH b \0EF b RaJlK QNP$+q\ J O FIPaQNP

J S b J R g C b EPoSNSUCVDGO J M RJ OSNHNP$+q\ J b J SNHUCVDmQ30 J HpS YE J Q^\GHNM J b \ Q Q^S J 0POHNF K FID J Q * OFI\ KGRaJ \GH #

MpeMpr =YT

YE

.

J Q 0POHNF K FID J QE\mSNP R PQNMQ K FI\GH0O J SNS J MSN\ b J J K MH^P J DeSYC RJ QNFIDeS9M&SYC R FIDGDGMQ ' RJ \mHNQ J&\ O CVOPaSUMQMpe

J S MprQNFVDqS b FIDGO OFVDGDq\ J Q g C b 0PaS^SYCVDmO J M RJ O&SUH^P$+q\ J b J SNHUCVDmQ30 J HpS YE

J Q^S b FIDmO b FIDGDGM J[K CVH R CH JR CXSUPFVD i # / #

YE =1

MpeMpr

YT =1

MpeMpr

Sve + Yep(0)

p(`). #3/

D20nCVPQNCVDeS2\GQUC J b CVDGQ2O J SNS J J K H J Q^QNPFVD b J QH JR CZSUPaFIDGQ #/ )' J S # 8 P R _ P J DeS

YE =1

MpeMpr

Yi(1 − B) + Yp(1 +B) + Ye(1 +B)

e−j kz`1 +Bej kz`1

e−jkz`1 +Dej kz`1

e−j kz` +Dej kz`, #3/ 6

+q\GP b FIDmD J OF' K S J S J Dq\ b J Q J K H J QNQNPaFIDGQ #3/ J S #

YE =1

MpeMpr

j sin kz`

[


Yi+Y1

2

Ye + Yr + 2Yp

Yi

]

+ cos kz`

[

Ye + Yr + 2Yp +Y1

2

(

1 +(Ye + Yp)(Yr + Yp)

Y 2i

)]

+Y1

2

[

j sin kz(`− 2`1)Yr − Ye

Yi

+ cos kz(`− 2`1)

(

1 − (Ye + Yp)(Yr + Yp)

Y 2i

)]

. #3/ #'

% CZDGQ RaJ O CVQ K CZH^SUPaO\ R P J H F `1 = `/2 O J S^S J J K H J QNQ^PFID b J _ P J DeS

YE =1

MpeMpr

j sin kz`

[


Yi

]

+ cos kz` (Ye + Yr + 2Yp)

+Y1

2

[

1 − (Ye + Yp)(Yr + Yp)

Y 2i

+ j sin kz`Ye + Yr + 2Yp

Yi+ cos kz`

(

1 +(Ye + Yp)(Yr + Yp)

Y 2i

)]

.

#3/, DmFVS J H+ \ g J D R g C > Q J DGO J b g M R M- J DeS OC K P RaR CVPaH J Y1 = 0

Rhg J K H J QNQ^PFID #3/ #' KdJ H J S b J H J SUH^FI\m_ J HR g J K H J Q^QNPaFID b J[Rhg C b 0PaS^SYCVDmO J CVOFI\GQpSUP$+q\ J b J SNHUCVDGQ 0 J HpS YT

+-, . , . 249;BTU< LNAD3=ADC:ETLo9XE >YC:TL 5VHU9ZM HUMS5V3 ZE8LZT< ;:C JE89 T >@<=LZC TUH

>@H 58<BA J EF b RaJ9K H^F K FVQNM b CVDGQ RaJK CVHNCIHUC KGJK H^MOM b J DeSED J SUP J DeS OF' K S J + \ J b J R C K HNF K CeCZSNPFID

b g FID b J Q KGR CVD J Q % CZDGQW\GD S JR Q Q^S J b J Q EF b J QQN\ K MH^P J \GHNQ EF b J QHUC b P CV\ J S M_ J DeSU\ JRRaJ J DeS M R PaOF < b CV\ KJ \m_ J DeSfC _IFIPaHc\GD J PD \ J DmO J DmFIDLDGM R P J C >GRJ Q^\GH R C2_ZC RaJ \GH b J YE

J Q EF b J Q-Q^FIDeS R PMQ

[ !"#$&% &# %#%#$ ' (#%#$

CV\ K HNF2 R Q b J _ PaS J QNQ J DGFID \GDmP/0`FIH J Q b J Q J > HNCVD J Q b J 0POH^F K FVD J Q CV\ LHUC VFIDGQ b J Q J > HNCVD J Q+q\GP QNFIDeS b P MH J DeSNQ b J O JR \GP b J[R C OC)_ PoSUM J S R C K H^MQ J DGO J b J Q M R M J DqSNQ OCVHUCZOSUMHNPaQ J H J D K CZHNFIP J EF b RJ K HNMQ J DeSNM b CVDGQO JWK CZHUCIHNC KGJ K H^F K FVQ J b JK H J D b H JWJ D9OF' K S JRJ Q"EF b J QQ^\ K MHNP J \GH^Qb CVDGQ R C OC _ PoSUM J S H JK FIQ J QN\mH R C 0`FIH6 \ R CZSNPFID PaDeSUM-VHUC RaJ b J Q K H^F >GR0 J Q CV\ R P0PaS J Q- R H JK H J D b J DIHUCZD b JLK CVH^SNP JLRaJ 0F b RJ K H^MQ J DeSUM b CVDGQ RJ HNC KGK FIH^S b \ OFVDqSNHUCZS ( D%3/ / 98 i # * / CVPDmQNP+q\ J b J DGF' > H J \ M R M J DeSUQ b \EF b RaJ b M&_ JR F KGK M b CVDGQ RJ O C K PaSNH J 6 ! !# !#&')(*!#"$ $"! /(81 ? '( ' 36 "

J Q K FVS 9 Q J Q QN\GH RaJ O C K CVOFI\GQ^SNP$+q\ J M&SYCVDeS RJ Q J Q PaOP +q\ J b CVDGQ RJ K CVHUCIHUC KGJ 6 3/ RJ QWM!+q\ CZSNPFIDmQ 0`FID b C J DeSUC RJ QWQ g J K H^P J DeS b J R C* J 0nCFID J S K CVH Q^\GPaS J RJ O C K b J K H J Q^QNPFVDCVOFV\GQ^SNP$+q\ J QUCZSNPQ 0nCVPaS 0R g M.+ \GCZSUPaFID b J[K HNF K CeCZSNPFID 6 3/ #

(∆ + k2)p(~r) = 0 , ∀~r ∈ D , #3/ C _ J O M.+ 6 / ,

k2 = k20

1 + (γ − 1)Kh

1 −Kv

≈ k20

[

1 +1 − j√

2

2

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

. #3/ O J S^S J M!+q\ CZSNPFID b J K HNF K CeCZSNPFID QNFVDqSTCVQ^QNF OPM J Q RJ QOFID b PoSUPFVDGQ2CV\ 20`H^FIDeSUP H J QQ^\GPa_ZCVDeS J Q CVDGD J J M.+ / #' #

ρ0c0vz = ρ0c0ve − ζz0p ,K FI\GH r ∈ (0, ae) , θ ∈ (0, 2π) , z = 0 , #/ 8 C

ρ0c0vz = −ρ0c0vR + ζz`p ,K FV\GH r ∈ (0, ar) , θ ∈ (0, 2π) , z = ` , #/ 8=>

ρ0c0vz = −ζz0p ,K FV\GH r ∈ (ae, a) , θ ∈ (0, 2π) , z = 0 , #/ 8 O

ρ0c0vz = ζz`p ,K FI\GH r ∈ (ar, a) , θ ∈ (0, 2π) , z = ` , #/ 8 b

Fζz0

J S ζz`

QNFIDeS RJ QfC b 0PaSNSUCVDGO J QlCVOFI\GQpSUP$+q\ J QfQ K MOP32+q\ J Q b J Q K CVHNFIPaQ J D z = 0 J S z = `SUHUC b \GPQNCVDeS

RJ Q J<J SUQ b J Q OFI\GO GJ Q R P0PaS J QS GJ HEF)_ PQ3+q\ J \GQ J QQN\GH RJ O C K CVOFI\GQ^SNP$+q\ J b CVDmQ R C O C _ PaSNM CVDGD J J M.+ / , GJ S

vr =y1

ρ0c0p K FI\mH r = a , θ ∈ (0, 2π) , z = `1 ,

#/ 8VJ vr = 0 K FI\mH r = a , θ ∈ (0, 2π) , z ∈ (0, `) , #3/ 8 0

RJ Q J8 J SUQ b J Q OFV\GO GJ Q R PEPoS J Q0S GJ H60F)_ PaQ+q\ J \GQ J Q R CZSNMHNC RJ QMSYCZDqS b M K H^PQ J D OF K S J b CVDGQ RJDGF' > H J b g FID b J OF' KGRJ J k C Q^\GH30nCVO J σ1

Q^\GH R C+q\ JRRaJTJ QpS R F O C R PQ^M JTR g C b EPoSNSUCVDGO J Q K MOP2+q\ J y1

O CVHNCVOSNMH^PQ J HfMSYCZDqSfSNH Q KdJ SNPaS J b J _ZCVDeS R C QN\GH 0nCVO J R CZSNMHNC RJ b J R C O C _ PaSUM eJRaRJ D g J QpS K CVQZSUM J b J O JRRaJ8* OPb CVDGQ R C OFID b PaSUPaFID # / 8 0 1 J Q2C b 0PaS^SYCVDGO J Q ζz0

J S ζz`

H J Q^SUCVDeS KJ SUPoS J Q OF K S J S J D \ b J Q2M!+q\ CZSNPFIDGQ 6 '6 3/ 6 O J Q2OFID b P *SUPaFIDGQ%CV\ 0`HNFIDeSNP H J Q # / 8 C * 0Q g MOHNPo_ J DeS

∂ p

∂n= −∂ p

∂z=

j k0ρ0c01 −Kv

vz ≈j k0ρ0c01 −Kv

ve − j k0ζz0 p # ) IC

K FV\GH r ∈ (0, ae) θ ∈ (0, 2π) z = 0

∂ p

∂n=∂ p

∂z= − j k0ρ0c0

1 −Kvvz ≈

j k0ρ0c01 −Kv

vR − j k0ζz` p # ) >

!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5

K FI\GH r ∈ (0, ar) θ ∈ (0, 2π) z = `

∂ p

∂n= −∂ p

∂z= −j k0ζz0 p

# ) VO K FI\GH r ∈ (ae, a)

θ ∈ (0, 2π) z = 0 ∂ p

∂n=∂ p

∂z= −j k0ζz` p ,

# ) b K FI\GH r ∈ (ar, a)

θ ∈ (0, 2π) z = ` ∂ p

∂n=∂ p

∂r= −j k0y1p ,

# ) J K FI\GH r = a θ ∈ (0, 2π) z = `1

∂ p

∂n=∂ p

∂r= 0 , # ) 0

K FI\GH r = a θ ∈ (0, 2π) z ∈ (0, `) F'( J b M PaD b P/+ \mM CV\ O C K PaSNH J 6 M!+ 6 *) ' K FI\mH \GD J FID b J CVH60FIDGP/+ \ JRJ O C K b J

K H J Q^QNPFVD CVOFV\GQ^SNP$+q\ J p(~r) b CVDGQ>\GD b F CVPaD J D R PEPoSUM K CVH>\mD J Q^\GH30nCVO J 0 J HEM J S KdJ \mS SNH J MOH^PaSQ^FI\GQ R C 0`FIH J PaDeSUM-VHUC RaJ QN\GPo_ZCVDeS J: i , )*) nRaJ 0nCVOS J \GH ej ω t J QpSFEPaQ J S RaJ _ J O&S J \GH ~r J QpSH JK H^MQ J DeSUMPaOP K CZH RaJ Q(SNHNFIPaQ%OF FIH b FVDGDGM J Q r θ J S z #p(~r) =

∫∫∫

DG(~r, ~r0)f(~r0) dD0 +

∫∫

S

[

G(~r, ~r0)∂

∂n0

p(~r0) − p(~r0)∂

∂n0

G(~r, ~r0)

]

dS0 . # )/

C0`FIDGO&SUPaFIDf(~r0)

H JK HNMQ J DqS J Rhg CVOSNPFID b J Q QNFV\GHNO J Q(H^M K CVHpSUP J Q >Rhg PDeSNMHNP J \GH b \ b F'ECVPD J D # b CVDGQRaJ O CVQ K H^MQ J DeS f(~r0) = 0

D0nCVPQNCVDeS(\GQNC J b J Q%OFID b PoSUPFVDGQ CZ\ 0`HNFIDeSUP H J Q #*) IC0 K FI\mH RJ O C K b JWK H J Q^QNPaFID RJ O FIP&

b g \GD J 0`FIDmOSUPaFID b J @ H JJ D QUCZSNPQ30nCZPQUCZDqSWQN\GH Rhg J DmQ J >GRJ b J Q K CVHNFVPQ b J R C9O C _ PaSUM CZ\ OFID b PoSUPFVDGQ O JQ^FIDeS RaJ Q J Q +q\ J O JRaRJ Q%H J S J Dq\ J QTCV\ O C K PaSUH J 6 M!+ 6 ) )'

∂

∂nG(~r, ~r0) = −j k0ζz0G(~r, ~r0) ,

K FI\GH r ∈ (0, a) , θ ∈ (0, 2π) , z = 0 , # ) )VC'∂

∂nG(~r, ~r0) = −j k0ζz`G(~r, ~r0) ,

K FI\mH r ∈ (0, a) , θ ∈ (0, 2π) , z = ` , # ) ) > ∂

∂nG(~r, ~r0) = 0 , K FV\GH r = a , θ ∈ (0, 2π) , z ∈ (0, `) , # *) )XO

KdJ H J S b g J K HNP J H R g PaDqSNM-IHNC RJ b J QN\GH 0nCVO J C KmK CVHUCZPQNQNCVDeS b CVDGQ R C H JR CZSNPFID# *)/ +q\GPH JK HNMQ J DqS JLR CK H J Q^QNPFVD CVOFI\GQpSUP$+q\ J[J D SNFI\mS K FVPDeS b J[R C OC)_ PoSUM b J R C CZDGP H J Q^\GPa_ZCVDeS J#p(~r) =

j k0ρ0c01 −Kv

[∫∫

Se

ve(~r0)G(~r, ~r0) dS0 +

∫∫

Sr

vR(~r0)G(~r, ~r0) dS0

]

− j k0

∫∫

σ1

y1(~r0)p(~r0)G(~r, ~r0) dS0 , # )

SeJ S Sr

MSUCVDeS RaJ Q2QN\mH30nCVO J Q H J Q KJ OSNPa_ J Q b J R C J > HUCVD J b J Rhg M- J S^S J \GH J S b \ HNMO JK S J \GH RaJ Q J<J SNQb J Q OFI\GO GJ Q S GJ H60F)_ PaQ+q\ J \GQ J Q"MSUCVDeS K FIH^SNMQ K CVH RJ 0nCVOS J \GH Kv

+ \mP PaDeS J Hp_ P J DeS b CVDGQ Rhg P K M b CVDGO J b \0P R P J \ J S K CVH R C0`FIDGOSNPFID b J @[H JJ D JRRaJ8* J b CVDGQ RJ DGF' > H J b g FVD b J k K FI\GH RJ Q J<J SNQ R CXSUMHNCV\ J S b CVDmQ RJ Q(C b 0PaS^SYCVDmO J QQ K MOP2+q\ J Q ζz0J S ζz`

CV\9DmPa_ J CV\ b J Q5 J > HUCVD J Q b J Q.0POH^F K FVD J Q J S RaJ \GHK MH^P K MHNP J DGFZS J H R C[Q EM&SUHNP J M- J S^S J \GH * H^MO JK S J \GH b CVDGQ R g J K H J Q^QNPaFID b \0O C K b JK H J Q^QNPaFIDCVOFI\GQpSUP$+q\ J O J O C K b M KdJ D b]b J O J Q%_ PaS J Q^Q J Q +q\ JR +q\ g J D Q^FIPaS RaJ \GH%OCV\GQ J g J K H J Q^QNPaFID b J R C"0`FIDGOSNPFID b J @ H JJ D J QpS b FVDGDGM J CV\ K CZHUCIHNC KGJ:6 M.+$ 6 *)'# 96 /)C J S > 1

[ !"#$&% &# %#%#$ ' (#%#$ '!# !#&$)(84*!#"' 5! ! 5(

J O C K b J K H J QNQNPaFID]CVOFI\mQ^SUP/+q\ J b CVDGQ R C O C _ PaSNM J Q^SF > S J Dq\ J D H JK FIH^SUCVDeS R g J K H J QNQNPaFID 6 )'#'b J R C 0`FIDGOSNPFID b J @[H JJ D b CVDGQ R g M.+q\ CZSUPaFID # ) 1$ F' K S J S J Dq\ b J>R C 0`FIH J b J O J SNS J 0`FIDGO&SUPFVD b J@[H JJ D R C K H J QNQ^PFID p(r, θ, z) C KGK CVHNCSTC R FVHNQTQ^FI\GQ R C 0`FIH J b g \GD b M_ JR F KGKJ J DeS QN\GH RJ Q 0`FIDGOSNPFIDmQK HNF K H J Q ψt

µν(r, θ, z)#

p(r, θ, z) =∑

µ,ν=0,1...

t=c,s

[

j k0ρ0c01 −Kv

(


µν >r

)

−j k0gµν(z, `1)∑

µ′,ν′,t′

pt′

µ′ν′(`1) < y1ψt′

µ′ν′|ψtµν >σ1

]

ψtµν(r, θ) ,

# ) 6

F R CEDGFVSUCZSUPaFID < ve|ψtµν >e

b MQNPID J>Rhg PDeSUMIHUC RaJ QN\GH Se

b \ K H^F b \GPaS ve(r, θ)ψtµν(r, θ)

< vR|ψtµν >rb MQ^PID J Rhg PDeSUMIHUC RaJ Q^\GH Sr

b \ K HNF b \GPoS vR(r, θ)ψtµν(r, θ)

J S < y1ψt′

µ′ν′ |ψtµν >σ1

Rhg PDeSUMIHUC RaJ Q^\GH σ1

b \K HNF b \GPoS y1(θ)ψ

t′

µ′ν′(r, θ)ψtµν(r, θ)

-C Q^F R \mSNPFID K H J D b]b FVDGO R C"0`FIH6 Jp(r, θ, z) =

∑

µ,ν,t

ptµν(z)ψ

tµν(r, θ) ,

# )'#'C _ J O

ptµν(z) =

j k0ρ0c01 −Kv

[


µν >r

]

− j k0gµν(z, `1)∑

µ′,ν′,t′

pt′

µ′ν′(`1) < y1ψt′

µ′ν′|ψtµν >σ1

. # ),

% CVDGQ R C QN\GPoS J b J R g J K FIQ^M fRJ O C K b J _ PaS J Q^Q J ve O C K _ P > HUCXSUFIPaH J QNFV\GHNO J J Q^S QN\ KGK FIQ^M

PD b M KdJ D b CZDqS b JR g CZD RJ θ %J S K CVHQN\mPaS J OF K S J S J D \ b J R C Q EM&SUHNP J b J Q H^ RJ Q9S J Dq\GQ K CZH RJ Qb J \ &0POH^F K FVD J Q R C>_ PaS J QNQ J vR

MeC RaJ J DeS PDmQNP RJ O C R O\ R b J Q%PaDeSUM-VHUC RaJ Q(Q^\GPa_ZCVDeS J Q b FIDmD J( ve,Rb MQ^PIDGCVDeS2PD b P MH J J DeS veFI\

vR

< ve,R|ψtµν >e,r=

∫ ae,r

0

r0 dr0ve,R(r0)

∫ 2π

0

dθ0ψtµν(r0, θ0) ; # )

b g C K H Q RJ Q J K H J Q^QNPaFIDGQ 6 *) C J S > b J Q ψtµνXRhg PDeSUMIHUC RJ QN\GH θ0 J Q^S"D \ RRJK FI\GH µ 6= 0 ZJ S b J Q^\GHNOHNFaS JRaRJWJ Q^S M-eC RJ J DeS2Dq\ RaRJ K FI\GH t = s µ = 0 CVDmD J J i #3/ J S # *)'1

cFI\GH J K HNP J H R C _ PaS J QNQ J vR+q\GP b M KdJ D b JRaRJ J b \ O C K CVOFV\GQ^SNP$+q\ J p(r, θ, z) K CZH J *S J DGQNPaFID b J R C H JR CZSUPaFID SrvR = −Yrp(`)

H JR PCVDeS Rhg C b 0PaS^SYCVDmO J Yr

b \ 0POH^F K FVD J HNMO JK S J \GH0CV\ K H J QNQ^PFIDGQ J S>_ PoS J QNQ J Q CVOFV\GQ^SNP$+q\ J R FIHNQ3+q\ J RaJ O C K CZOFI\GQpSUP/+ \ J J QpS PD b M KdJ D b CVDeS b J r -R C_ PaS J QNQ JvR(r) J Q^SMOH^PaS J( M!+ 6 #

SrvR(r) = −Yr(r) < p(r, `) >θ , # )

FYr(r)

J QpS Rhg C b 0PaS^SYCVDmO J CVOFI\GQpSUP$+q\ J OPH^O\GPoS2FI\m_ J H^S b \ H^MO JK S J \GHDGFID * \GDGP/0`FVH6 J QN\GH R C Q^\GH30nCVO Jb J R C J > HUCVD J M!+ 6 8 1 -C K H J QNQ^PFID0F J DGD J Q^\GH R g CVD RJ θ < p(r, `) >θ

b JR C K H J QNQNPaFID p(r, θ, `)

!#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5

Q^\GH RJ HNMO JK S J \GH J Q^S b FIDGDmM J[K CVH

< p(r, `) >θ=1

2π

∫ 2π

0

p(r, θ0, `) dθ0 =∑

ν

pc0ν(`)ψ

c0ν(r) ,

# *) 8

R g PaDqSNM-IHNC RJ QN\GH θ0 b J Q 0`FIDGO&SUPaFIDGQ K H^F K H J Q ψtµν

M&SYCVDeSDq\ RRaJ Q K FI\GH t = s J S K FV\GH µ 6= 0 J S RaJ Q 0`FIDGO&SUPFVDGQK H^F K H J Q ψc

0ν

MSUCVDeSPaD b M KJ D b CVDeS J Q b J R g CZD RJ θ PaDGQNP XJ D QN\GPo_ZCVDeS RJ J HUCVPaQNFIDGD J J DeS +q\ J O JR \mP K H^MQ J DeSUM b CVDmQ RJK CVHUCIHUC KGJ 6 6XRJ K HNF b \GPaSQ^O C R CVPH J < vR|ψc

0ν >J QpS b FIDGDGM K CZH


∑

ν′

pc0ν′(`)Yr < ψc

0ν′ |ψc0ν >r .

#

J H JK FIHpS b J O J S^S J[J K H J QNQ^PFID b CVDGQ R g J K H J Q^QNPaFID # ) 6 b J[R C K H J Q^QNPaFID]OFID b \mPaS #

p(r, θ, z) =∑

µ,ν,t

[

j k0ρ0c01 −Kv

(g0ν(z, 0) < ve|ψc0ν >e

−Yr

Sr

g0ν(z, `)∑

ν′

p0ν′(`) < ψc0ν′ |ψc

0ν >r

)

δtcδµ0

−j k0gµν(z, `1)∑

µ′,ν′,t′

pt′

µ′ν′(`1) < y1ψtµν |ψt′

µ′ν′ >σ1

]

ψtµν(r, θ) ,

# /

F RJ QlS J H J Q < ψc0ν |ψc

0ν′ >r

b MQNPID J DqS RJ QfPDeSUMIHUC RaJ QlQ^\GH R C QN\GH 0nCVO J Sr

b J Q K H^F b \GPaSNQ ψc0ν(r)ψ

c0ν′(r)

OC R O\ R MQ b CVDGQ Rhg CVDGD J J i ) M!+ 3/'/)C > 1 J Q(S J H6 J Q < ψc

0ν |ψc0ν′ >r

J K HNP CVDeS b J Q2OFV\ KGR C J Q J DeSUH J b P MH J DeSUQ.0F b J Q ν 6= ν ′QNFVDqS SUH Q

KdJ SNPaSUQ b J _ZCVDeS RaJ Q%CV\ SUH J Q J S KdJ \ _ J DeS b FVDGO SNH J DGM R PIMQ nRaJ Q FVH b H J Q b J VHUCVD b J \GH(Q^FIDeS b FIDGDmMQ b CZDGQRaJ K CVHUCIHUC KGJ *)'1C K H J QNQ^PFID CVOFI\GQpSUP$+q\ J p(r, θ, z) J Q^S b FIDGO H J S J Dq\ J QNFV\GQ R C 0`FIH6 J&`J D10nCVO&SUFIH^PQUCZDqS Rhg PDe_ J HNQ J b JR C Q^\GH30nCZO J S = πa2 #

p(r, θ, z) =∑

µ,ν,t

[

j k0

1 −Kv

ρ0c0S

(g0ν(z, 0)S < ve|ψc0ν >e −Yr0νg0ν(z, `)p

c0ν(`)) δtcδµ0

−j k0gµν(z, `1)∑

µ′,ν′,t′

pt′

µ′ν′(`1) < y1ψtµν |ψt′

µ′ν′ >σ1

]

ψtµν(r, θ) ,

# )

FYr0ν = Yr

a2

a2r

< ψc0ν |ψc

0ν >r

7 ' /((8!#"' 5=(84*%' (8!#"$1 (%TJ J +q\ J b CZDGQ RJK CVHNCIHUC KGJ # ) > VR g J K H J Q^QNPaFID b J(R g C b EPoSNSYCZDGO J M RaJ OSNHNP/+ \ J b J SNHUCVDGQ 0 J HpS

b \ Q QpS J J Q^S MOHNPoS JYE =

1

MpeMprYT .

[ !"#$&% &# %#%#$ ' (#%#$

J O C K b J K H J QNQ^PFID J S RJ O C K b J _ PaS J QNQ J D g M&SYCVDeS K CVQ \GDGP 0`FIH6 J Q b CZDGQ R C O C _ PaSNM lRhg C b 0PaS *SYCVDmO J CZOFI\GQpSUP/+ \ J b J SNHUCVDGQ 0 J HpS HNC KGK FIH^S b \ b M > PaS b JWR g M J SNS J \GH J DOFI\mH^S * OPaHNO\mPaS CVOFI\mQ^SUP/+q\ J QN\mH R CK H J QNQ^PFID J D z = ` J QpS\mD J 0`FVDGOSNPFID b J r %TJ J +q\ J b CVDGQ RaJ]K CZHUCIHNC KGJ26 # R g C b EPoSNSUCVDGO J CVOFI\GQpSUP$+q\ J b J SNHUCVDGQ 0 J HpS DGFVH6 J ,/ , 6* ) ) ) J QpS b M2GDGP J9J D H J KGR C CZDqS RJ Q+q\ CVDeSNPaSUMQ"+q\GP b M KJ D b J DqS b J9R C _VCZHNP C >GRJ r K CVH RaJ \GH_ZC RJ \GH EF J DGD J Q^\GH R C]Q^\GH30nCVO J b \ SUHUCZDGQ b \mOS J \GH P KGR P$+q\GM `K CVH J S J DmQNPFVD b J R C]H JR CXSUPFVD ) / QUC _IFVPH #

YT =Se < ve >e + < Ye(r)p(r, 0) >e

< p(r, `) >r

, # F

Ye(r) =< Ye >e +δYe(r)J S p(r, 0) =< p(0) >e +δp(r, 0) .

CVH%Q^\GPaS J J D C b F K SYCZDqS R C J b M CZHNO GJ +q\ J b CVDGQ RaJ K CVHUCIHUC KGJ:6 # J S J D K FIQNCVDeS

Ve =< ve >e=2

a2e

∫ ae

0

ve(r0)r0 dr0 , # 6

J S

Ye =< Ye >e=2

a2e

∫ ae

0

Ye(r0)r0 dr0 , # #'

P R _ P J DeS

YT =SeVe + Ye < p(0) >e

< p(`) >r

. # , J O C R O\ R b J Rhg C b 0PaS^SYCVDmO J CVOFV\GQ^SNP$+q\ J b J SUHNCVDGQ 0 J H^S DGMO J Q^QNPaS J b FIDmO b J OC R O\ RJ H RaJ Q_ZC RJ \GHNQ EF J DGD J Qb J[R C K H J Q^QNPFVD CVOFI\GQpSUP$+q\ J Q^\GH R C J > HNCVD J b J Rhg M- J S^S J \GH z = 0

J S b \ HNMO JK S J \GH z = `1

2 D b J O C R O\ RaJ H RJ Ql_VC RJ \mHNQ EF J DGD J Q < p(0) >eJ S < p(`) >r

eR C _ZC RJ \GH 0F J DGD J < p(z) >e,rb J[R C K H J Q^QNPFVD p(r, θ, z) QN\GH R C Q J O&SUPaFID b J HNC IFID ae,r ae,r

b MQNPID CVDeSae

FI\ar

J QpSTMOHNPoS J< p(z) >e,r =

∑

µ,ν,t

ptµν(z) < ψt

µν >e,r

=∑

µ,ν,t

ptµν(z)

1

πa2e,r

∫ ae,r

0

r0 dr0

∫ 2π

0

dθ0ψtµν(r0, θ0) ; #

b g C K H Q RJ Q J K H J Q^QNPaFIDGQ 6 ) C J S > b J Q ψtµνeRhg PDeSNM-IHNC RJ Q^\GH θ0 J Q^SDq\ RRaJ%K FV\GH µ 6= 0 J S K FV\GH t = s

µ = 0 C _ZC RJ \GH 0F J DGD J < p(z) >e,r

b J _ P J DeSC R FIHNQ

< p(z) >e,r=∑

ν

pc0ν(z) < ψc

0ν >e,r , #

O J +q\GP b FIDmD J C R FIH^Q K FV\GH < p(0) >eJ S < p(`) >r

< p(0) >e=∑

ν

pc0ν(0) < ψc

0ν >e , # 8

< p(`) >r=∑

ν

pc0ν(`) < ψc

0ν >r . # 6

3 !#"$%!#&')(*'+',"-(.&"')(/(0(1.'23456!78(59:;5 5

J H JK FIHpS b J O J Q J K H J Q^QNPaFIDGQ b CVDGQ Rhg J K H J Q^QNPFVD b J2R g C b EPoSNSYCZDGO J CZOFI\GQpSUP/+ \ J b J SNHUCVDmQ30 J HpS YT

b FIDmD JC R FIHNQ

YT =

SeVe + Ye

∑

ν

pc0ν(0) < ψc

0ν >e

∑

ν

pc0ν(`) < ψc

0ν >r

. # 6 /

g C b EPoSNSYCZDGO J M RJ OSUH^P$+q\ J b J SUHNCVDGQ 0 J H^S J Q^STC R FVHNQ b FVDGDGM J K CVH

YE =1

MpeMpr

SeVe + Ye

∑

ν

pc0ν(0) < ψc

0ν >e

∑

ν

pc0ν(`) < ψc

0ν >r

. # 6 )'

J OC R O\ R b JTR g C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SUHUCZDGQ30 J H^S YED J DmMO J QNQ^PaS J b J OFIDGD C SUH J + \ JTRJ QS J H J Q

K FI\GH RaJ Q+q\ JR Q t = c J S µ = 0 R H J Q^S J b FIDGO %J KGR POPoS J H RJ Q OF J\ OP J DeSUQ pc

0ν(z)K FI\mH*+ \ JR g C b EPoSNSUCVDGO J

YE

Q^FIPaS J DeSUP H J J DqS b M&S J H60PDGM J J H JK FIHpS b J R g J K H J QNQNPaFID b J[R C K H J QNQ^PFID # ) b CVDGQ Rhg J K H J Q^QNPFVD # b FIDGD J

pc0ν(z) =

j k0

1 −Kv

ρ0c0S


c0ν(`)]

− j k0g0ν(z, `1)∑

µ′,ν′,t′

pt′

µ′ν′(`1) < y1ψc0ν |ψt′

µ′ν′ >σ1, # 6

+q\GP b J _ P J DeS J D 0nCZPQUCZDqS2\mQUC J b J[R g J K H J Q^QNPaFID 8

pc0ν(z) =

j k0

1 −Kv

ρ0c0S


c0ν(`)]

− j k0ρ0c0g0ν(z, `1)Y1

∑

ν′

pc0ν′(`1)ψ

c0ν(a)ψ

c0ν′(a) ,

# 6 6

FY1 = σ1/(ρ0c0)y1

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QNPO J S9M R M J DeS J Q^S\GD J 0 J DeS J CVDGDq\ R CZPH J J S 4Yt

QNP Rhg M R M J DeS J Q^S\GD HNMQ J CV\ b J +q\ CZSUH J SN\ >J QO R PD b HNP$+q\ J Q CVDGD J J J S HNM.0`MH J DGO J$ 8 i * ) J S * J SNS J M!+q\ CZSNPFIDEOFID b \GPaS J D K FIQUCVDeSQN\GOO J Q *Q^Pa_ J J DeS z = ` J S z = `1

b J \ H JR CXSUPFVDGQ%PD b M KdJ D b CZDqS J Q J DeSUH J pc0ν(`)

J S pc0ν(`1)

$C HNMQ^F R \ SUPFVDb J O J Q QpS J b g M.+ \GCZSUPaFIDGQ b FIDGD J0RJ Q[OF J\ OP J DeSUQ pc

0ν(`)J S pc

0ν(`1) O J +q\GP KdJ H6 J S b g J D b M b \GPH J

RaJ Q(OF J&\ OP J DqSNQ pc0ν(z)

J S b FIDGO RJ QOF J\ OP J DeSUQ pc0ν(0) DmMO J QNQNCVPH J Q C _ J O RaJ Q pc

0ν(`)K FI\mH RJ O C R O\ R b J

R g C b EPoSNSYCZDGO J M RJ OSUH^P$+q\ J b J SNHUCVDGQ 0 J HpS YE

+-, . , 3BADCT 5VL 58A 5V73B9XE8HC<BA <65 JE8<>@H<E8A EF3BADCT 5VLX5VA < J 3EFH>@<BMS5VLZC J QlHNMQN\ R SYCZSNQ"S MFIHNP/+q\ J QfF > S J Dq\GQ K FI\GH RJ Q b J \ 0F b RJ Q b MS J H60PDmMQ b CZDGQfO J O C K PoSUH J 0F b RJ

> CZQNM[QN\GH R C K HNF K CeCZSNPFID b g FID b J Q KGR CVD J Q%M.+ $#/ # J S EF b RJ > CVQNM[QN\mH R C 0`FIH \ R CXSUPFVD PDeSNM-IHNC RJ M.+ # 6 )' Q^FIDeS F > Q J H^_VMQ J S OF' K CVH^MQH J Q KdJ O&SUPo_ J J DeS b CVDGQ RaJ Q K CVHNCIHNC KGJ Q]C J S > J Q HNMQN\ R SYCZSNQS MFIH^P$+q\ J QQ^FIDeSOF' K CVHNMQ b J Q K H J EP J H^Q%HNMQ^\ R SUCZSUQ J K MH^P J DeSYCZ\ b CVDmQ RaJ K CVHNCIHUC KGJ O

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SUP/+ \ J Q b J SUHNCVDGQ30 J H^S b J[R C OC _ PoSUM b J OFV\ KGR C J DmFVSUMQ VT

b M2GDGPQ2OF J QN\mPaS #

VT =γRPR

jωYT ,

FγR

J S PR

Q^FIDeSTH J Q KdJ O&SUPo_ J J DeS RJ HUC KGK FIHpS b J Q O C K CZOPaSNMQ O C R FIHNP32+q\ J Q CpJ S Cv

CZQNQNP/+q\ J Q b JRhg CVPH J S R C K H J Q^QNPaFID]QpSYCZSNP$+q\ J b CVDmQ RaJ Q%OFID b PoSUPFVDGQ%Q^SUCVD b CVH b]b J H^M.0`MH J DmO J

cFI\GH b J Q OF' 0F b PaSNMQ b g J KGR FIPoSYCZSNPFID b J Q H^MQ^\ R SUCZSUQ RJ _IF R \ J b J SNHUCVDGQ 0 J HpS VTJ Q^S DGFVH6EC R PaQNM

K CVH>QUC _ZC RJ \GH CZ\ > CVQNQ J Q"0`HNM.+q\ J DGO J Q J D R g C > Q J DmO J b g M R M- J DeSUQ CVOFI\mQ^SUP/+q\ J Q J D K CVHNFVPQ J S b g J<J SNQS GJ H60P$+q\ J Q b M2 DmP K CVH i # )' #

VTbf0 = V + Vbfe + Vbfr .

J Q OFV\GH >dJ Q b JR C2V\GH J # # H JK H^MQ J DeS J DeS RJ QL_IF R \ J Q DmFIH6EC R PaQNMQ VT/VTbf0

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b J \ EF b RJ Q K Q^P$+q\ J QTC _ J O \GD OFV\ KGRJ \GH2QUCVDmQ M R M J DqSCVOFI\GQpSUP$+q\ J[J D K CZHNFIP J Q"OFV\GH >dJ Q b J Q 2V\GH J Q #-, # 8 H JK H^MQ J DeS J DeS RJ Qc_VF R \ J Q CVOFV\GQ^SNP$+q\ J Q b J SUHNCVDGQ 0 J H^S DGFIH C R PQ^MQ

VT/VTbf0

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L2

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J S C3

H JK HNMQ J DqS J DeS RJ QO C _ PaSNMQ Rhg CVH^HNP H Jb J Q KdJ SNPaSNQ%M R M J DeSUQ CVOFV\GQ^SNP$+q\ J Q

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b J \ 0F b RJ Q K Q^P$+q\ J QTC _ J O>\mD]OFI\ KGRaJ \GH2OF K FIH^SUCVDeS J D K CVH^FIP \GD J 0 J DqS J CVDGDq\ R CZPH J FV\m_ J H^S J

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\GHfO C+q\ J 2I\GH J b J ## # 8eRaJ QfHNMQN\ R SYCZSNQF > S J Dq\GQ C _ J O RJ Q b J \ (0F b RJ QfQNFIDeSSUH Q K HNF O GJ Q J Q MOCVH^SNQ J DeSUH J O J Q HNMQN\ R SYCZSNQ Q^FIDeS K HNMOPQ^MQ b CVDGQ RaJ K CVHUCIHUC KGJA>

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Q JR FID b J \ &0F b RaJ Q K QNP/+q\ J QC _ J O \mD OFI\ KGRaJ \GH%OF' K FIHpSYCVDeS J D K CVH^FIP+q\ CZSNH J SN\ >J Q%O R PaD b H^P$+q\ J QFI\m_ J H^SNQ-

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Q JR FID b J \ &0F b RaJ Q K QNP/+q\ J QC _ J O \mD OFI\ KGRaJ \GH%OF' K FIHpSYCVDeS J D K CVH^FIP+q\ CZSNH J SN\ >J Q%O R PaD b H^P$+q\ J QO CVHIMQ K CVH2\GD_VF R \ J CVH^HNP H J

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C]OF K CZHUCVPaQNFID b J Q OFI\GH >J Q b J9R C2I\mH J # # C _ J O9O JRRaJ Q b J Q42I\GH J Q #-, # 8 0FIDeSUH J +q\ Jb CZDGQ R C K CVHpSUP J > CZQNQ J b J(R C eC( J b J 0`HNM.+ \ J DGO J Q O C+q\ J OF K FIQUCZDqS CVOFI\GQ^SNP$+q\ JJ D K CVH^FIPqPDG\ J DGO J0`FIHpS J J DqS R C _ZC RaJ \GH b \ _IF R \ J DGFVH6EC R PaQNM VT/VTbf0

D HUC K P b J OC R O\ R b \]_VF R \ J DGFIH C R PQ^M[CV\ > CZQNQ J Q 0`H^M!+q\ J DGO J Q VTbf/VTbf0

eKJ H6 J S b J 0P J \ OF' K H J D b H J O J SNS J PaD \ J DGO J g J K H J Q^QNPaFID #/ #' J D Q^\ KGK FIQUCZDqS kz`

KdJ SUPoS J S OF' K S J S J Dq\ b J O J +q\ JRJ Q C b 0PaS^SYCVDGO J Q b J Q 0POHNF K FID J Q Ye Yr

J SR g C b EPoSNSYCZDGO J b J K CVH^FIP Yp

Q^FIDeS KdJ SNPaS J Q OFVD b \GPoS MOHNPaH J[Rhg C b 0PaS^SYCVDGO J CVOFI\GQ^SNP$+q\ J b J SUHUCZDGQ30 J H^S b \Q QpS J CV\ > CVQNQ J Q 0`HNM!+q\ J DGO J QTQ^FI\GQ R C0`FIH J Q^\GPa_ZCVDeS J$#

YTbf ≈ j kz` Yi + 2Yp + Ye + Yr + Y1 ,

+q\GP OFIHNH J Q K FID b R CAQ^F'( J b J R g C b EPoSNSUCVDGO J CVOFV\GQ^SNP$+q\ J b J SUHNCVDGQ 0 J H^S YTbf0 =jω VTbf0

γRPR

QUCZDGQOF' K FIQNCVDeS[CVOFV\GQ^SNP$+q\ J QN\mH RJ Q K CVHNFVPQ b J R C OC)_ PoSUM M.+ J S b J Rhg C b 0PaS^SYCVDmO J b g J DeSUHNM J Y1

b J QKdJ SNPaSUQOF K FIQUCZDqSNQ J D K CVH^FIP J _VF R \ J b J SNHUCVDmQ30 J HpS(DGFVH6EC R PaQNM2CV\ > CZQNQ J Q 0`HNM.+q\ J DGO J Q VTbf/VTbf0KdJ \mSC R FIHNQ%Q g MOH^PH J

VTbf

VTbf0

≈ 1 +γRPR

jω VTbf0

Y1 .

J SNS J J K H J Q^QNPFVD b \ _IF R \ J b J SNHUCVDmQ30 J HpS DGFIH6EC R PQNM KJ H6 J S b J 0nCVPH JWKGR \GQNP J \GH^Q H J ECVH+q\ J QQ^\GH RJ QOFI\GH >dJ Q b J Q 2I\GH J Q # , # 8 #

^\mQ+q\ g ]K H Q b J / cR C b MOH^FIPQ^QUCVDmO J0R PDmM CVPaH J`J DAMO GJRaRJ R FeCVHNPoS 0P$+q\ J b \_IF R \ J b JSNHUCVDmQ30 J HpS(P K FIQNM J2K CVH R C K HNMQ J DGO J b JR C0 J DeS J FI\m_ J H^S J EFVDqSNHNM JK CZH R C 2I\mH J # , OFIH^H J Q K FID b> P J D (J D MO mJRRaJ R F'ICVHNPoS 0P$+q\ J CV\ OF' K FVH^S J J DeSHNMQNPQpSUP 0LCXSNS J D b \ b J O JRRaJ8* OP Y1

H^M JR F > Q J H^_ZCZSUPaFID +q\GPQ g C KGKGR P$+q\ J MeC RaJ J DeS b CZDGQ RJ O CVQ F RJ OF' K FIQUCVDeS(CVOFI\mQ^SUP/+q\ JWJ D K CVHNFIP J QpS\mD]H^MQ J CV\ b J +q\ CZSUH J SU\ >dJ Q CVPaQ ^\GQ3+ \ g +q\ JR + \ J Q2O J DqSUCVPD J Q b J [Q J \ RJ J DeS 2I\mH J #

+q\ CZD b RaJ Q KdJ SNPaSUQ M R M J DqSNQ0CVOFI\mQ^SUP/+q\ J Q b M > FI\mO GJ DeSEQN\GH \GD J OC)_ PoSUM J D > CVQNQ J Q 0`HNM.+ \ J DGO J QR g J8 J S b \ _VF R \ J b J O JRRaJ8* OP J Q^S K HNM K FID b MHNCVDeS b J _ZCVDeS R g J8 J S9HNMQNPaQ^SUP 0 b J R C 0 J DqS J FI\ b J QSN\ >dJ Q O J +q\GP J Q^S2H JK HNMQ J DqSNM[QN\GH RaJ Q(OFI\GH >dJ Q b J Q.2I\mH J Q # J S # 8 K CVH RaJ 0nCVPoS +q\ J RJ _IF R \ Jb J SNHUCVDmQ30 J HpS2DGFIH C R PQ^MS J D b b CZDGQ RJ Q > CVQNQ J Q 0`HNM!+q\ J DGO J Q _ J HNQ \GD J _ZC RJ \GH(OFIDGQ^SUCVDeS J +q\GP J QpSMeC RaJ CV\_IF R \ J b J O J S^S J O C _ PaSUM

CVHCVP RRJ \GHNQ P R J Q^ST_ PaQNP >mRJ +q\ J Rhg PD \ J DmO J b J O J QWM R M- J DeSNQTCVOFI\GQpSUP$+q\ J Q J D K CVH^FIPcQ^\GH R C _ZC RaJ \GHb \E_IF R \ J b J SNHUCVDmQ30 J HpS VT/VTbf0

J S b FIDGOQN\GHO JRRJ b JWR g C b EPoSNSUCVDGO J M RJ OSUH^P$+q\ J b J SNHUCVDGQ 0 J HpS YE =YT/(MpeMpr)

mJ Q^S%P K FIHpSYCVDeS J O CZH JRaRJ KJ \mS b M K CVQ^Q J H 6 b K FI\GH RaJ S KdJ b g M R M J DeSUQ%CVOFI\GQpSUP$+q\ J QM&SU\ b PaMQ2POP g PaD \ J DGO J PaD b Po_ P b \ JRRJ b J O C+q\ J M R M J DqS CVOFV\GQ^SNP$+q\ J J D K CVH^FIPM&SYCVDeSLD J S^S J J DeS ECVH3+ \mM JR C EPaQ J J D 1[\m_ H J b J R C(EM&S F b J K H^F K FIQNM J b CVDGQTO J O C K PaSNH J K FI\GHO CVHNCVOSNMHNPaQ J HWPD b Pa_ P b \ JRRaJ J DeSRaJ Q KdJ SUPoSUQ(M R M J DqSNQ2CVOFV\GQ^SNP$+q\ J Q b J R g FVH J P RRaJ CZH^SUP32 OP JRaRJ Q J >GRJ b FVDGOWSUFI\ S 0nCVPaS J De_ PQUC J C >mRJ b CZDGQR C K HUCZSNP$+q\ J R H J QpS J SUFI\ S J 0`FVPQ 9K HNF K FIQ J H \GD K HNFZSUF OF RaJLK FI\GH J S^SUH JLJ D K HUCXSUP$+q\ J O J S^S J 0MS F b JJ S RaJ DGPa_ J CV\ b J[K H^MOPaQNPaFID2+ \ g JRRJ K FI\GH^HUCVPoS KdJ H6 J S^SUH J b g CZS^S J PD b H J J SNS J M&SU\ b J[J K MH^P J DeSYC RJ D g CK CZQ K \ SUH J9K FV\GHNQ^\GPa_ P J Q K FI\GHLC RRaJ H ^\GQ3+q\ g R C J Q^\GH JEK HNF K H J J DeS b PaS JEJ D HUCVPaQNFID b J b P \ O\ R SUMQS J O DGP/+q\ J Q )":'"' 0,' 5 ' M!#' /((8!#"' 5=(84* '%(8!#"$1 (

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% CVDGQ RaJ K CVHNCIHNC KGJ TWT R C QNF R \mSUPaFID K H^F K FVQNM J J Q^S PaQNQ^\ J b J R CAHNMQNF R \mSUPaFID b J R g M.+q\ CZSUPaFID b Jb P\GQ^PFID b JER C O C RJ \GH T T RaJEK HNF O J Q^QN\GQ b J b P\GQNPaFIDSNHNFI\ _VCZDqS>QNFVD FIHNPIPD J b CVDGQ RJ O C K b JK H J QNQ^PFIDECVOFV\GQ^SNP$+q\ J p qRJ O C K b J S J K MHUCZSN\GH J τ1 b J _VCZDqS b J QN\GH^OH^FSQUCXSUPQ 0nCVPH J" b J QOFID b PaSUPaFIDGQCV\ R P0PaS J Q b J % PH^PO GRaJ S F'0F' D J Q J S b J Q2OFVD b PaSNPFIDmQ%PDGPoSUP C RJ QDq\ RRaJ Q$#

(

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τ1 = 0 K FV\GH ~r ∈ S , t > 0 , 3/3 τ1 = 0 K FV\GH ~r ∈ D , t = 0 , 3/ O

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g(~r, t)#

τ2 = g ∗t

(

γ − 1

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,

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g = 0 K FI\GH ~r ∈ S , t > 0 , ) g = 1 K FI\GH ~r ∈ D , t = 0 . )ZO

cFI\mHc_IMHNP32 J H R g M.+q\GPa_ZC RaJ DGO J b J O J Q b J \ K HNMQ J DqSUCZSUPaFIDGQ P R Q^\ \ S b J H J ECVH+q\ J H +q\ g JRRaJ QcQNFIDeS R g \GD JJ S Rhg CV\mSNH J b J Q(OCVQ K CZH^SUPaO\ R P J HNQ b g \GD& J K H^FF R J IMDmMHUC R O J Q(QNF R \mSUPaFIDGQ K CVH^SNPO\ R P H J QCVQNQ^F OPM J QM&SYCVDeS RJ Q - J Q J[K HNFF R J IMDGMHNC R OFVDGQNP b MH^M J Q^S b FIDGDmM K CVH$#

(

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τg(~r, t) = f(~r, t) K FV\GH ~r ∈ D , t > 0 , TIC

τg(~r, t) = 0 K FV\GH ~r ∈ S , t > 0 , WTτg(~r, t) = τi(~r)

K FV\GH ~r ∈ D , t = 0 . TVO J[K HNF R J CV\_ZC RJ \GHNQ K H^F K H J Q CVQNQ^F OPM J QpS b J[R C0`FIH J

(∆ + k2m)ψm(~r) = 0 K FI\GH ~r ∈ D , C

ψm(~r) = 0 K FV\GH ~r ∈ S . C 0`FVDGOSNPFID

τg(~r, t)KdJ \mSC R FIHNQ SUH J b MOF' K FIQ^M J b J[R C0nCFID QN\GPo_VCZDqS J

τg(~r, t) =∑

m

< ψm|τg > ψm(~r) , #'F

< ψm|τg > J Q^S R C K H^F J O&SUPaFID b J τgQ^\GH RJ _ J OS J \GH K HNF K H J ψm

#

< ψm|τg >=

∫∫∫

Dψm(~r)τg(~r, t) d~r .

!

C K HNF J OSUPaFID b J[Rhg M!+q\ CZSNPFID WTIC' QN\GH R C!GCVQ J b J Q%_ J OS J \GH^Q K H^F K H J Q b FIDGD J1

c0`h

∂

∂t< ψm|τg > − < ψm|∆τg >=< ψm|f > ,

F mJ D 0nCZPQUCZDqS \GQUC J b \9S MFIH J b J H JJ D J S%OF K S J S J Dq\ b J Q(OFID b PoSUPaFIDGQ(CZ\ 0`H^FIDeSUP H J Q T (QUCXSUPQ 0nCVPaS J Q K CZH ψmJ S τg OFID b PaSUPaFIDGQ b J0% PHNPaO GRJ S F'0F' D J Q < ψm|∆τg >=< ∆ψm|τg >= −k2

m < ψm|τg > ,O J +q\GP b FVDGD J C R FVHNQ RaJ K HNFF R J K H^F J SNM

(

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m

)

< ψm|τg >=< ψm|f > K FI\GH t > 0 , -,IC < ψm|τg >=< ψm|τi > K FI\GH t = 0 , -,

b FVDqS R C Q^F R \ SUPFVDQ g MOH^PaS

< ψm|τg >= e−c0`h k2mt

[

< ψm|τi > +

∫ t

0

c0`h ec0`h k2mt0 < ψm|f > dt0

]

, U

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O J +q\GP b FIDmD J[Rhg J K H J QNQ^PFID b J[R C"0`FIDGO&SUPaFIDτg(~r, t)

#

τg(~r, t) =∑

m

e−c0`h k2mt

[

< ψm|τi > +

∫ t

0

c0`h ec0`h k2mt0 < ψm|f > dt0

]

ψm(~r) . '

( DGFVS J H +q\ J O J SNS J J K H J QNQ^PFID IMDGMHNC RJ>KJ H6 J S b J H J SNHNFI\m_ J H R C 0`FIDGO&SUPFVD b J H JJ D G(~r, ~r0; t)b \ K HNF R J( / J D K FIQUCZDqSf(~r, t) = δ(~r − ~r0)δ(t)

J S τi(~r) = 0 ,

O J +q\GP b FIDmD JG(~r, ~r0; t) =

∑

m

c0`h e−c0`h k2mtψm(~r0)ψm(~r) . 8

B $

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τ1 =∑

m

e−c0`h k2mtψm(~r) < ψm >

γ − 1

βγ

∫ t

0

ec0`h k2mt0

∂

∂t0p(t0) dt0 ,

3/

F< ψm >=

∫∫∫

Dψm(~r) d~r

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τg J D K FIQUCVDeS

f(~r, t) = 0 J S τi(~r) = 1 ,O J +q\GP b FIDmD J

g(~r, t) =∑

m

e−c0`h k2mtψm(~r) < ψm > . /'/

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τ2(~r, t) = g ∗t

(

γ − 1

βγ

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=γ − 1

βγ

∫ t

0

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∂t0p(t0) dt0

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m

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γ − 1

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∫ t

0

ec0`h k2mt0

∂

∂t0p dt0 .

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1

c0`h

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1

c0`hδ(t)

6,

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n=1

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exp−σn2π2t

a2 ,

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r = 0\An = −Bn

]

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−σn2π2t

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a= 1 ,

/'/ 8

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+!M*" _@1 ( .H. *G ;="P"" !4.2* 0/" 1C.!" sin

nπr

a

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\ % ) ]1 +!6* G

τ\ T ) )T-] _

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∂t,

\ % ) #]

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\ T ) 8 ] )65M $ ( *τG>* ( ! D ;=1!43_61 ( E. *F1

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*:>&" ,a! ( !" < g >

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9

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n=1

6

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1

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9X2

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X

I%* " @\σ = c0`h

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.

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)

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(σt)2)

,\) ^]

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x

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∫ ∞

x

e−t2 dt .

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(σt)1/2A

V+

1

3σtA2

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(σt)2)

.\) )"]

59;= ,,! *G:$ ( " < τ >

*τ,:$" (

V

< τ >=< g > ∗t∂ θg

∂t,*" _@*@ 2B* ( ! H* @"!_

< τ >= jω < g(ω) > θg(ω) = jωγ − 1

βγpg < g(ω) > .

5ME$ ( " < τ >

;D1,!%_ *@ B* ( ! H* @" !_< τ >≈ γ − 1

βγpg

[

1 −X +3

4

1

3X2 + O

(

X4)

]

.\) MT-]


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)

g = 0&"

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g = 1&"

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1

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1

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=2

π2

+∞∑

n=0

1

(n + 1/2)2

1

1 + (n+ 1/2)2π2X2 ,\) U ]

X

I%* " @\σ = c0`h

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A2

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.

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I8* " *<;D 2 ;= ,,!" \ T)WT U ]C_

< g >= 1 − 2

`(σt)1/2

[

2√π

+ 4

+∞∑

n=1

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2

1

(σt)1/2

]

,\) "]

1 +!6& ( C*G** !< τ >

_ [ P"!P" -+H ( G '!30/1_< τ >≈ γ − 1

βγpg(1 −X) .

\ ) -]

# % % )

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a) ( 7 E*2 ( C.!1 ! * !$+* * ( ! B*<;DC.* _@B F ( >\ T )-];D1!42 '1?"* 1?! *!/+ #

(

∂ 2

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σ

∂

∂t

)

g = 0"

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g = 0&"

r = a , t > 0 ,\ ) ) ]

g = 1&"

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W ( ( +G,1* (>( -3_ +" .!" +*G1 F ( B" -2*" %@g(r, t) = ϕ(r, t) ,

:0/" 1C.! ϕ(r, t)

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*G 0/ 1!" g;=1!2* 1_@*<;D \ T)*) "]C_

< g >=

+∞∑

n=1

4

λ2n

exp−λ2

n

a2

A2

V 2 σt ,\) ) ^]

B&",A/V

$ 2/a

_@"!4+*@ B* ( ! B* @" !_

< g >=+∞∑

n=1

4

λ2n

1

jω +λ2

n

a2

A2

V 2 σ

,\) ) )"]

1&+!6*" G" +:E ;=1,! ,. -J B* G ( .τ_

< τ > = jωγ − 1

βγpg < g >

=+∞∑

n=1

4

λ2n

1

1 +λ2

n

a2 X2

,\) )T-]

X

,%*" G@\σ = c0`h

]X2 =

A2

V 2

c0`hjω

.

5AB," !" " 1"I7 ( +@ .. 30/" ( +*25M:1#*W$ ( " < g >

,7*" *<;D ;= ,!" \ T ) T ^] _< g >≈ 1 − 2√

π

A

V(σt)1/2 +

1

4

A2

V 2 σt +1

24√π

A3

V 3 (σt)3/2 ,\) ) ]

1&+!6& ( * * *!G:$ ( " < τ >

< τ >≈ γ − 1

βγpg

[

1 −X +1

4X2 +

3

4

√π

1

24√πX3

]

.\) )"]


K $ &

56* ( ! G*<;=. * D ,W . ! *B**! ( !" `x, `y, `z

)&56.&"I%$" ( ,30`1$V

A=

2

1/`x + 1/`y + 1/`z,

C%.,.R1

CR2

," -%* @ !@

R1 =`x`y

R2 =

`x`z

.

56BF ( >\ T) -] [1*" %1,! ;=1!(

∂ 2

∂x2 +∂ 2

∂y2 +∂ 2

∂z2 − 1

σ

∂

∂t

)

g = 0\)*),-]

&"x ∈ (0, `x)

_y ∈ (0, `y)

_z ∈ (0, `z)

t > 0

_g = 0

" x = 0

`x , y = 0

"`y , z = 0

"`z , t > 0 ,

\)*), ]g = 1

" x ∈ (0, `x) , y ∈ (0, `y) , z ∈ (0, `z) , t = 0 .

\) ),13]5MF,".!

gI21 ,1 " 0/ ( H*<;= &0/" 1!" " $!#,@#

g(x, y, z, t) = ψx(x, t)ψx(y, t)ψx(z, t) ,

ψx(x, t)

_ψy(y, t)

Cψz(z, t)

" -*" .!" G* F ( [\ T ) ,-]H&"3+ ;D#>**! ! ',2&1C.!$ ( -x_yz)

T ,:,1 T 3<U9XBT9 X YX6ZS5 @9;:LZT5M$ ( B+W$"" (

V* 0/" 1!"

ψx

_ψy

ψz

," -*" +@#H* $"" & ( - ( ?*@F\ T)*) S] #

< ψx,y,z >=2

π2

+∞∑

n=0

1

(n+ 1/2)2 exp−(n + 1/2)2π2 4

`2x,y,z

σt .\)*) U ]

56,* !%*ψx,y,z

*" a"2 ;= ,,! *g)&5M>$ ( %G$"" (

V*

g;=1,!%"_.131 0`!_

< g >=8

π6

+∞∑

m=0

+∞∑

n=0

+∞∑

p=0

1

(m + 1/2)2(n + 1/2)2(p+ 1/2)2 e−π2A2

V 2 σt

exp(m+ 1/2)2 + (n+ 1/2)2R2

1 + (p+ 1/2)2R22

(1 +R1 +R2)2,

\)*) "],"!*@ B* ( ! H* @",!_

# ( #$ &# 3 )

< g > =8

π6

+∞∑

m=0

+∞∑

n=0

+∞∑

p=0

1

(m + 1/2)2(n + 1/2)2(p+ 1/2)2

1

jω + π2A2

V 2

(m + 1/2)2 + (n+ 1/2)2R21 + (p+ 1/2)2R2

2

(1 +R1 +R2)2 σ

,\) )-]

1&+!6*" G" < τ >

< τ > =γ − 1

βγpg

8

π6

+∞∑

m=0

+∞∑

n=0

+∞∑

p=0

1

(m + 1/2)2(n + 1/2)2(p + 1/2)2

1

1 + π2 (m+ 1/2)2 + (n+ 1/2)2R21 + (p+ 1/2)2R2

2

(1 +R1 +R2)2 X2

.\) T-]

T , , 9ZTUC 5VHU97Z @O69ZC EV5 5XEYX LXE 5DEFL9Z:A o9XE@36X ;WX KL TULNOXW ( ( +29:H !" ( *@_:H," !" " 1,9. ( +@0. 0/" ( *25M1I%* *!4.G* *C$""& ( -a\ T ) T U ] )@56+$ ( "

< ψx,y,z >;D1!4$"

< ψx,y,z >≈ 1 − 2√π

2

`x,y,z(σt)1/2 .

\) T ^]5MG,* !+*

ψx,y,z

* #E$ ( " *< g >

#

< g > ≈ 1 − 2√π

(σt)1/2A

V+

4(R1 +R2 +R1R2)

π(1 +R1 +R2)2 σt

A2

V 2

+−8R1R2

π3/2(1 +R1 +R2)3

(σt)3/2A3

V 3 ,\) T#)"]

1&+!6*" G" +:a$ ( < τ >

_

< τ >≈ γ − 1

βγpg

[

1 −X +4(R1 +R2 +R1R2)

π(1 +R1 +R2)2 X2 +

−8R1R2

π3/2(1 +R1 +R2)3X3

]

.\) TFT-]

K

59;= ,,! '* B:$ ( " < τ >

& C.,H1!_*@ +W* ( ! W* @" !_:0/" ( BP .>\ T ) ))WT"] #< τ >= θg(ω)EP ,

\) T ]

Ep

$


EP =+∞∑

n=n0

DNn

MnX2 ,

\)WT]" !

EP = 1 −X +G1X2 +

3

4

√πG2X

3 ,\)WT ,"]

O7. ( D_n0

_Nn

_Mn

_G1

CG2

*& * -*HP" ( C.!%* R* ( ! )59;=! *!1n&,, -.% !+! *!1)56aJ3 )>*" EH$,B*F1* ! -.B$,!:H&"W,!BP" ( .,!B** ( ! H*<;=*)

D n0 Nn Mn G1 G2

6/π2 1

n2

n2π2

9

1

3 ! #"$%&$"

2/π2 ' 1

(n + 1/2)2(n + 1/2)2π2

(*)+ #"$,%&$" - 1

λ2

n

λ2n

4

1

4

1

24√

π

(*)+ /.01328/π2 40',5 76 1

λ2

n(m + 1/2)2(m + 1/2)2π2 + λ2

nR

(2R + 1)28R + πR2

π(2R + 1)2R3 − 6R2

3√

π(2R + 1)3

8:9 9 ;; 2 ; 2"<=8/π6 40',5 ',5 ' 6 Nmnp > Mmnp >> 4(R1 + R2 + R1R2)

π(1 + R1 + R2)2−8R1R2

π3/2(1 + R1 + R2)3

> Nmnp =1

(m + 1/2)2(n + 1/2)2(p + 1/2)2>> Mmnp = π2

(m + 1/2)2 + (n + 1/2)2R2

1+ (p + 1/2)2R2

2

(1 + R1 + R2)2

X Y2[\& ) 7,2* * ! -.1" ,. -.%, [aP" ( !H*'* ( ! W*<;DC.* )


5M/" 1C.!" * , .!!,!1!," -+/" 1.!" #* ,</ 1!" 8*, ( *! @1?! *!*G, ( ! ,B, 1) K !

5A"/" 1C.! Jν(z)

_3$1z ∈ C

_, #/" 1.!" *$ 1 ! * !%7*, ( ! 7&1+*<;="*,ν_@ !" O*G ;D&?.!"

∂ 2f

∂z2 +1

z

∂ f

∂z+

(

1 − ν2

z2

)

f = 0 . ' +))(^]5A*/ 1!"

Iν(z)_-3$1

z ∈ C_"* !P" ++/" 1C.! E*",@1?! *! ( *! @*+ ( ! ,, 1*<;D*, ν !" O*G ;D&@.!"

∂ 2f

∂z2 +1

z

∂ f

∂z−(

1 +ν2

z2

)

f = 0 . ' +)-,"] . $/ 10 2 $3

4 "!#* 5/ 1!" +* 68,71" 8? Zν(z)

CBν(z)

* ;D",*ν_@1"/" 1C.! O", * -+" !C.2* ;D! P".!" O !$ -. (MT _:9);9<,)(_=9)-9 ))(C*,#"

∫

zν+1Zν(z) dz = zν+1Zν+1(z) , ' +)WT"]∫

zZν(az)Bν(bz) dz =bzZν(az)Bν−1(bz) − azZν−1(az)Bν(bz)

a2 − b2, ' +) ]

∫

z [Zν(az)]2 dz =

z2

2

[

[Zν(az)]2 − Zν−1(az)Zν+1(az)

]

. ' +);9"]5A>/" 1.!" *?,

Zν(z)&" * '" G:a,",!B!4$ -. (3T _:@ ) U (") T"∂ n

∂z(zνZν(x)) = zνZν−n(z) . ' +)-A"]

(B, U

2R EG4)?HE+5IKJMI1 JG+ N5/1 @H/1 4602DJM+.460H1@B/ /131 /N

K $/ 2 ! 56B*$ ( -H ( .!B*G:>/" 1C.!" O*?, ( * ! @

Iν(z);D1!4 ,_ U );9)-, U "

Iν(z) =ez

√2πz

[

1 − 4ν2 − 12

1!8z+

(4ν2 − 12)(4ν2 − 32)

2!(8z)2 − ...

]

+e−z±jπ(ν+1/2)

√2πz

[

1 +4ν2 − 12

1!8z+

(4ν2 − 12)(4ν2 − 32)

2!(8z)2 + ...

]

. ' +) U ];D ., @I3_2 *$ ( - ( ..!%* : /" 1C.! * 8,

Jν(z);=1! ,?_

U )-9))( A"

Jν(z) =

√

2

πz[Pν(z) cosϕ−Qν(z) sinϕ] , ' +) @-]3$"1

Pν(z) ≈ 1 − (4ν2 − 12)(4ν2 − 32)

2!(8z)2 +(4ν2 − 12)(4ν2 − 32)(4ν2 − 52)(4ν2 − 72)

4!(8z)4 − ... , ' +)-]

Qν(z) ≈4ν2 − 12

1!8z− (4ν2 − 12)(4ν2 − 32)(4ν2 − 52)

3!(8z)3 + ... , ' +) ]ϕ = z − (ν + 1/2)π/2 . ' +)13]

56B*$ ( - ' ) @-]& % !M;D1!Jν(z) =

1√2πz

[

ej ϕ[Pν(z) + jQν(z)] + e−jϕ[Pν(z) − jQν(z)]]

,

,"!Jν(z) =

1√2πz

[

ej ze−j (ν+1/2)π

2 Aν(z) + e−j zej (ν+1/2)π

2 Bν(z)

]

, ' +)("]

Aν(z) = Pν(z) + jQν(z)

Bν(z) = Pν(z) − jQν(z) .

K $/ 2 ! Kh

/ 0 +/ 3

4 "!4+ . &G*G. "" a_@H (

Kh

;=1!4 ' ,))( @-]Kh =

2

kha

J1(kha)

J0(kha), ' +)( (S]

kha =

1 − j√2

√

k0/`h a .

Kh! "$#&%' ( *) +

4 "!z = kha

_z " H1!4#,"+:?/" (

z = (1− j )α+ ;D !" ' +))(-]8* " *" 1

Jν(z) =1√2πz

[

eαej αe−j (ν+1/2)π

2 An(z) + e−αe−jαej (ν+1/2)π

2 Bn(z)

]

. ' ))(B,"] ( (

αI2P". *<_ ;= &" !

e−α ,%C.!B2 ;= ,,! ' +)(B,]+* $?! Jν(z) =

1√2πz

eαejαe−j (ν+1/2)π

2 [Pν(z) + jQν(z)] ,

"!43_ " ;D",*?9, ,

Jν(z) =ej z

√2πz

e−j (ν+1/2)π

2

[

1 − 4ν2 − 12

1!8j z+

(4ν2 − 12)(4ν2 − 32)

2!(8j z)2

]

. ' ))(MT-](W! !<$/" 1C.! +*?,

J0(z)CJ1(z)

;D1!4$"

J0(z) =ej z

√2πz

e−j π

4

[

1 +1

8j z+

9

128(j z)2

]

, ' ))( ]

J1(z) = −jej z

√2πz

e−j π

4

[

1 − 3

8j z− 15

128(j z)2

]

. ' ))(B9"]0 [,&",. -12*C$""& ( -._

z = khR_* ;= ,,!" '*

Kh

_@! $?! Kh =

2

j khR

(

1 − 1

2j khR− 1

8(j khR)2

)

. ' ))( A-]50;D ,!" [*

KvI2: ( ( ? 1H*

Kh ' ( -3

kh@

kv)

R

(MT8(

< L2N5/ @B/ 1 JMEGLO0H1 4)E#-I /1A@B/CQ L ?\N5LODF/

<

E aIH 1 ( ( -W* @.P". !T ) ,H ,"@P- .! [*<;=" * H )5A1 ( &". -N.*!: * R ;=" * ,F1" ,$_0 ,"@P- .! ,F C.,R1 *<;=" *@! )5M+4J#. $"2!1!&" - ( ( G1 N. $"2* [email protected]". T ) _1!6& ( * , ,.! @ 3! ( .!" ! O 2C.`!)

H ( !B. ( _6 @ .! G* "@P-.!" *> ;D1" ,.!F! 3!E 1!4.*@ G * ( ! D 1" ' ))(^]C)9B" - ,! ,* E13*<;= 1?! */ ( " B ?. ( !4.H@H* ( !1" W*<;D* ( !,J 1%&1!$

YeYr_ ;= * ( ! 1,"" B. - ! ( * ;D ( " $ ( -G*E$?!.,

ve ' ) T-] _ 11!0*@ B G*E* ( ! ;D* ( !,J 1H1"I.!%G*B. /I%* [ I ( ' ) ]C) 3 / 0 /

5MB @!" B! 3! ( "P HP" $" @ ( $" ( G* !*" -1H*" *@ [email protected]". T ))(#1

c0

∂ ~v

∂t+

1

ρ0c0~grad p = `v ~grad div ~v − ` ′

v~rot ~rot ~v ,

ρ0c0 div ~v +γ

c0

∂

∂t(p− βτ) = 0 ,

(

1

c0

∂

∂t− `h ∆

)

τ =γ − 1

βγ

1

c0

∂ p

∂t.

5M1 @ ( '*$?!,@,!1:!~v,%1!4%1 (>( G:F &"!4.! '* ;D [1 @ ( *G$?!,." ! @!

~vv

2*<;= [1 ( '*B$?!,G: ( ! @!~v`

#

~v = ~vv + ~v` , ~rot ~v` = ~0

div ~v` 6= 0 , div ~vv = 0C

~rot ~vv 6= ~0 . ' )(^]4 ! 1"P" * ( $" ( : ( ! @!,RC " !" @!OI P"!P"_+ ;D&?.!" *W3$?! 4 . ' T )(S]2& $C.,EE * &?.!" ' ) )-9 1! *,"J]2*.,.GB ?I ( * TF&@.!" ' T)("_ T );,aC>T)WT-]* $?! !#

1

c0

∂ τ

∂t− ρ0c0

γβdiv ~v` =

1

β

1

c0

∂ p

∂t, ' )-,"]

(MT 9

R EG46?HL)L JG+5460 @ 460H@H/1 LO13+ ?2N.L 0H/1

(

1

c0

∂

∂t− `h ∆

)

τ =γ − 1

βγ

1

c0

∂ p

∂t, ' )WT-]

(

1

c0

∂

∂t− `v ∆

)

~v` = − 1

ρ0c0~grad p , ' ) ]

(

1

c0

∂

∂t− ` ′

v ∆

)

~vv = ~0 , ' );9]3$"1 ~rot ~v` = ~0

div ~vv = 0

) + ?, ( +* &?.!" 0& ( *.," $0 @ .! *+,"@P".!" a&" $,!:

p_τ_~v`

C~vv

) 56G131<&"+a$!:τ,21C.*@ a&/ 1 @_ !,)-9))( " )50;D13I ! ,. -J O*O. ( &. .,

τ& " .,[ ! ( O,"F: /" ( R* [ (>( *<;= F. ( &. F1",!%

τa*<;= E. ( &. E -"!

τh? !0" -&1!$ ( -," !" +* @ .! ( "P

[

1

c20

∂ 2

∂t2−(

1 + `vh1

c0

∂

∂t

)

∆

]

τa = 0 , ' )-A"]

`vh = `v + (γ − 1)`h_

[

1

c0

∂

∂t− `h

(

1 + (γ − 1)(`v − `h )1

c0

∂

∂t

)

∆

]

τh = 0 ,

,"![

1

c0

∂

∂t− `h ∆

]

τh ≈ 0 . ' ) U ]50;D13I ! ,. -J 9J"*9,!"

p = pa+ph

C $?!.,8@,.!1!, ( ! @!~v` = ~va+~vh.! /" -% ( ( 2&@.!" ' ) A-] ' ) U ]G ;=1,! ,. -J G*B ( .)

. 0L 0L ?/ 5M 1 ,1 a*" .!" &"2$!

p = pa + ph

_~v` = ~va + ~vh

_τ = τa + τh

~vv

, ( " :E" .!" *<;D&@.!" 2*G@P-!" C%* *! ! '1:,!? ) 23 ( "! _1" 1 @ 80$!:01,!%77 " !% ' p _ ~v`_τ]C_-B" .!" E9* >* ;D 2& C.,a** !E*!,1 ( -B*EN !" [*E: "!! ( )60 C3_<,&",B* E:N," !"

τ = τa + τh* ;= @!" ' ) T-] _1 ( G '*2 @!" ' )-A-]C ' ) U ] _1 *!4 "

p = pa + ph , ' )-@-]3$"1

pa ≈ γβ

γ − 1

(

1 − `h1

c0

∂

∂t

)

τa ≈ γβ

γ − 1τa , ' ) "-]

ph ≈ γβ(`v − `h )1

c0

∂

∂tτh << pa . ' ) ]

( '

5MG,&",*G12,.. ' ) " _ ]*@ ;= @!" ' ) ]C_ 1 ( .H * @!" ' ) A-] ' ) U ]C_$",! @2+$?!4.,%@,.!1!, ~vaC~vh_* #

~v` = ~va + ~vh , ' ))(-]3$"1~va ≈ − 1

ρ0c0

γβ

γ − 1

[

(

1

c0

∂

∂t

)−1

+ (`v − `h )

]

~grad τa

≈ − 1

ρ0c0

γβ

γ − 1

(

1

c0

∂

∂t

)−1

~grad τa , ' )(<(S-]

~vh ≈ γβ

γ − 1`h ~grad τh . ' ))( (3 ]

(W! ! _&*@ H* ( ! /& -.! _ &"W E" * @ ( " ! ej ω t ' ω ,,, -.F:.!" *H ;=" * S]7,H,"@P" -#*@ #%* ( ! D _?# @ .! ' ))(^] " ' )( ( ]7* -$#

(∆ + k2a) τa(~r, t) = 0 ∀~r ∈ D , ∀t , ' ))(B,"]

(∆ + k2h) τh(~r, t) = 0 ∀~r ∈ D , ∀t , ' ))(MT-]3$"1

τ = τa + τh_@C

k2a ≈ k2

0

1 + j k0`vh

≈ k20[1 − j k0`vh ] , ' ))( ]

k2h ≈ − j k0

`h[1 − j k0(γ − 1)(`v − `h )] ≈ − j k0

`h, ' ))(B9"]

p = pa + ph ,3$"1pa ≈ γβ

γ − 1(1 − j k0`h )τa ≈ γβ

γ − 1τa , ' )( A"-]

ph ≈ j k0γβ(`v − `h )τh << pa , ' ))( A ]~v` = ~va + ~vh ,3$"1

~va ≈ j

k0ρ0c0

γβ

γ − 1[1 + j k0(`v − `h )] ~grad τa ≈ j

k0ρ0c0

γβ

γ − 1~grad τa , ' )( U -]

~vh ≈ γβ

ρ0c0`h ~grad τh , ' ))( U ]

(∆ + k2v)~vv(~r, t) = ~0 ∀~r ∈ D , ∀t , ' )( @"]3$"1

k2v = − j k0

` ′v, ' ))( -]

div ~vv = 0 . ' )-, -]


6/ / ! 0 / 'L

50;D" * @ ( " !%+,"@P+" Fz1"!. 9*@ 9 @!*+*!,!@.!) ' @ !*# ( $?! ]H1 *@ B E1 ?! *!? E @!W..H,!P"!*_* ; #

z_6* E" P

`B*E ""

a)656 E,/ ( " B ?., ( !.

z = 0Cz = `

@W* ( !1" ) 5M( !1" " F!4.>

z = 0/" 1C.! F ( 1!9

z = `/" 1C.!" F 1

' @P", T)(^] )-56 ( !1,"" # ( ,. 7! ( &",# z = 0

+$?!,~ve

* 7H1 ( . -.+*!,%&", P"!P")560. "" *B. 9IM. M&,! " ;D@!6*A1"1 ! ( !$?! AM ( !? _* . "I. O ;D *N* R,!" & .,R1 !*,R1 ( ( @! : # >I2@,! !/" ( *@ a1 <O1.!" * ' 1;=I "# *!N a ( -a$,!: 1/" 1C.! *R1",*" >.*!r] _6! * *@ - *>:O1",*" >! ( . θ @!," *> ( C.,!_6C,"@P".!4$"B !$ -2 ;D#

z* . )50;D C%* W1"1 W! ( !. ( !%2'$"!,! @P"G*2@,"!

z = 0z = `

IW,! 1 ( .G@+ ;D! ?*1C.! R*<;= * ( !4. 1B* "!Yp

*" G@Yp =

S

ρ0c0

1 + j√2

√

k0(γ − 1)√

lh , ' )-, (^]' ;=" *W1" ,.!H. -+ O! 1!* 1H ( #1+@!_? C.#*%$!1"!4.W," - !4$"[* 2 ?., ( !.+

z = 0Cz = `

]C)(W! ! _ [ @!" ' )( ,_ ))(MT C )( @-]C_&" -H,1!HW1" *!4.! 2 /" -.! ,W,! $ -.#

τa + τh = 0 , vaz + vhz + vvz = 0 , var + vhr + vvr = 0 ,&"

r = a , ∀z ∈ (0, `) , ' )-, ,"]C

S < vz(0) >S +Yp < p(0) >S= Sve ,

z = 0 , ' )-,T""]

S < vz(`) >S −(Yr + Yp) < p(`) >S= 0 ,&"

z = ` . ' );,T ];D .,G@,_ ( ,*<;=" * ka

_kh

Ckv

$" G1!4." 2>/" ( k2

a = k2ar + k2

z , ' )-, ]k2

h = k2hr + k2

z , ' )-, 9"]k2

v = k2vr + k2

z . ' )-,<A-]5M !" '* ;D&@.!" ' ))(B,"]# 1?"* " H1 ! * !%_ 1 ( * & ,,. _,2*G:>/" (

τa = AaJ0(karr)

J0(kara)(e−jkzz +Bej kzz) , ' )-, U ]

W1" ,J B

. -0* ( ! # :21" *!!" E 6/" -.! , ' )-,T ]M z = `) C+" .!" &"

τa ( *G**!

pa

C~va

_1 ( G '*2:!" ' )( A-]8 ' ))( U -] #

! %( ( ( ( +

pa ≈ Aaγβ

γ − 1

J0(karr)

J0(kara)(e−jkzz +Bej kzz) , ' )-,<@-]

vaz ≈ Aakz

k0ρ0c0

γβ

γ − 1

J0(karr)

J0(kara)(e−jkzz −Bej kzz) , ' )-,-]

var ≈ Aakar

j k0ρ0c0

γβ

γ − 1

J1(karr)

J0(kara)(e−j kzz +Bej kzz) . ' ) T-]

5A2" .!" G* ;D&@.!" ' )(MT"]C_S1 ( 7 a*8: ( ! ,81" *!4.! ?/" -.! , ' );, ,]C_;=1!

τh ≈ −AaJ0(khrr)

J0(khra)(e−jkzz +Bej kzz) . ' ) T8(^]

56:!" ' )( A ] ' ))( U ]& ( C -%",#*<;= [** !G+ ,,! %* ph

C~vh

#

ph ≈ −Aaj k0γβ(`v − `h )J0(khrr)

J0(khra)(e−j kzz +Bej kzz) ≈ 0 , ' ) T ,"]

vhz ≈ Aaj kzγβ`hρ0c0

J0(khrr)

J0(khra)(e−j kzz − Bej kzz) , ' ) TFT-]

vhr ≈ Aakhrγβ`hρ0c0

J1(khrr)

J0(khra)(e−jkzz +Bej kzz) . ' ) T ]

59;= @ .! ' ))( @-] _S 1*" 1 ! * !%_S1 ( .9. *8+.!" ' );, -] div ~vv = 0G*E1 ? vvθ = 0

_<* " ' a ( &" ∆,, -.N!1! a:1! !% " >/" 1C.! 1:!3]

(∆ + k2v)vvz = 0

(∆ − 1

r2 + k2v)vvr = 0 ∀r ∈ (0, a) ∀z ∈ (0, `) .

5MF,".! O*G ;= @!" ' ))( @-]&"+a1 ( . -.vvz

_ ,2* 1G*G:>/" (

vvz = AvJ0(kvrr)

J0(kvra)(e−j kzz − Bej kzz) , ' ) T 9"]

Av

,2*C. ( ! B@+E* ! ( B1" * !.!" /" -.! , ' );,<,"] #

Av = Aa−kz

k0ρ0c0

γβ

γ − 1. ' ) T A-]

5AE,: .! ' );, "] div ~vv = 0* " G&"

vvr

#

vvr = Avj kz

kvr

J1(kvrr)

J0(kvra)(e−j kzz +Bej kzz) . ' ) T U ]

9 @ <_ G,2*2 !" ' ) T _ ) T _ )WT U ]*@ 2:a.,"!,! ( B1" *!4.! 5/ ! ' );,<,"]8& ( * * . ( !

kz

#


k2z

1

kvr

J1(kvra)

J0(kvra)− kar

J1(kara)

J0(kara)− j k0`h (γ − 1)khr

J1(khra)

J0(khra)= 0 . ' )WT @"]

4 !kara

,9&.!43_^8. "I * /" 1C.! *, kar

& #* $"" + J1(kara)

J0(kara)≈

kara

2

) ( .2. ?O* W1*? k2

a = k2ar + k2

z

Ck2

h = k2hr + k2

z

_?:G.!" >1* -.G*" H"k2

z [1 +Kv + j k0`h (γ − 1)Kh] − j k0`h k2h(γ − 1)Kh = k2

a , ' ) T-]'#. ( Kv

Kh

" -%*" @Kv,h =

2

kv,hra

J1(kv,hra)

J0(kv,hra). ' ) -]

H > 1" *!4.!" E,. *@* * [,&/, 1_8F" P">13.1C.,!I.!%`h

C`v

$ - 3! ( .!4$" ( `h = 6.10−8 ( C

`v = 4.10−8 ( )#56 J 3 ))( *" $ R* ( k2

a/k20

k0`h (γ − 1)

/, 1% ?. ( ( GC>T ("_-A ) EGI / 0HDF/

k2a/k

20

( ("_-,)( −8 ( T_ U ))( −5

(γ − 1)k0`h _ ))( −10 ("_ ))( −5

XZY\[2& )(7,+*B. ( 2@! -%* :a,: .! ' )WT"]C)56$,a*" E* a1NJ 3 ( ,. -F",* , ,.! @,3 ! ( .! ,! $ -.#

k2a ≈ k2

0 , k2h ≈ −j

k0

`h, 1 ± j (γ − 1)k0`h ≈ 1 , ' ) (S]

F,.!" ' ) T-]*C$! -2*" 1k2

z

(

1 +2

kvra

J1(kvra)

J0(kvra)

)

= k20

[

1 + (γ − 1)2

khra

J1(khra)

J0(khra)

]

, ' ) ,"]H >: ( ' . ""

a*Q. &[CN P" ( ( O*5/&? 1 "! ' ! 8/,! "R ( ! 6/ 1F*E1" F* . &S]W," -B. F:R* !,!@ .! ! * @ !*E,, P"!P"32 O,P-,*O*H1%!H N ( $! ,B*$ -+*@ +81"1 ! ( !4._"

|kz| << |kh| ,* ;D

k2hr ≈ k2

h , |kz| << |kv| ,*<;=

k2vr ≈ k2

v .50;D ,!" ' ) ,"]* kz

*" * " 1k2

z = k20

1 + (γ − 1)Kh

1 −Kv, ' ) T-]

Kv,h =

2

kv,ha

J1(kv,ha)

J0(kv,ha), ' ) F ]

,"!_ " ;="*,2( , ,a*%" P"+1.1.!,.!

( (3 (*

k2z ≈ k2

0

[

1 +1 − j√

2

2

a√

k0

(√

` ′v + (γ − 1)

√

`h )

]

. ' ) 9"]

59;D* ( !,J 1F1",!%>*F.. /,* ?, ( IE.&"I* *!*>: ( (

Sve ' ",! ..!" P")(^]9 ;=13,+! ,. -J W*H,,!" ( " ',# B1C.!" O*!. " :"I.!< p(`) >S

H;D1!4YT =

Sve + Ye < p(0) >S

< p(`) >S

, ' ) A-] :$ ( " *

p;D1,!

< p(z) >S=2

a2

∫ a

0

p(r, z)r dr ≈ 2

a2

∫ a

0

pa(r, z)r dr .

( .% O*+, ! ( .! 9 +*@ 8W.P. W,1* - ' ,!" ) (^]F*<;D F ;=,,! * ;D13IF*O,!" Q -.,"!? ph ' ) ,"] _ !#@>? '1>1,*F,!" . * $, )650;=,,!" *> ;=13,G! ,J . F*F,,!" .J

p& *" 1C.,. ( " pa

*" ' );,<@"]C)5M ( G*p;=1!42",

< p(z) >S≈ Aaγβ

γ − 1(1 − j k0`h )

2

kara

J1(kara)

J0(kara)(e−j kzz +Bej kzz) ,

!6*" _@ '*C$""@ %B&",#*"/" 1C.! +*?, J1(kara)

J0(kara)≈ kara

2' kara < 1

] _

< p(z) >S≈ Aaγβ

γ − 1(e−j kzz +Bej kzz) . ' )"]

5AN$?!4.,ve

*a ( ( _ F !/" ( B E1C.!" G",! a,!$ -G ; z_@I2* ( ! G@+a1" * !.!" /" -.! , ' );,-]8 z = 0

#

Sve = S < vz(0) >S +Yp < p(0) >S , :$ ( " * $?!.," P"!4.*! @vz

;D1!4< vz(z) >S=

2

a2

∫ a

0

vz(r, z)r dr =2

a2

∫ a

0

[vaz(r, z) + vhz(r, z) + vvz(r, z)]r dr .

5MH,,! B&1!$G*%$!4.vaz

_vhz

vvz

_ ' ) );,"]C_ ' &&) ) -] ' ) ) 9_ ) A-] _01 ( [. * 1 (γ − 1)k0`h

IN NC.! *C$ ( ' ) ) 8(^]C_91 *! - "1" !* ? O$?!4.,> -.,"!? " P"!4.*! @vhz

I &!.E* $ -vaz

vvz

_61 ! ( *W P!PM*@ 11?*# ;=! -P"9,1* -)50;D!" F*#2$?!, ( < vz(z) >S

,%#* @

< vz(z) >S≈ Aakz

k0ρ0c0

γβ

γ − 1(1 −Kv)(e

−jkzz −Bej kzz) . ' ) @-]


50; * ( !4J 1H!4..!$"Yi

*'1?! *,B,2* @Yi =

S

ρ0c0

kz

k0(1 −Kv) =

S

ρ0c0

√

[1 + (γ − 1)Kh](1 −Kv) , ' )-],"!_ " ;="*,2( , ,a*%" P"+1.1.!,.!

Yi ≈S

ρ0c0

[

1 +1 − j√

2

2

a√

k0

(√

` ′v − (γ − 1)

√

`h )

]

. ' )-9 -]5MF1" IJ -

B& %" C.,* . ( ! G*<; :E1" *!!" /" -! , ' );,-]8 z = `

#

B = e−2j kz`Yi − (Yr + Yp)

Yi + (Yr + Yp). ' )-9 (^]

! ! _ 1 ( a. ? * ,!" ' );9 "]% ' )-9 (^] _< ;D* ( !,J 1a1"I.!%F*E. /IYT

;D1,!YT = j sin j kz`

[


Yi

]

+ cos j kz` (2Yp + Ye + Yr) . ' )-9 ,"] ,.W!" O*H ; * ( !4. 121"I.!%%*%. /I#,+!* -.! " ;=,!" RCJ !*@ >1*<;D *: ' ) @ "] )A5M ( * E*<;=" * : ,*" 1> > -a&"F131*> ;D* ( !,J 1E1"I.!%>*F. /,* ?, ( ( !1," ! %1" G*@ F13

( " $ ( -2* ( ( +* ( !1" +IW! ( ! " 1!6*<;= [ !," : <)


5M <;= 2" * 1" ,.! @!* +*!,!.!), 1 !2,9 @,"!,!P"!*_0 C.9*1" 1 B! ( !.#$?!1 ( !%H& $ -.* !.%@% 1 1 G*<;D* ( !. 1,1! ? /" 1C.! * 1? >1! -*E$?!1"!4.>*>1!! ( W* 1? >1! -* 1" *1C.!" ( !_* !*$`-" >$2:H @!" * % ;D" *1,!%29 : F,!P"!*# @!*+$?!1 ( !#&"* ( ( 0131,!,!% ? 2H @! >,0 >: E*<; * ( !4. 1&1! :

ζ @!* % R*!,!@.!) ( _ )8,<9 8,<A , )50;= ,C#*21C.W _8!& *R RP". *2@,!2@P". );,))(W* H:,&/ 1 @! _?I*H,1!#1CH !" *<; * ( !4. 1%&1! :B*@,"!? !8 R 1 ( * ;D1,!. >*a1" *!4.!" / !" @!8 ! ( *<;= B$?!., ' _! ( !4J -8 ( !! @!*B*!,!@.!) )5M @!* +,9* 0 E@1 ( ! ! !! ( !7 F a! @ ! `!4. ( !P!*7 y = 0 ! ( H* ;D G$?!4., !)/" ( ~V0

( $#E1 : ' P" )(^] )

%x

&('*),+.-/'*01-2),+1'34+ &('*),+.5468796:04;,-2+

~V0

y

<(=?>.@ BA(DCFEHGHIBJLKNMLOHPQ8R%*OHISHTUJ!EHIR"EHKVR"GHIVE*Q8I$P8IW8X YXNK:Z[R"P8P8JLE,QFPOHW\OHES[TUJLEDWR"]^R_GHILAàI SHT"JLEbGHX YEHRcSYJLW$T"JdGHR"WINKVQ8R"MÊeGgfhOHEHI MÊ[GHIJLKNMÔ[PQ1R*OHI SHTUJLE[IiZYJLW8jMÊHR*OHIkR"E[KNR"G[INE,Q1I IVQBTlfmJnI

OyE[M^W8joJLTJLOdSHT"JLEdIVPQ.KVMÊ8pMÊHG[OgqnSHJLWrKVMÊ,sÎNE,Q8R"MÊgqHJtsLINKFT_IuSHT"JLE

xOyA vIFTUJBP8M^WwQ1ILqnT"IvSHW8M[T"NjIxGHI

W8X YInR"MLEyPSzXVKNOHT"JLR"WI$SzMLP8XINPwQuR_EHGHXNSINE[GYJLE,QxGHITUJQ1WM^R"PR"VjoIFKVMnMLW8GHMÊ[EHXNIGHIT|fhINPSYJLKVIzA

àISHW8M HT"VjoIBSzMLP8XIVPQuGHXYE[R~SYJLWxT"IVPuX&9OHJ!Q1R_MÊHPuG[R_zXNW8IVE,Q1R"IVT"T_INP\IVQuW8INT"J!Q1R_MÊHP|iA @,x#|iA;, ,pK&4Av A-,^qYJLO *OHINT_T"IVPxPMÊ,QvJLP8PMnKVR"XNIVPvT_INP.KVMÊHGHRQ1R"MLEHP\JLO pW8MÊ,Q1R_NWINP

τ = τa + τh = 0 , ~v = ~v` + ~vv = ~V0 , y = 0 , ∀x , A)((


IQvG[INPxKNMÊHG[R_Q1R_MÊHP G fmJLE[E9O[TUJ!Q8R"MÊ # T|fhR"E HEHR|A` f XVK JLWwQ R_EHPQ1JLE,Q:JLEHXGHIQ8INjSzXVW1J!Q8OHW8I

τSINOnQ Q1W8IXNKVW8R_QoKNMLjojIT"J POHSINW8SM^PR_Q1R_MÊGgf O[EHI MÊHG[I

SHT"JLEHI$R_EHKNR_GHINE,Q1Iτ+ IQvWX YXVK:ZHR"I τ− pT"I JLKQ1INO[W ej ω t INPwQxM^joR_P:

τ = τ+ + τ− = (τ+a + τ+

h ) + (τ−a + τ−h ) . BA;,L

àJ KNMÊHG[R_Q1R_MÊτa = −τh

GHIsLJ!E*QvVQ8W8IBP1J!Q8R"P JLR_Q8I$INEy = 0

IVQxSMÔHW.Q1MÔnQxq T_INP.Q1IVW8jINP

τ+ IQ τ− PMÊ,QXVKNW8RQ1Pτ+ =

[

ej kayy + A+h ej khyy

]

e−jkxx ≈ ej kayye−jkxx , BA ^J^

τ− =[

Rae−j kayy + A−

h e−j khyy]

e−jkxx , BA Y

MeTlfmJLjSHT_R_Q1O[GHIGHIiTlf MLEHGHIiR"E[KNR"G[INE,Q1Iτ+a

IVPQBK:ZHM^R_P8R"IiXV],JLT_Ii#DT|fhOHEHR_Q8XLqzMeT|f JLjoS[T"R_Q8OHGHIA+

h

GHIiT|fhMÊHG[IQ8ZHINWjoR9O[I.R"EHKVR"GHIVE*Q8I.INPwQ(E[XN]^T_R"]^XVI GHIVs!JLE,Q KNIVT"T_I GHI\T|fhMÊHGHI\J!KNMÔHPwQ1R9O[I.IVQcM

Ra

G[XNP8R_]ÊHI\T"IrKNM9I KNR_INE,QGHIW8X HInR"MÊGHITlf MÊ[GHIJ!KNMÔHPwQ1R9O[I |Ra|

R"E pXNW8R_INOHWx# Tlf O[EHR_Q8XBjoJLR_PxsLM^R"PR"EG[IKNIVT"T_I KVR?A àIVPFEHM^j HW8IVPGgfhMÊHG[IP8J!Q1R_P pMÊ,QxJLODW8INT"J!Q1R_MÊHP POHR_s!JLE,Q8INP

k2a = k2

x + k2ay , k2

h = k2x + k2

hy , BA ,

MDT_INP*OYJ!E*Q8R_Q8XNP SM^WQ1JLE,Q.T"IVPrR_EHGHR"KVINPxIQyW8INS[W8XNPINE,Q1IVE,QvT_INP KVM^jSzM^P8JLE,Q1IVP P8OHRs!JLE,Q

xIVQyqka

IVQkhXQ:JLE,QxGHMÊHE[XNPxW8INPSzIVKVQ8R_sÎVjoIVE,QuSYJ!W.T"INP\I[S[W8INPP8R_MÊHP|iA(,(IVQ|iA)(Â

ÌNPFPM^T"O[Q8R"MÊ[PvGHObS[W8MHT_NjISMÔHWvT"Is9RQ1INPP8IiSYJ!WQ1R_KNOHT"JLR"WITUJLjR"EYJ!R"W8I!qHP8O[SzIVW8SM^P8RQ1R"MLE GgfhMÊHGHIVPFR"E KVR"GHIVE*Q8IdrIVQxW8X YXNK:ZHR_Io wq

~v` = (~v+à + ~v+

`h) + (~v−à + ~v−`h) BAL

PMÊ,QvG[MÊHEHXVINPxSYJLW X&AiA)( LJiIVQ

v±`x = −j kx

[

j

k0ρ0c0

γβ

γ − 1τ±a +

γβ

ρ0c0`h τ

±h

]

, BA-A^J^

v±`y = ±[

j kayj

k0ρ0c0

γβ

γ − 1τ±a + j khy

γβ

ρ0c0`h τ

±h

]

, BA-A Y

IQvSMÔHW.T"JksnRQ1IVP8P8I$Q8MÔHW [R"T"T_MÊHEYJ!R"W8I~vv = ~v+

v + ~v−vSYJLWB X& A iA)( @i# iA-, ,

v±vx = ±A±v

kvy

kxe±jkvyye−jkxx ,

BA LJ^

v±vy = A±v e±jkvyye−j kxx ,

BA YJ sÎVK

k2v = k2

x + k2vy ,

BA-@ÎQM gq#DT|fhR"EHPwQ:JLWuGHI TUJ s9R_Q8INP8PI INE,Q1WM^SHR%*OHI R_EHKNR_GHINE,Q1I!qTUJ s9R_Q8INPP8IiQ1MÔ[W HR_T"T"MLEHEYJLR_W8IBR"EHKVR"GHIVE*Q8I

~v+v

IVPQNqGYJ!EHP\TUJiP8OHRQ1I!qHEHXN]LT"R"]LXNI$INEyW8IV],JLWGyG[IBT"JksnRQ1IVP8P8IBJLKVMÔHPwQ1R%*OHIR_EHKNR_GHINE,Q1I!AÌNPuKVMÊHGHRQ1R_MÊHPx# TUJ2pW8MLE*Q8R"VW8I

y = 0SzMÔ[WuT|fhXNKNJLWQvGHIBQ1INjSXNW1JQ1OHWI

τqzIQFSMÔHWxT"INPvKNM^jSM^P1JLE,Q8INP

SYJ!W1JLT_T"NT_INP IQxEHM^W8joJLT_I$JLO SHTUJ!Ey = 0

GHIT"Jis9RQ1INPP8ISYJLWwQ1R_KNOHT"JLR"WI~vqYPNfhXNKVW8R_sLINE,QvW8INPSINKVQ8R_sLINjINE,Q

1 +Ra + A−h = 0

A ,

(1 +Ra)kx

k0

1

ρ0c0

γβ

γ − 1− jA−

h

γβ

ρ0c0kx`h − A−

v

kvy

kx= 0

BA( ,

−kay

k0

1

ρ0c0

γβ

γ − 1(1 −Ra) − jA−

h

γβ

ρ0c0khy`h + A−

v = V0 . BA( (t

INPcQ1WM^R"P GHIVW8EHR_NWINP W8INT"J!Q1R_MÊHPKVMÊHGHO[R_sÎVE,Q q9SYJLW XNT_R"jR"EYJQ1R"MLEGHIVP(SYJ!W1JLjVQ8W8INPA−

h

IQA−

v

q*#$TUJ$W8IVTUJ!Q8R"MÊP8OHRs!JLE,Q1I

j (γ − 1)k0`h (1 +Ra)(k2x + khykvy) + (k2

x − kaykvy) +Ra(k2x + kaykvy)

= kvyk0ρ0c0γ − 1

γβV0 ,

A)(B,^

P8M^RQ qnKNM^jS[Q1I$Q8INE*OG[INPxW8INT"J!Q1R_MÊHP A*rIVQ A @,qYJ sLINKkx << khy

IQkx << kvy

q

khy =

√

−j k0

`h− k2

x ≈√

−j k0

`h

IVQkvy =

√

−j k0

` ′v

− k2x ≈

√

−j k0

` ′v

, A)( ,

kay1 − Ra

1 +Ra=

1 + j√2

√

k0k0

[(

1 −k2

ay

k2a

)

√

` ′v + (γ − 1)

√

`h

]

+k0ρ0c01 +Ra

γ − 1

γβV0 .

A)(*` J SYJLWM^R$INE

y = 0q\J!EHR"jXNIbGgfhOHEHIes9R_Q8INP8PI

V0

q.INPQJLT_M^W8PDKNJLW1JLKQ1XVW8R"PXNI SYJLW P8MLE JLGHjR_QQ:JLE[KNIP8SXNKVR :*OHI @[q ?(Â nA L

ζ = ρ0c0V0 − vay(0)

pa(0),

A)(^M X&AiA)( A^J[q iA)( LJ,

pa(0) ≈ γβ

γ − 1(1 +Ra)e

−j kxx ,

vay(0) ≈ − 1

ρ0c0

γβ

γ − 1

kay

k0(1 −Ra)e

−j kxx ,

KNI*OHR|q[KNM^jS[Q1IFQ8INE*OG[IBT"JkW8IVTUJ!Q8R"MÊ A)(*q[KNMÊHG[OHR_Qx#y @nqHX A [A)(!

ζ ≈ 1 + j√2

√

k0

[(

1 −k2

ay

k2a

)

√

` ′v + (γ − 1)

√

`h

]

. A)( A,

5 g <

IVQ8Q8I JLEHEHInIDW8IN]LW8MÔHSI*OHIVT%*OHINPiKNJLT"KVOHT"PGHMÊ,QiT_INPkWXNP8O[T_Q:JQ1PPMÊ,Q O[Q1R_T"R_P8XNPBGYJLE[PiT"I K:ZHJLSHR_Q8W8I j J!R"P*OHREgfDPMÊ,QvSHJLP\GHXVsLINT"MLSHSzXVPuJYEG[IBE[IBSHJLP\INEJLT"MÔ[W8GHR_WrT"I$S[W8M^SM^PNA

/ 5/ 2 gµν(z, z0)

àIDWINSM^WQ G[IDTUJ pM^WjoIdG[IDP8M^T_O[Q1R_MÊ HA-, ^BGYJLEHPiTlf X&9OHJ!Q1R_MÊ GHIDSHWM^SYJL],JQ1R"MLE GHO SHW8M[T"NjIdGHIWINIVE HA-,*rGHMÊ[EHI

∑

µ,ν

ψµν(r0, θ0)

[

∆~w +∂ 2

∂z2 + k2

]

ψµν(r, θ)gµν(z, z0) = −δ(~w − ~w0)δ(z − z0) ."!:A$#

ÌuSHWMnGHO[R_Q(P8K J!TUJLR_W8I\GHIFKVIVQ8Q8IuX&9OHJ!Q1R_MÊdSYJLWψµ′ν′(~w)

qnKVMÊHGHOHRQ q*KNMLjoS[Q8IvQ8INE*ODGHIuT"JWINTUJQ1R"MLEoGgfhM^WwQ1ZHM ]^MÊYJ!T"R_Q8XuGHIVP$pMÊHKVQ8R"MÊ[PxS[W8M^SHWINPVq

∫ a

0

∫ 2π

0

ψµν(r, θ)ψµ′ν′(r, θ) dθ dr = δµµ′δνν′

"!:A;,ÎVQ.G[I$T|fhX *OYJ!Q8R"MÊdG[I$SHW8M^SHJL],J!Q8R"MÊ HA-,<A^J,cGHO SHW8M HT"VjoIuJLOds!JLT_INOHWP SHWM^SHW8IVPNqY#kTlf X&*OYJ!Q1R_MÊdPOHR_s!JLE,Q1ISzMLOHW.T"IVP.KNM9I KNR_INE,Q1P

gµν

GHOyGHXVsLINT"MLSHSzIVjoIVE,Q [

∂ 2

∂z2 + k2zµν

]

gµν(z, z0) = −δ(z − z0) , ∀z ∈ (0, `) ,%!:A ,

J sÎNKk2

zµν = k2 − k2wµν .

"!:A * KNIVQQ1IeX *OYJQ1R"MLE "!:A ,q.R"TFKVMÊ,s9R"INE,QGgf JLP8PMnKVR"INW T_INP KNMÊ[GHR_Q8R"MÊHPDIVE pWMÊ,Q1R_NW8IVP KNMÊ,Q1IVE*OHINPGYJLEHPyT"ISHW8M HT"VjoI HA-,*

(

∂

∂z− j k0ζz0µν

)

gµν(z, z0) = 0 ,SMÔHW

z = 0 ,"!:A&!J,

(

∂

∂z+ j k0ζz`µν

)

gµν(z, z0) = 0 ,SzMÔ[W

z = ` .%!:A'

#

IP 9PQ1VjoIBGgfhX *OYJ!Q8R"MÊ[Pk"!:A [q !:A'^ INPwQuKVINT_OHR JLO *OHINTaP8J!Q1R_P JLRQ.OHEHIpMÊHKQ1R"MLEGHI$W8IVINEIVEIVP8SYJLKVIYE[Ra#OHE[IBG[R"jINEHPR"MÊDGYJ!EHP\T"I$GHM^joJLR_EHIFGHIYMÔHWR"INWF @[q+A[A #Â [|AàJdP8M^T_O[Q1R_MÊ GHIiKVIiP 9PQ8NjISINOnQVQ1WIiGHXVQ8INWjoR_EHXNIkIVEbQ1W8MLO[s!JLE,QGHINO P8M^T_O[Q1R_MÊHPFR_EHGHXVSzIVEHGYJLE,Q1IVP

g1(z)IVQg2(z)

G[IFT|fhX *OYJ!Q8R"MÊoZHM^jM^]^VEHIuJ!P8P8M9KNR_XNIu#T|fhX *OYJ!Q8R"MÊ %!:A ,qn#\WMÊHP *R_INEWg = g1g

′2 − g′1g2pR"EHGHXVSzIVEHGYJLE,QxGHI

z EHMÊyE*OHT|qHIQxP1J!Q8R"P J!R"P1J!E*Q\K:ZHJLKNOHE[I#iOHEHIBKNMLEHGHR_Q8R"MÊ J!OpW8MLE*Q8R"VW8INP"!:A&LJnq AYàJ

pMÊ[KVQ1R_MÊgµν(z, z0)

P8MLT"O[Q8R"MÊDGHOySHWMHT"VjoIi%!:A [q !:A&LrPNfhXNKNWR_QvJLT_M^W8P

gµν(z, z0) = −g2(z)

∫ z

0

g1(x)δ(x− z0)

Wgdx− g1(z)

∫ `

z

g2(x)δ(x− z0)

Wgdx .

"!:A-A^ÌNP P8M^T_O[Q1R_MÊHP

g1

IQg2

G[IxTlf X&*OYJ!Q1R_MÊ ZHM^jM^]^VEHILq,P8J!Q1R_P JLR_P1JLE,QcW8IVP8SINKQ1R_sLINjINE,Q.J!O KNMLEHGHR_Q8R"MÊ[P IVEpWMÊ,Q1R"VW8IVP"!:A&!J[q YqHSINO[sLINE,QFQ1WIBXVKNWR_Q1IVP\P8MÔHP\T"J2pMLW8jIFP8OHRs!JLE,Q1I

g1(z) = cos(kzµνz + ϕ0µν)IQ

g2(z) = cos(kzµν(z − `) − ϕ`µν) ,

Mϕ0m

qϕ`m

P8MÊ,QxGHMÊHE[XNPxSYJLW.T"IVP\KNMÊHG[R_Q1R_MÊHP.J!OpW8MLE*Q8R"VW8INP"!:A&LJnq

tanϕ0,`µν = − j k0ζz0,`µν

kzµν.

"!:A LÌ\W8MÊ[P *R"IVE

Wgs!JLO[QvJLT_M^W8P

Wg = kzµν sin(kzµν`+ ϕ0µν + ϕ`µν) ,

E YEgq\Ggf JLSHWNPyT|fhInSHW8IVP8PR"MÊ %!:A A,dGHIgµν(z, z0)

q.PRz < z0

q\JLT"MLW8Pz0 /∈ (0, z)

IQTlf R_E,Q1XN]LW1JLT_I∫ z

0

g1(x)δ(x−z0) dxINPwQ~Q1MLO MÔHWP~E*OHT"T_ILA uIrjoVjoI!qPR

z > z0

q!JLT_M^W8P T|fhR"E,Q1XV]^W1J!T"I ∫ `

z

g2(x)δ(x−z0) dxIVPQxQ1MÔ MLOHW8P.E*OHT_T"ILA R_EHP8Rlq[TUJ>pMÊ[KVQ1R_MÊ

gµν(z, z0)IVPQxGHMÊHE[XNIBSYJLW

gµν(z, z0) = −cos(kzµνz + ϕ0µν) cos(kzµν(z0 − `) − ϕ`µν)


"!:A-@^J^PRz < z0

qYIQxSYJLW

gµν(z, z0) = −cos(kzµνz0 + ϕ0µν) cos(kzµν(z − `) − ϕ`µν)


"!:A-@ YPRz > z0

A

/ / / ÌNPInSHW8IVP8P8R_MÊHPy HA;, LJIQ GHINP pMLEHKVQ8R"MÊHPBSHWM^SHW8IVPkSINW8jIVQQ1IVE*QGgfhXNKNWR"WI T"IVPI[S[W8INPP8R_MÊHPkGHIVP

SHWMnG[OHR_Q8P.P8K J!TUJLR_W8INP.G[IpMLEHKVQ8R"MÊHP.S[W8M^SHWINP< ψt

µν |ψt′

µ′ν′ >r

PMÔHPxT"IVPpM^W8jINP.POHR_s!JLE,Q1IVP

< ψcµν |ψc

µ′ν′ >r =< ψsµν |ψs

µ′ν′ >r=δµµ′

bµνbµν′

∫ ar

0

Jµ(kwµνr0)Jµ(kwµν′r0)r0 dr0"!:A^

< ψsµν |ψc

µ′ν′ >r =< ψcµν |ψs

µ′ν′ >r= 0 ,"!:A$# ,

( +

P8M^RQ< ψc

µν |ψcµν >r=

1

b2µν

a2r

2

[

J2µ(γµνar/a) − Jµ−1(γµνar/a)Jµ+1(γµνar/a)

]

,%!:A # #tJ,

IVQ

< ψcµν |ψc

µν′ >r=1

bµνbµν′

aar

γ2µν − γ2

µν′

[γµν′Jµ(γµνar/a)Jµ−1(γµν′ar/a)

−γµνJµ−1(γµνar/a)Jµ(γµν′ar/a)] ,%!:A # #Y

P8Rν 6= ν ′

A MÔHWOHEHIoW1JLSHSM^WwQ

ar/a = 0, 96qaT"INPBSHWMnGHO[R_Q1PBP8KNJLTUJ!R"W8IVPSMÔHWBT"IVP *OHIVT"P

ν = ν ′s!JLT_INE,QINE,s9R"W8MLE

[q ,nAgÌVP$SHW8M9GHOHRQ1PFI[S[W8R"joJLE,Q$GHIVPKNMÔHS[TUJL]ÎVPFR"E,Q8INW8jM9GYJLOP8MÊ,QBJLTQ1IVW8EHXVPFIVQGHIiTlf MLW8GHWIkGHI0, 075

qKNMÊ[P8R"G[XNW8XVP PXNSYJLWXNjINE,Q q R"T_P EHIPMÊ,QdGHMÊ[KSYJ!P EHXV]^T"R_]Î J [T"INPVA INSINEHGHJLE,Q q TUJeKNMÊ,Q1WR HOnQ1R"MLE GHIKNIVPQ1IVW8jINP$GYJLE[P$T"IkKNJLT"KVOHTT|f JLGHjR_QQ:JLE[KNIJLKNMÔ[PQ1R*OHI GHIkQ1W8JLEHP pIVWQIVPQR"E pXNW8R_INOHWIi#

5.10−3 G:uq T"INP$S[W8M GHOHRQ1PkPK JLT"JLR"WINP "!:A$# # InSHWR"joJLE,Q GHINPiKVMÔHSHT"JL]ÎVPiINE,Q1WIyG[R_zXNW8IVE,Q1PijMnGHIVPkP8MLE*QGHMLEHKDEHXN]LT"R"]LXNPkJ!OSYJLW8JL]^W1J!SHZHI HA W8IVTUJ!Q8R"MÊ [A LA

`cfhInSHWINPP8R"MLE %!:A*q

k2zµν = k2 −

γ2µν

a2 ,

SzIVO[QxKNMÊ[GHOHR"WIB# O[EHIBs!JLT_INOHW.R_j J!]^R"EYJ!R"W8IxSMÔHWkzµν

PRγ2

µν/a2 > Re(k2)

qYKNI?9O[RKNM^WW8INPSzMLEHG #iGHIVPMÊHGHIVP\XVs!JLEHIVP8KNIVE,Q1INPuP8OHRsLJ!E*Q

zA

`cf JLGHjR_QQ:JLE[KNIζzµν = ζz0µν = ζz`µν

P8O[WxT_INP.SYJ!W8M^R_P.P8R_Q8OHXNIVP\INEz = 0

IQz = `

INPwQvJLT"M^WP.XNKVW8R_Q8I

ζzµν ≈ 1 + j√2

√

k0

[

γ2µν/a

2

k20

√

` ′v + (γ − 1)

√

`h

]

SMÔHWRe(k2) ≥

γ2µν

a2

ζzµν ≈ 1 + j√2

√

k0

[

√

` ′v + (γ − 1)

√

`h

] SMÔHWRe(k2) ≤

γ2µν

a2 .

%!:A #B,^

! / 0 cT_OHP8R_INOHWPpM^W8jINP\SINO[sLINE,QvVQ1WIINE,s9R"P1J!]^XNIVPvSMÔHW.T_I$K:ZYJLjSGHI$s9R_Q8INPP8IBGHIT|fhXNjIVQ8Q8INOHW

ve

OHE[IBs9R_Q8INPP8IOHEHR)pM^W8jI

ve = Ve

q OHE[IBs9R_Q8INPP8IEHMÊyOHEHR)pM^W8jILqnR"jSzM^PXNI$SYJ!W.T"INP\KNJLW1JLKQ1XVW8R"PwQ1R9O[INP.jXNK J!EHR%*OHIVPxG[IBT"JijoIVj HW1J!EHILq

ve(r) = VeJ0(ker) − J0(keae)

J2(keae),

%!:A # ,

GHMLE*QuTUJis!JLT"IVOHWvjM ÎVEHEHIIVPQVe

IVQuMke = ω

√

MS/TMS

XQ:JLE,QvTUJ j JLPP8IP8OHW JLKNR9O[IIVQTT"JiQ8INEHPR"MÊ GHIT"J jINj HW1JLE[I$GHOjR"KVW8M^S[ZHMÊHIFXVjoIQ8Q8INOHW1A

OHE[I s9R_Q8INPP8IcEHMÊBOHEHR)pM^W8jILqVXNKNWR_Q8I PMÔHPaTUJpM^WjoI G f OHEBGHXsÎNT_M^SHSINjINE,Q P8OHWaT"J HJLP8IcGHIVP pMÊ[KVQ1R_MÊHPSHWM^SHWINP

ψc0ν(r)

qHT"IVP.joM9GHIVP.P8OHSXNWR"INO[W8Pv# #XQ:JLE,QxEHXN]LT"R"]LXNPNqve(r) = Aψc

00 +Bψc01(r) ,


PM^R_Qve(r) =

A

a0b00+

B

a0b01J0(kw01r) ,

"!:A$# *T_INP\KNMLEHPQ1JLE,Q1INP

AIVQBXVQ1JLE,QxGHXVQ8INW8jR"E[XNINP\SMÔHW.P8J!Q1R_P JLR_W8IFT_INP\KNMÊ[GHR_Q8R"MÊHPrP8OHRsLJ!E*Q8INP

ve(ae) = 0IQ

< ve >e= Ve ."!:A$# L

IVPvG[INOyKVMÊHGHRQ1R"MLEHP%!:A # ^rSINWjoIQ8Q8INE,QxGgf XVKNWR"W8IA

B= −b00

b01J0(kw01ae)

IVQVe =

A

a0b00+

B

a0b01

2

γ01

a

ae

J1(kw01ae) ,

KVI9O[R G[MÊHEHIA =

Ve

1 − αa0b00

IVQB = − Ve

1 − α

a0b01J0(kw01ae)

,"!:A$# A^

J sLINKα =

2

γ01

a

ae

J1(kw01ae)

J0(kw01ae).

"!:A$# L` fhInSHWINP8PR"MÊGHOyK:ZYJLjSyG[Is9RQ1INPP8I

ve

G[IBTlf XVjoIQ8Q8INOHW.IVPQxGHMÊHKGHMLEHEHXNISYJ!W

ve(r) =Ve

1 − α

[

1 − J0(kw01r)

J0(kw01ae)

]

."!:A$# @^

EHM!Q1INW*OHIuKVINPrQ8W8M^R_PrInSHWINPP8R"MLEHP SHWM^SzMLP8XNIVP SMÔHWrT_IvK:ZHJLjoS GHIvs9R_Q8INPP8Ive

E[IuSINO[sLINE,Q\SYJLP Q1WIKVMÊHP8R_GHXNWXNIVP$KNM^jjoI!I[JLKQ1IVPÂ EbIVzIVQNqgTUJjoIVj HW1J!EHIXQ:JLE,Q nIkPOHW$T"IVP M^W8G[PNq PMÊbjMÔ[sÎVjoIVE,QEHI$SINOnQ\VQ1WI$KNIVT"OHRzGgf O[EyS[R"PQ8MÊDSHT"JLE SHWINjR"VW8IFI[S[W8INPP8R_MÊ A fmJ!O[Q1WIBSHJLWQNq[T"IuK:ZYJLjSDGHIFs9R_Q1IVP8PIBG[IT"JjINj [W1JLEHIGHMLR_Qv#oT"J pM^R_P\GHXNSINEHG[W8IG[INPuKNJLW1J!KVQ1XVW8R_PQ1R*OHINPxjoXVK JLE[R%*OHINPxGHIKNINT_T"I KNRIQuG[INPuK:ZHJLjoS[PJLKVMÔHPwQ1R%*OHIVPvPOHW\P8IVPvGHIVO JLKVINPVA

/ / / ! / 0

ÌNP\SHW8M9GHOHRQ1P.PK JLT"JLR"WINP< ve|ψt

µν >e

GHXNSINEHGHJLE,QuGHIT"JsnRQ1IVP8P8Ive

P8MÊ,QxGHMÊ[EHXNPxSYJLW

< ve|ψtµν >e=

∫ 2π

0

∫ ae

0

ve(r0, θ0)ψtµν(r0, θ0)r0 dr0 dθ0 .

àJks9R_Q1IVP8PIve

IVPQ\P8O[SHSzMLP8XNIBR"E[GHXNSINEHGHJLE,Q1I$GHIT|f JLEH]LT"IθqHP8R HR"IVE*OHI$PINOHTT_INP Q8INW8jINP.KVM^W8WINPSzMÊ[GYJLE,Q

#µ = 0

P8MÊ,QxEHMÊyE*OHT"PBpX A HA-,LJkIVQ R"T s9R"IVE,Q

< ve|ψcµν >e =

2πδµ0

a0b0ν

∫ ae

0

ve(r0)J0(kw0νr0)r0 dr0%!:A # ^J,

< ve|ψsµν >e = 0 .

"!:A$# W8M^R_PFInSHWINPP8R"MLEHP$INE,s9R"P8JL]Î JHT"IVP$SzMÔ[WFTUJ snRQ1IVP8P8I

ve

PMÊ,QGHMÊHE[XNINP$GHJLEHP$T"ISYJLW8JL]^W1J!SHZHIkSHWXNKVXNGHIVE*QNqT_INPuInSHW8IVP8P8R_MÊHP$GHIVP$SHW8M9GHOHRQ1PxP8KNJLTUJLR_W8IVP

< ve|ψc0ν >e

SzMÔ[WuK:ZYJ<*OHI>pMLW8jIGHIve

P8MÊ,Q$G[MÊHEHXVINPFGYJ!EHPT_INP\SYJLW8JL]^W1J!SHZHINP\POHR_s!JLE,Q1PVA

( ' ( ( ( ' ' (

! #"$&%'%')(*+ -,/.1023

` J s9R_Q8INPP8IyGHI T|fhXNjIVQ8Q8INOHWiIVPQoP8OHS[SzM^PXNIyOHEHR)pM^W8jIve = Ve

A ` fhInSHWINP8PR"MÊ %!:A # ^J,G[O SHWMnG[OHR_QP8KNJLTUJLR_W8I

< ve|ψc0ν >e

G[MÊHEHIBJLT_M^W8P

< ve|ψc0ν >e=

2π

a0b0νVe

∫ ae

0

J0(kw0νr0)r0 dr0 ,

P8M^RQ< ve|ψc

0ν >e=1

a0b0νSeVeV0ν ,

%!:A-, ,M

V0ν

IVPQxOHE5 JLKVQ8INOHW\P1J!EHP\GHR"jINEHPR"MÊ[P$9O[RgsLJ!O[Q

V0ν =

1P8Rν = 0

a

ae

2

γ0νJ1(kw0νae)

P8Rν 6= 0 .

%!:A-, #tJ,

#4 56! #"7&%7%'8*+.*9(+* :,/.103; -<.%'=610?>:&%A@B10$@C"7=D02 :%E"7 :F(+&%3=&@B*+ -F(&%G)>:93H)I0$*

` Jks9R_Q8INPP8IBGHIT|fhXNjIVQ8Q8INOHW.IVPQxP8O[SHSzMLP8XNIBEHMÊyOHEHR)pM^W8jIFIQvS[W8INE[GT"J pM^W8jI X&A !:A # ,

ve(r) = VeJ0(ker) − J0(keae)

J2(keae).

E J!R"P1J!E*QFOHP1JL]LIiGHINPW8IVTUJ!Q8R"MÊ[Pi'9A& ,9A$#Nlq @[A #Â-,lq nA'HA #NIQ @[A HA-,~GHIiTUJdWX&pXNWINEHKVIy # IVQGHIT|fhX *OYJ!Q8R"MÊ HA-,<@,qzTlf I[S[W8INPP8R_MÊ %!:A # ^J,vG[ObSHW8M9GHOHRQ$P8KNJLTUJLR_W8I

< ve|ψc0ν >e

IVPQXVKNW8RQ1IiPMÔHP$TUJojoVjoIpM^W8jI *OHIBT|fhInSHWINP8PR"MÊ %!:A-, ,

< ve|ψc0ν >e=

1

a0b0ν

SeVeV0ν ,

J sÎNK

V0ν =

1P8Rν = 0

a

ae

2

γ0ν

ke

k2e − k2

w0ν

kw0νJ1(keae)J0(kw0νae) − keJ0(keae)J1(kw0νae)

J2(keae)

P8Rν 6= 0 .

%!:A-, #

KJ ! #"$&%'%')=D&>:.1L=)%'(+0M>-&%63.!+&%?0$.+02&%GN>-O@PQ R"7=

àIK:ZYJLjS GHI s9R_Q8INP8PIve

G[IoTlf XVjoIQ8Q1IVOHWBINPQGHXVsLINT_M^SHSX P8O[WT"IVPjM9GHINPBSHW8MLSHW8IVPGHIoTUJ K J s9R_Q1XIVQSHW8IVEHGyTUJ pM^WjoIpX A !:A$# @,

ve(r) =Ve

1 − α

[

1 − J0(kw01r)

J0(kw01ae)

]

.

ÌNPkjMnG[INPPOHSXNW8R_INOHWPi# #dXQ:JLE,QiEHXV]^T"R_]^XNPBGYJLE[PkKNIQ8Q1IDInSHW8IVP8P8R_MÊgqT_INPSHW8M9GHOHRQ1PPK JLT"JLR"WINPEHIdP8MLE*QK JLT_KNOHT_XNP$*OHISMÔHW.T"IVP\joM9GHINPB [q ,rIVQ nq$#^qhKtA

cE JLR_P1JLE,QaOHP8JL]Î GHINP W8IVTUJ!Q8R"MÊHP'nA' ,nA #Nlq*'9A&[A$#NÎVQ( @nA HA;,,GHIcTUJ.W8X pXNWINEHKVIF # !,IQ G[INP X *OYJ!Q8R"MÊHP HA-,LJxIVQ HA;, @,qT|fhInSHWINPP8R"MLE %!:A # ^J,aGHOkSHWMnG[OHR_Q P8KNJLTUJLR_W8I

< ve|ψc0ν >e

IVPQ XVKNWR_Q1I(P8MÔ[P~TUJxjoVjoIpM^WjoI*OHIT|fhInSHW8IVP8PR"MÊ %!:A-, ,

< ve|ψc0ν >e=

1

a0b0νSeVeV0ν ,

J sÎVK

V0ν =

1PRν = 0

−J0(kw01ae)

[

1 +1

1 − α

[

J1(kw01ae)

J0(kw01ae)

]2] PR

ν = 1 .

%!:A-, # K

/ / pc0ν(`)

pc0ν(0)

Ì$KNM9I KNR_INE,Qpc

0ν(`)IVPQvGHMÊHE[XBSHJLWB HA'^

pc0ν(`) =

S < ve|ψc0ν >e

1 −Kv

j k0

S

ρ0c0

1

g0ν(0, `)+ Yr0ν

g0ν(`, `)

g0ν(0, `)

."!:A;, ,L

` fhInSHW8IVP8PR"MÊ GHI T"J pMÊHKQ1R"MLEg0ν(z, z0)

INPwQ GHMÊHEHXVIbSYJLWdT_INPdWINT"J!Q1R_MÊHPb"!:A-@LJ[q YA MLjoS[Q8IQ1IVE*O G[ITlf XV],JLT"RQ1XyGHINP J!GHjoRQ8Q1JLEHKNIVP

ζz0µν

IVQζz`µν

T"INP 9OHJLE,Q1R_Q8XNPϕ0µν

IVQϕ`µν

P8MÊ,Q XN]^JLT"IVPdIVQg0ν(0, `)

IQg0ν(`, `)

PNfhXNKNWR_sLINE,Q

g0ν(`, 0) = g0ν(0, `)

= − cos2 ϕ0ν

kz0ν sin(kz0ν`+ 2ϕ0ν),

"!:A;, ^g0ν(`, `) = g0ν(0, 0)

= − cosϕ0ν

kz0ν sin(kz0ν`+ 2ϕ0ν)cos(kz0ν`+ ϕ0ν) ,

"!:A;,,J sÎVK

ϕµν = ϕ0µν = ϕ`µν

GHMÊHE[XBSHJLW X A !:A ^JLR_EHP8R INEySYJLWQ8R"KVOHT"R_INW

1

g0ν(`, 0)= −kz0ν

[

(1 − tan2 ϕ0ν) sin kz0ν`+ 2 tanϕ0ν cos kz0ν`]

=

[

j kz0ν + jk2

0

kz0νζ2z0ν

]

j sin kz0ν`+ 2j k0ζz0ν cos kz0ν` ,"!:A;, L

IQg0ν(`, `)

g0ν(`, 0)= cos kz0ν`− tanϕ0ν sin kz0ν`

= cos kz0ν`+k0

kz0νζz0νj sin kz0ν` ,

"!:A;,<A^PM^R_Q

1 −Kv

j k0

S

ρ0c0

1

g0ν(`, 0)=

[

Yi0ν +Y 2

p0ν

Yi0ν

]

j sin kz0ν` + 2Yp0ν cos kz0ν` ,"!:A;,L

IQg0ν(`, `)

g0ν(`, 0)= cos kz0ν`+

Yp0ν

Yi0ν

j sin kz0ν` ,"!:A;,<@^

MYp0ν =

S

ρ0c0ζz0ν

IQYi0ν =

S

ρ0c0

kz0ν

k0

(1 −Kv) .

< ψc0ν >e,r

E JLR_P1JLE,Q.O[P1JL]ÎGHIKVINPxGHINOyG[INW8E[R"NWINP\W8IVTUJ!Q8R"MÊ[P GYJLEHP\Tlf I[S[W8INPP8R_MÊ "!:A;,<,^rGHIpc

0ν(`)qHR_T s9R"INE,Q

pc0ν(`) =

S < ve|ψc0ν >e

YT00ν + Yr0ν

[

Yp0ν

Yi0ν

j sin kz0ν`+ cos kz0ν`

] ,%!:A-, ,

MYT00ν =

[

Yi0ν +Y 2

p0ν

Yi0ν

]

j sin kz0ν` + 2Yp0ν cos kz0ν` .%!:A ,

`cfhInSHWINPP8R"MLEGHIpc

0ν(0)IVPQvGHXNG[OHR_Q8I$GHIo HA'<@,

pc0ν(0) =

j k0

1 −Kv

ρ0c0S

g0ν(0, `)

[

S < ve|ψc0ν >e

g0ν(0, 0)

g0ν(0, `)− Yr0νp

c0ν(`)

]

,

*OHRGHMÊHE[ILqYINE5 JLR_P1JLE,Q\OHP8JL]ÎGHIo%!:A-, ,^q

pc0ν(0) = pc

0ν(`)

[

g0ν(0, 0)

g0ν(`, 0)+ Yr0ν

j k0

1 −Kv

ρ0c0S

g20ν(`, `) − g2

0ν(`, 0)

g0ν(`, 0)

]

.%!:A #

ÌNP I[S[W8INPP8R_MÊHPB%!:A-, ,cIVQB"!:A;,* GHIVPpMÊHKQ1R"MLEHPg0ν(`, 0)

IQg0ν(`, `)

SzIVW8jIVQ8Q8INE,Q.GgfhXNKVW8R_W8IFT"JW8IVTUJ!Q8R"MÊP8OHRs!JLE,Q1I

g20ν(`, `) − g2

0ν(`, 0)

g0ν(`, 0)=

sin kz0ν`

kz0ν.

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0ν(0)qHR"T s9R_INE,Q

pc0ν(0) = pc

0ν(`)

[

Yr0ν + Yp0ν

Yi0ν

j sin kz0ν` + cos kz0ν`

]

.%!:A ,

/ 5/< ψc0ν >e,r

àIVP?pMLEHKVQ8R"MÊHP$S[W8M^SHWINPψc

µν

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< ψc0ν >e,r

PMÊ,QFJLT_M^W8PrGHMÊHEHXVINPxSYJLW

< ψc0ν >e,r=

2

a2e,r

∫ ae,r

0

1

a0b0νJ0(kw0νr)r dr ,

P8M^RQ

< ψc0ν >e,r=

1

a0b00

P8Rν = 0

1

a0b0ν

a

ae,r

2

γ0νJ1(kw0νae,r)

P8Rν 6= 0 .

%!:A *

/ 5/ / < p(`) >r

2 0

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R_E*Q8W8M9GHOHRQkGYJLE[PiT"IdKNJLT"KVOHTrGHIDT"JSHWINP8PR"MÊ jM ÎNE[EHI< p(`) >r

POHWiT"IdWXNKVINS[Q8INOHW OHEHIDSMÊHG[XNW1JQ1R"MLEur(r)

GHITUJ pM^WjoI$POHR_s!JLE,Q1I ur(r) =

J0(krr) − J0(krar)

J2(krar),

"!:A LM

krINPwQvG[XYEHRaSMÔHW.T_I$W8XNKVINS[Q8INOHWxGHITUJkjNjI J-VMÊ59O[I

keSMÔHW.Tlf XVjoIQ8Q1IVOHWB X& A !:A # ,A

àJ jM ÎVEHEHIBSzMÊ[GHXNWXNI< p(`) >r

IVPQuJLT"M^WP XNKVW8RQ1I

< p(`) >r =1

Sr

∫

Sr

ur(r0)p(r0, `) dS0

=2

a2r

∑

ν

pc0ν(`)

∫ ar

0

ur(r0)ψc0ν(r0)r0 dr0 ,

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< p(`) >r=∑

ν

1

a0b0ν

pc0ν(`)U0ν ,

"!:A A^J sÎVK

U0ν =

1P8Rν = 0

a

ar

2

γ0ν

kr

k2r − k2

w0ν

kw0νJ1(krar)J0(kw0νar) − krJ0(krar)J1(kw0νar)

J2(krar)

P8Rν 6= 0 .

"!:A L

d 5

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. $/ / / / 0 `cfhR"E[KNINWwQ1RQ1OHGHI Q 9SIdINPwQiT|fhR"EHKVINWwQ1R_Q8OHGHIoGgf O[EHIDjoIVP8OHWIdInSHWR"jXNIdPMÔHPiT"JpM^W8jI G f OHE XVK JLWwQ |Q nSILA

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V (y)A

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R"E[P8R|q9P8MLR_Q(OHEHI$]^W8JLEHGHIVOHWYG[XVQ1IVW8jR"EHXVI$#SYJ!WQ1R_WrGHI

nJLO[Q8W8INPr]^W1JLE[GHINOHWP

X1

qX2

qA_AAqXn

SYJ!WrTUJpMÊHKQ1R"MLE

fQ8INT_T"I*OHI

Y = f(X1, X2, ..., Xn) ,P8M^RQyIQxi

GHIVPINPQ8R"joJ!Q1R_MÊHPFWINPSzIVKVQ1RsÎVjoIVE*QGHIYIVQXi

qgTlf R_EHKNIVWQ8R_Q1O[GHI |Q nSIiKNMLjoSM^P8XVIuc(y)

INPwQGHMÊHE[XNISYJLW

V (y) = u2c(y) =

n∑

i=1

(

∂ f

∂xi

)2

u2(xi) ,[A$#

M T"IVPu(xi)

P8MLE*QxT_INP.R"E[KNINWwQ1RQ1OHGHIVP |Q nSIGHINPxXVs!JLT_OYJ!Q8R"MÊHPxi

G[INPXi

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UGHIBT|fhINPwQ1R"joJ!Q8R"MÊ

yGHITUJi]^W8JLEHGHIVOHW

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SYJLW\T"JkW8INT"J!Q1R_MÊ

U = ku(y) ,[A-,^

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#


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pqR"T J!O[Q

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qMp2

qMp3

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Mp1 =

√

M12M13

M23

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f JLSHW8VP\T"INP\WINTUJQ1R"MLEHP #LA$# LrIVQ ,9A @^

MpeMpr = ZEYT ,JtsLINK

ZE =−1

jωCref

uA

uB

u′Bu′A

,

(* $ ( ( ( ( ( (

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qi = 1, 2, 3

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∑

[

(

∂ Mp1

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V (x) +

(

∂ Mp1

∂xi

)2

V (xi) +

(

∂ Mp1

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]

.[A *

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V (x) .[A'^

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Mp1 SHJLWQxT"IVPvKVMnI KNR"IVE,Q1PuG[IKVM^W8WINKVQ8R"MÊGHIBSHW8IVP8P8R_MÊIQFGHIQ1IVjoSXNW8J!Q1O[W8ILq T_INP\SYJLW8JLjoQ1WINPxT"R"XVP\OHEHR

*OHIVjoIVE*QJLO jR"KNWM^SHZHMLEHINPdPMÊ,QDQ1MÔ[PDKNM^jSHW8R_PdINE,Q8R"NWINjINE,Q GYJLE[PDT|fhInSHW8IVP8PR"MÊ GHIbTlfmJLG[joRQ8Q:J!EHKNIJLKVMÔHPwQ1R%*OHIGHINPdjR"KVW8M^S[ZHMÊHIVPb X& A$,nA-,<A[q,nA-,<@, JLR"E[P8R#9O[I GYJLEHP T|fhInSHW8IVP8PR"MÊ G[O s^M^T_OHjoIyIVQdGHIT"JT_MÊH]ÔHIVOHW Q1M!Q:JLT_I$GHITUJkK J s9R_Q8XpM^W8jXNI$SYJ!W.T"INP\jR"KVW8M^SHZ[MÊHINP.IQ\T"I$KVMÔHSHT_INOHWVA

! T~WINPQ8Ii#dP1J sLM^R"WFPR~T"Js!JLT"IVOHWFGHIT|fhI K JLKVR_Q8XGgfhOHEbjR"KNWM^SHZHMLEHILqGHXVQ8INWjoR_EHXNIkSHJLWFTUJjoXQ1ZHM9GHIkG[IWXNKNR_SHW8M9KNRQ1X!qINPQBR_E YOHIVEHKNXVI SYJLWT_INPSYJLW8JLjoQ1WINPGHINPBGHIVO JLO[Q1WINPjR"KNWM^SHZHMLEHINPo KVI *OHRcEHI GHIs9W1JLRQSYJ!P VQ1WI T"IrK JLP JFS[W8R"MLW8R?A`cfhXVQ8OHGHI$*OYJLT_R_Q1J!Q1RsÎ W1JLSHR_GHI*OHRHPOHR_Q~GHI.T|fhI KNJLKNRQ1X

Mp1q!InSHWR"jXNI PINT"MLEiOHE

jMnG[NT"I\SHZ 9P8R*OHIFP8R_joS[T"R$HXLq,JLR_EHP8R:*OHIvGHINPrJLSHSHT_R"KNJ!Q1R_MÊHP E*OHjXNW8R*OHINP(P8R"jSHT_INPcSzIVW8jIVQQ1INE,Q(GHIxGHMÊHEHIVWGHIVPvXVT"XVjoIVE*Q8P\GHIW8XNSMÊHPILA` f I oKNJLKNRQ1X

Mp1

GHOjoR_KNW8MLSHZHMÊHI#IVPQuGHMÊHEHXVISYJ!WvTlf I[S[W8INPP8R_MÊ [A ,qKVM^jS[Q1IQ1IVE9OGHID #Â$# ^qPM^R_Q

Mp1 =

√

M12M13

M23=

√

ZE12ZE13

M23

√

YT12YT13 ,

MyT"IVP\JLGHjR_QQ:JLEHKVINP\JLKVMÔHPwQ1R%*OHIVP\GHIBQ8W1JLE[P pINWwQYT ij

P8MÊ,QxXNKVW8R_Q8INP\Ggf JLSHWNP\T|fhInSHWINP8PR"MÊ ,9A$#B,L

YT ij = jωVij

γijPij

CTHij + Yi + Yj ,[A-A^

MVij = V + Vfi + Vfj + V +

i + V +j

IQCTHij

GHXVSzIVEHGDGHINP.KVMÊHGHRQ1R_MÊHP(JLj HR"JLE,Q1INPVqnG[Ods^M^T_OHjoIVij

IQGHIT"J POHW J!KNI$Q1MLQ1JLT"I

Aij

ARlA ( àJ G[XNW8Rs^XVISYJ!WQ1R_INT_T"IuG[I

Mp1

SYJLW.W8JLSHSM^WQx#iTlfmJLG[joRQ8Q:J!EHKNIY1

GHMÊHE[IJ!T"M^WP

∂ Mp1

∂Y1=

√

ZE12ZE13

M23

1

2

[

√

YT12

YT13

∂ YT13

∂Y1+

√

YT13

YT12

∂ YT12

∂Y1

]

,

J sÎNK ∂ YT12

∂Y1

=∂ YT13

∂Y1

= 1qHKVI?*OHRaKVMÊHGHOHRQx#

∂ Mp1

∂Y1=

1

2Mp1

(

1

YT12+

1

YT13

)

.[A L

uIFT"JjoVjoI# J-NMLEgqnT"JGHXVW8R_sLXNIFSHJLWQ8R"INT_T"IxGHIMp1

SYJ!W W1JLS[SzM^WwQ.JLO DJLG[joRQ8Q:J!EHKNIVPY2

IVQY3

PNfhXNKVW8RQ∂ Mp1

∂Y2

=1

2Mp1

(

1

YT12

− 1

YT23

) IVQ ∂ Mp1

∂Y3

=1

2Mp1

(

1

YT13

− 1

YT23

)

.[A-@^

(* $ ( ( ( ( ( ( +

àIVPJLGHjR_QQ:JLEHKVINPuJLKNMÔ[PQ1R*OHINP$GHIQ1W1J!EHP pIVWQYT ij

G[XNSINEHGHIVE,Q#oTUJ pM^R_PuGHIVP$SYJLW8JLjoQ1WINPFGHIVPFjR KNW8MLSHZHMÊHIVPIQBGHINPBKNMÊ[GHR_Q8R"MÊHPJ!j HRUJ!E*Q8INPSINEHGYJ!E*QK:ZYJ<*OHIjINP8O[W8ILA ÌNPBjR"KNWM^SHZHMLEHINP$MÊ,QBGHIVPK JLW8JLKVQ8XNW8R_PQ8R%*OHINPrSzIVOdGHR_zXNWINE,Q1IVP.IVQrT"INP(X *OHR_SzIVjoIVE*Q8P O[Q8R"T"R_P8XVP(SMÔHW(T"INP(XVQ1JLT"MÊ[EYJL]ÎVPrSINW8jIVQQ1IVE*QGHI JLR"WI INE PM^WQ8I$9O[I.T"INP KVMÊHGHRQ1R_MÊHPJ!j HRUJ!E*Q8INP~s!JLWR"IVE*QSzIVOG f OHE[IxjINPOHW8I\#FTlfmJ!O[Q1WI jINPOHW8IVP(#SHW8IVP8P8R_MÊiKNMLE*Q8W^T_XNI!q!T_MnKNJLTnKVT"R_j J!Q8R"PXtA R"EHPR|qT"IVPJLG[joRQ8Q:J!EHKNIVPJLKVMÔHPQ8R%*OHIVP GHI(Q8W1JLEHP pINWwQ

YT ij

P8MLE*QP8OHSHSM^PXNINP(SzIVOoG[R_zXNW8IVE,Q1INP INE,Q8W8I\INT_T"INPVA,ÌNP GHINO GHINWEHR"VW8INPcGHXVW8Rs^XNIVP(SHJLWQ8R"INT_T"IVPP8MÊ,Q(GHMÊHK"*OYJLPR E9O[T"T"IVPvIQFTlf I oKNJLKNRQ1X

Mp1

GHOjoR_KNW8MLSHZHMÊHI#SINOnQFGHMÊ[KVQ8W8IKNMÊHPR"GHXVW8XVIKNMLjojIR"EHGHXVSzIVEHGYJLE,Q8IGHINPvJLGHjR_QQ:JLEHKVINP G[INPxjoR_KNW8MLSHZHMÊHIVP#,kIVQ [AvEHI JLS[SHT"R_K J!Q8R"MÊyE*OHjoXVW8R9O[IijoMÊ,Q8W8I *OHIiT_INPFG[XNW8Rs^XVINPSYJLWwQ1R"IVT"T_INP ∂ Mp1

∂Y2

IQ ∂ Mp1

∂Y3

J!Q8Q8INR_]ÊHINE,QJLObj J [R_jkOHj

10−3∂ Mp1

∂Y1

SMÔHWuGHINPGHR XVW8INE[KNINP I9Q1WNjINPbINE,Q8W8I T"IVP$JLGHjR_QQ:JLEHKVINPY1

qY2

IVQY3AràJ SYJLWwQ1R"KVR"SYJQ1R"MLEG[I KNIVP Q1IVW8jINPdGYJ!EHPdT|fhR"E[KNINWwQ1RQ1OHGHIyKNMLjoSM^P8XVI

uc(Mp1)SINO[QDG[MÊHK Q1W8I

EHXN]^T_R"]^XVILAR"R|A ( (* (* ( àIVPrKNJLW1JLKQ1XVW8R"PwQ1R9O[INPr]LXNM^jXVQ8W8R%*OHIVP(G[INPrjR"KVW8M^S[ZHMÊHIVP(EgfhMÊ,Q*OgfhOHEHIuSYJLWQ8R"KVR"SYJ!Q8R"MÊiWXNGHO[R_Q1IvGYJLEHPT"INPW8XVP8OHTQ:J!Q8P Ggf XQ:JLT_MÊHEYJ!]ÎLAcàJ S[W8M<pMÊ[GHINOHW IQ T_I sLM^T"O[joIpW8MLE*Q1JLT.IVQoEHIySHW8XVP8IVE*Q8INE,Q*OHISzIVOGHI s!JLWRUJ!Q8R"MÊ[PdGgfhOHE jR"KVW8M^S[ZHMÊHIe# T|f JLO[Q8W8ILAvE PR"jSHT"IK JLT_KNOHT$SINWjoIQ GHI PVfmJ!P8P8O[W8INW*OHI T"INPK JLW8JLKVQ8XNW8R_PQ8R%*OHINPc]^XVM^jXVQ1WR%*OHIVP GHINPcjR"KNWM^SHZHMLEHINP,FIQ $MÊ,QcOHE[IxKVMÊ,Q1WR HO[Q8R"MÊiE[XN]^T_R"]ÎNJ HT"I(GYJLEHPT|fhR"EHKVINWQ8R_Q8OHGHIFKVM^joSM^PXNI

uc(Mp1)A

4 M^R_QX1

qX2

qX3

KNINPSYJLW1J!joQ1W8IVP]LXNM^jXVQ8W8R%*OHIVPNqaT"IVPG[XNW8Rs^XVINPkSYJ!WQ1R_INT_T"INP$G[IMp1

SYJ!WW8JLSHSM^WQJLO IVPQ1R_j JQ1R"MLEHP

x1

qx2

qx3

GHINPX1

qX2

qX3

PVf XVKNWR_sÎVE,Q

∂ Mp1

∂x1=

1

2Mp1

[

1

M12

∂ M12

∂x1+

1

M13

∂ M13

∂x1

]

,[A ^J,

∂ Mp1

∂x2=

1

2Mp1

[

1

M12

∂ M12

∂x2− 1

M23

∂ M23

∂x2

]

,[A

∂ Mp1

∂x3

=1

2Mp1

[

1

M13

∂ M13

∂x3

− 1

M23

∂ M23

∂x3

]

.[A LKt

àIVP SYJLW1J!joQ1W8IVPd]^XVM^joXQ1WR%*OHINPXi

XQ:JLE,Q P8O[SHSzMLP8XNPJ s^M^R_WdOHEHIeR_E YOHIVEHKNIbX&9O[R_s!JLT"IVE,Q1IbP8O[WDT"INPSHW8M9GHOHRQ1P

Mijq*T"IVP G[R_zXNW8IVE,Q1INP GHXVW8R_sLXNIVP(SYJ!WQ1R_INT_T"INP ∂ Mij

∂xi

P8MLE*Q(KNMLEHP8R_GHXNWXNINP(KNM^jjI.X *OHR_s!JLT_INE,Q1IVPNAàIVPG[XNW8Rs^XVINPSHJLWQ8R"INT_T"IVPFSYJLWW1J!SHSzMLWQJLO SYJ!W1JLjVQ8W8INPGHIVPjR"KVW8M^S[ZHMÊHIVP ,DIVQ SzIVO[sÎVE*QGHMÊHKVQ1WIEHXN]^T_R"]^XVINP.GYJ!EHP\T"I$K J!T"KNO[TgGHIT|fhR"EHKVINWwQ1R_Q8OHGHI$KVM^joSM^PXNI

uc(Mp1)A

R"R_R|A ' (*( ( ( ( ZYJ<*OHIrS[W8M9GHOHR_Q GgfhI KNJLKNRQ1XNP

Mij

GHXQ1INWjoR_EHX GYJLEHP T"IVP KVMÊHGHRQ1R"MLEHP JLj HR"JLE,Q1INPrPij

qTij

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PqTgJ YEGHIcGHXQ1INWjoR_EHINWT_INPI oKNJLKNRQ1XVPR"EHGHRs9R"GHOHIVT"T_INP

GHINPdjR"KVW8M^S[ZHMÊHIVPNA ZYJ<*OHIbSHWMnGHO[R_QMij

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CPi

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Mp1

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KNRQ1XLq[SzIVO[QxVQ1WIKNMÊHPR"GHXVW8XVIBKVM^jjoI$R_EHGHXNSINE[GYJLE,Q1IGHIVPvSYJ!W1JLjVQ8W8INP.G[INPvJLO[Q1WINP\jR"KNWM^SHZHMLEHINPVA


àJ KNMÊ,Q1WR HOnQ1R"MLE GHINPiSHJLW1JLjVQ8W8IVPiGHIdQ 9SzIXi

GYJLE[PiT|fhR"EHKVINWQ8R_Q8OHGHI KVM^jSzM^PXNIu2

c(Mp1)[A*PI

WXNGHOHRQxJLOyP8INO[TaQ8INWjoI(

∂ Mp1

∂x1

)2

V (x1) .[A #,

#4 J .* "'0$ :I( "7 :.*O&% 102B"'02&% >: :=&% (+* :F(H3&*" >: 3&%7( 02 > -*@PD07"7 #"7( %'(+0M> 9@P@B #"7=

Mp1

RlA ( ( # àJSHW8IVP8PR"MÊ

Pij

qnT"JBQ1IVjoSXNW8J!Q1O[W8ITij

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iP8O[W

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′B/u

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Pij = P0 + ∆Pij , Tij = T0 + ∆Tij , Hij = H0 + ∆Hij ,

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P0

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X[A'Â

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q∆X13

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q∆X13

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V (∆Xij)GHI KVINPiQ1IVW8jINP P8MÊ,QXN]^JLT"IVP

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uc(Mp1)PVf XVKNWR_QxJLT_M^W8P

[

(

∂ Mp1

∂∆X12

)2

+

(

∂ Mp1

∂∆X13

)2

+

(

∂ Mp1

∂∆X23

)2]

V (∆Xij) .[A # #

R"RlA ( (&( (ÌNPs!JLWRUJLE[KNINPGHINPW1JLSHSM^WwQ1PBGHIoQ8INEHPR"MÊHP

(ur/ue)ij = (uA/uB)ij(uB′/uA′)ijXVQ1JLE,Q1PkXV],JLT_INPNqaT"J

KNMÊ,Q8W8R [O[Q1R_MÊoG[INP(ur/ue)ij

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∆Xij

GHMLEHEHXNIBSHT_OHP.ZYJLO[Q[A$# #tA

! / 0 2 / / / !

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(* #

`cfhI KNJLKNRQ1XMp1

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X A#Â @,A àJ GHXNWR_s^XVISHJLW Q1R_INT"T_IuGHI

Mp1SYJLW.W8JLSHSM^WQx#

U0GHMLEHEHIGHMÊHK

∂ M1

∂U0

=Mp1

U0

,

KNI*OHRGHMÊ[EHIOHEHIKNMÊ,Q8W8R [O[Q1R_MÊ GHIUpol

GYJLEHP\Tlf R_EHKNIVWQ1RQ1OHG[I$GHIMp1

G[I(

∂ Mp1

∂U0

)2

V (U0) =M2

p1

U20

V (U0) ,[A #B,^

V (U0)XQ:JLE,QxTUJs!JLWRUJLE[KNI$GHI

U0

A fmJ!O[Q1WI SYJLWQNq GHIVP S[ZHXNEHMLjoVEHINPaSYJLW8JLP8RQ1IVPGgf MLW8R"]LR"EHI XNT"IVKVQ8W8R%*OHI MLOJ!KNMÔHPwQ1R9O[I R_E,Q1INWwsÎVEYJLE,Q GHOHW1J!E*Q

TUJ\joIVP8OHWIrSYJ!WQ1R_KNR_SzIVE*Q #uTlf R_EHKNIVWQ8R_Q1O[GHIcP8OHW T"INP W8XVP8OHTQ:J!Q8P GgfhXVQ1JLT"MLEHEYJL]Î!A INPSHZ[XNEHM^jNE[INPPMÊ,QT"I HWOHR_QJLj HR"JLE,Q qT"IVP pOHR_Q8INP$JLKVMÔHPwQ1R%*OHIVPNqT"JdGHR"PwQ1M^WP8R"MLEgq T"J GHRUJ!SHZHMÊHR_ILqzT"INPuIVzIVQ1PXVT"INKQ1WR%*OHINP$GHJLEHPTUJ ]^JLW8GHIGHINP\jR"KVW8M^SHZ[MÊHINPVA

cE YEgqgGHMLR_sÎVE,QVQ1WIiKNM^jS[Q1J HR"T_R"PXNPNqGYJLE[P$T"IKNJLT"KVOHTGHIVP$R"EHKVINWwQ1R_Q8OHGHIVPNqgT_INPFIVW8W8IVOHW8P$G fmJLWW8MÊ[GHR"PVq TUJW8XVSzXQ:J HR_T"RQ1XFIQuT_INP\INWW8INO[W8PuG[OHINPuJLOjM9GHNT_IFQ1ZHXVM^W8R9O[IO[Q8R"T_R"P8X$SMÔHW\WINSHWXNP8IVE,Q1INWuT"IP 9PQ8NjIDpINW8WINOHWP8OHW\T"JkSHW8R_P8IINEyKNM^jS[Q8I$GHINP\IVzIVQ8PxQ8ZHINWjoR*OHINP\IVQ.s9R_P *OHINO IQ\GHINPxMÊHG[INP\W1JLG[RUJLT_INP:AàIVP\R"EHKVINWQ8R_Q8OHGHINP\POHW\KNINP\SYJ!W1JLjVQ8W8INP\PMÊ,QvPINOHT_INjINE,QxXVs!JLT_OHXNP\GHIjoJLEHR"VW8IFR_E,Q1OHRQ1R_sLILA

!

àI ]^W8JLSHZHR *OHIiGHI TUJ Y]Ô[W8I [A$# WINSHWXNP8IVE,Q1ILq INEpMÊHKQ1R_MÊeGHI TUJpW8X *OHINEHKVILq T|fhR"EHKVINWwQ1R_Q8OHGHIkXVTUJLW]^R"I!qJ sÎNKBOHE JLKVQ8INOHWxGgfhXNTUJ!W8]^R_P8P8IVjoIVE,Qy X A [A^qPOHWxT"IjoM9GHOHT_I G .G[ITlf I oKNJLKNRQ1XGgfhOHEjR"KVW8M^S[ZHMÊHI` M[Q8INE*OHIVPJLO ` .A IVQQ1IkR_EHKNIVWQ1RQ1OHG[ISINO[QuVQ8W8IKVM^jSYJLW8XVI#oKNINT_T"IBM[Q8INE*OHIkJ!O c` pQ1J gA[ALA E YJ!P8P8IVPpWX *OHINE[KNINPIVQ OHP *Ogf #INE,s9R"WMÊ !q Tlf R_EHKNIVWQ8R_Q1O[GHI GHO ` KNM^WW8INPSzMLEHG # KNINT_T"IGHO c` A FJLE[PFT"IVPvZYJ!O[Q1IVP pW8X 9O[INEHKVINPNq T|fhR"EHKVINWQ8R_Q8OHGHIGHMÊHE[XNIkSYJ!WvT_I` IVPQ$POHSzXVW8R_INOHWIGHISHT"OHPGgfhOHEG[R nR"VjoIFG[IBG #iKNIVT"T_I$GHO ` #! LA

102

103

104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

"$#&%'(*),+$-,)/.10243

5 6798:<;=7 ><?;@ ABC D EF

<(=?>.@ [A$# G ! EHKNIVWQ8R_Q1O[GHIXNTUJ!W8]^R_IuP8O[WvT_I$joM9GHOHT_IFGHIT|fhI K J!KNR_Q8X$GgfhOHEjR"KVW8M^S[ZHMÊHI$` A


*G nq #*G # nq

#nq$# G # nq-A nq #*nq G # A nq # A[q # G nqnq$# G nq nq$#

~@ [AG ! EHKVINWQ8R_Q8OHGHINPxXNT"JLW8]LR"INP JLEHEHMÊ[KNXNIVPvSHJLW\T"I c` SzMÔ[W.GHINPxjoR_KNWM^SHZHMÊ[INP\` A

vEHIJ!EYJLT$9PI$GHITUJkSYJLWwQ.W8IVTUJ!Q8R_sLI$MnKVKNOHSXNI$SYJ!W.T"INP.G[R_zXNW8IVE,Q1P\SYJLW8JLjoQ1WINP.GgfhR"E!HOHINEHKVIGYJLEHP.KVIVQ8Q8IR_EHKNIVWQ1RQ1OHG[IyXVTUJLW]^R"I!qJLO YJLPP8INPoIVQoJLO ZYJLO[Q8INP pW8X 9O[INEHKVINPNq(INPwQoR_T"T_OHPQ8W8XNIDSHJLWiT"IVP]LW1JLSHZ[R *OHINP GHIVPY]LOHW8IVP [A IVQ [A [A INPd]^W8JLSHZHR *OHINPoSzIVW8jIVQ8Q8INE,QDGHIGHXVQ8INWjoR_EHINWoT"INPoSYJLW8JLjoQ1WINP SMÔHWoTUJ joIVP8OHWIGHIVP *OHIVT"PxOHEPM^R"E SYJ!WQ1R_KNOHT_R"IVW GHM^RQ.VQ8W8IjoR_PNA

P: 4%T: 5%

H: < 1%

CP,C

T :2%

Vbf

:12%

F0,Q :< 1%

Vf,Vsup :21%

Param. Cav. :17%

Param. Elec :13%

Cth :2%

Fuites, Bruit :< 1%

Repet., Arr. :8%

Diaphonie, Disto. :10%

Ondes Rad., Visco. :4%

<(=?>.@ [AG JLWQ8R"KVR"SYJ!Q8R"MÊiG[INP SYJLW1J!joQ1W8IVPcGgf R_E!YOHIVEHKNIxGYJLE[P(Tlf R_EHKNIVWQ1RQ1OHG[I.XNTUJ!W8]^R_IU(Mp)

###"%$LA E YJLP8PINPpWX *OHINE[KNINPVqTUJSYJLWQFTUJoS[T"OHPvR"jSzM^WwQ:JLE,Q8IGYJ!EHPFT|fhR"E[KNINWwQ1RQ1OHGHIXNT"JLW8]^R_IIVPQ$M9KNKVOHSXNIkSYJ!W

T_INPxSYJLW1J!joQ1W8IVP\]^XNM^jXVQ8W8R 9O[INPNq SHT_OHP\SYJLWwQ1R"KVOHT"R_NWINjINE,Q\T"I$s^MLT"OHjI pW8MÊ,Q:J!TGHOjR"KVW8M^SHZ[MÊHILA cEIVzIVQNqT_I\sLM^T"O[joI/pW8MLE*Q1JLT GgfhOHE jR"KVW8M^SHZ[MÊHI\INPwQ(Q8W8NP GHXVT"R"KNJ!Q #G[XVQ1IVW8jR"EHIVWNq,T_IxG[RUJLjVQ8W8I.R"E,Q8INW8E[IvEgfhXVQ1JLE,Q SYJ!PjINPOHW1J [T"IJ sÎVKSHWXNKVR"P8R_MÊ nA [A #A

MÔ[WGHINP pWX *OHIVEHKNIVP S[T"OHPXVT"Is^XNIVPNq,T_INPR_EHKNIVWQ1RQ1OHG[INPPOHWT"IVPSHZHXVEHM^jNEHIVP GHI.GHR_PQ1MLW8P8R_MÊkIQ P8OHWwQ1MÔnQGHI$G[RUJLSHZ[MÊHR"IvSHW8IVEHEHIVE*QxGHIVP.SHW8MLSzM^WwQ1R_MÊHPrQ8W8NP.XVT"Is^XNIVP\GYJLEHP Tlf R_EHKNIVWQ8R_Q1O[GHIuXVTUJLW]^R"I!A[àJkGHR"JLSHZHMÊ[R"IuIVPQNqIVE I IQ OHE SHZHXVEHM^jNEHIbGHXVT"R_K J!Qd# KNJLW1J!KVQ1XVW8R_P8INW K JLWDR_TvSINO[QyJ sLM^R"WdG[R_zXNW8IVE,Q1INPDMLW8R"]LR"EHIVPb KNJLSYJLKVR_Q8XSYJ!W1JLPR_Q1IINE,Q1WI T_INPDjR"KVW8M^SHZ[MÊHINP IVQ INE,Q8W8IeT"IVPdK& [T"INP1A.àJ SYJLWwQDR"jSM^WQ1JLE,Q1IS[W8R"PIbSYJLWdKVINP IVzIVQ8PSYJ!W1JLPR_Q1IVPFjMÊ,Q1W8I 9O f OHE[IiXVQ1O[GHIiInSXNW8R_joIVE,Q:JLT_IkSHT"O[P$JLSHSHWM pMÊHG[R"Is9R_P1JLE,Q#djR"IVOT"INPuK JLW8JLKVQ8XNW8R_P8IVWIVPQvEHXNKVINP8P8JLR"WILA

E YEgqnR"TzJLSHSYJLW8J"Q *OHI$T|fhR"E!HOHINEHKVIuGHI$Tlf R_joSXNGHJLEHKNIxGHODjR"KVW8M^SHZ[MÊHIuPOHW.T|fhI KNJLKNRQ1Xvs!JLOnQ.P8OHWwQ1MÔnQ


P: 3%T: 4%H: < 1%CP,C

T :< 1%

Vbf

:3%

F0,Q :2%

Vf,Vsup :5%

Param. Cav. :6%

Param. Elec :5%

Cth :< 1%

Fuites, Bruit :< 1%

Repet., Arr. :21%

Diaphonie, Disto. :30%

Ondes Rad., Visco. :19%

<(=?>.@ [A G ~J!WQ1R_KNR_SYJ!Q1R_MÊbGHIVPSYJ!W1JLjVQ8W8INPBGgfhR"E!HOHINEHKVIoGHJLEHPBT|fhR"EHKVINWQ8R_Q8OHGHIiXNT"JLW8]LR"IU(Mp)

# #"%$!A

SYJLWT_Irs^MLT"OHjI JLO YJLPP8IVP pWX *OHIVEHKNIVPNA9àJ pW8X *OHIVEHKNI\GHI.WXNPMÊYJLEHKVI\IVQ TlfmJLjM^WwQ1R"PP8IVjoIVE*Q~EgfhMÊ,Q 9O f OHE[IQ1WNPcSzIQ1RQ1I\SYJLWwQ(GHJLEHPcT|fhR"EHKVINWwQ1R_Q8OHGHI.XVTUJLW]^R"I!A vEHI\KVMÊHEYJLR_P8P8JLEHKNI\Q1WNP SHW8XVKNR_P8IxGHI\KNINP SYJLW8JLjVQ1WINPcEgfhINPwQGHMÊHKiSHJLP$EHXNKVINPP1JLR_W8I IVQT_INP$jXVQ1Z[MnGHIkJLKVQ8OHINT_T"INjINE,Q$IVEeOHP1JL]LILq HR_INE *OgfhR"jSYJLW J!R_Q1IVPuPINj HT_INE,QBJLT_M^W8PP8O P8JLE,Q1INPxR"KVR|A

E[MLQ1IVW *OHIBKNIQ8Q1IXVQ8OHGHIBGHINPxR"EHKVINWwQ1R_Q8OHGHIVPuJ XVQ8XI IVKVQ1O[XNIkJ s!JLE,Q 9O[IEHIBP8MLR"INE,QvW8IVjoR_P.INEKNJLOHPIT"IVPoI[S[W8INPP8R_MÊHPdEHMLW8joJLT"R_P8XNIVP SzMÔ[WoTlfmJLG[joRQ8Q:J!EHKNI JLKVMÔHPwQ1R *OHIGHI Q1W1J!EHP pIVWQ

YT

IQ T_INPjoM9GHNT_INPP8OHWT"IVP *OHINT_P$INT_T"INPuW8INSM^PINE,Q A uIVSHOHR_PNqR"T INPwQBJLSHSYJLWO 9O[IT_INPFI IQ1P$Q1Z[INW8jR *OHIVP$P8MÊ,QSHWR"P$IVE KVM^jS[Q1IiGHI JNMLE R"EHIHJ!KVQ1I IQ 9O[IyT_INPiI IQ1Pis9R"P *OHIVOEHIDGHMLR_sÎVE,QoSHJLPiVQ8W8IDEHXV]^T"R_]^XNP ,A ÌVPiR"EHKVINWQ8R_Q8OHGHINPP8OHW.T"JiSHW8R_P8I$INEyKNMLjoS[Q8I$GHI$KNIVPvG[INO I IQ1PxINPwQvG[MÊHK$EHIQ8Q1IVjoIVE,QvPMÔHP XVs!JLT_OHXNIGYJ!EHP\KNIVQQ1IXVQ8OHGHIGHIVPR"EHKVINWwQ1R_Q8OHGHIVP P8OHW.T_INP.XQ:JLT_MÊHEYJL]LINP IVEyS[W8INPP8R_MÊ GHI$joR_KNWM^SHZHMÊ[INP SYJ!W.TUJjXVQ8ZHMnG[IFGHI$W8XVKNR_SHW8M9KNRQ1XLA ~JLWInINjSHT_ILq[Tlf IVW8WINOHW(KNM^jjoR_P8IvP8OHW Tlf InSHW8IVP8P8R_MÊ GHIFT"JKNM^WW8IVKVQ1R_MÊdQ1Z[INW8jR *OHI

CTH

INPwQ\R"KVRzXsLJ!T"OHXVI# # GHI P8Js!JLT"IVOHWdpP8M^RQkIVE*s9R_W8MÊ nA<A yG # " $dSzMLOHWOHE joR_KNW8MLSHZHMÊHI ` q KNI *OHR(INPwQkQ1WNPkIVEGHINPP8MÔ[PGHIoT|fhINW8WINOHWKVM^jojR"PI INE WX JLT_R_Q8Xy SHT_OHPG[I [A GPOHWT"J s!JLT_INOHWGHIVPI oKNJLKNRQ1XVPM[Q8INE*OHIVPT"M^WP GgfhOHEyXVQ:J!T"MÊHEHJL]Î$SYJLW.WXNKVR"SHWMnKVR_Q1X!qHK4A [A'Â

g

\

\ d

IVQ8Q8IJ!EHEHInIGHXVsLINT"MLSHSzIOHE KNJLT"KVOHT.GHIySHW8M9GHOHRQP8K J!TUJLR_W8IDGHMÊ,QoT"I W8XVP8OHTQ:J!QoINPwQ OnQ1R"T_R"PXDGYJLEHPT"IK:ZYJLSHRQ1WI y nA! nA! ,rjoJLR_P *OHREgf INPQxSYJ!PxG[XVsÎVT"M^S[SzXBJ YEyGHIEHISYJLP\IVEJLT_MÔHWGHR"W(T"I$SHWM^SzMLPNAàI SHW8M9GHOHRQkP8KNJLTUJ!R"W8I

< y1ψt′

µ′ν′ |ψc0ν >σ1

q~GHJLEHPiT|fhInSHWINPP8R"MLE nA#"$ ,GHINPiKVMnI%oKVR"IVE*Q8Ppc

0ν(z)qINPwQ

GHMÊHE[XBSHJLW < y1ψ

t′

µ′ν′|ψc0ν >σ1

=

∫

σ1

y1(θ0)ψc0ν(a)ψ

t′

µ′ν′(a, θ0) dS0 ,

M& T|f JLGHjR_QQ:JLEHKVIFP8SXNKVR$ *OHIy1

GHIT|fhXNT_XNjINE,Qx# K JLW8JLKVQ8XNW8R_P8IVW pINE,Q1IMÔ Q1O('zIVP: INPwQpMÊ,Q8R"MÊ GHIT|f JLEH]LT"IOHEHR 9O[INjINE,QxIVQ\T"IVP/pMÊHKVQ8R"MÊ[P.SHW8M^S[W8INP

ψc0ν

PMÊ,QuR_EHGHXVSzIVEHGYJLE,Q1IVP\GHITlfmJ!EH]^T"Ik X A)"HA*LJkIQ+' A MÔHW(GHXNKVW8R_W8IuT"JPOHW JLKVI

σ1qθ0sLJ!W8R"IxGHI k#

2πIVQz0G[I`1#`1 + δz

qδzXQ:JLE,Q(Q1W8VPrSIVQ8R_Q G[IVs!JLE,Q

`1qYKNI *OHRGHMÊ[EHI

< y1ψcµ′ν′|ψc

0ν >σ1= a δz ψc

0ν(a)

∫ 2π

0

y1(θ0)ψt′

µ′ν′(a, θ0) dθ0 .-, A #

àIVPSHJLW1JL]LW1JLSHZ[INP.,iA$#IVQ/,iADGHXQ:JLR_T"T"IVE,Q$T"I K J!T"KNO[T GHI KNISHW8M9GHOHRQP8K J!TUJLR_W8IiGYJLE[PBT_I K JLPBM& T|f JLG10joRQ8Q1JLEHKNIF# KNJLW1J!KVQ1XVW8R_P8INW\IVPQxW8IVP8SINKVQ8R_sLINjINE,QuO[EHI pINE,Q1IJLEHE*OHT"JLR"WIFGgf JLGHjR_Q8Q1JLEHKVI$Ggf IVE,Q1W8XVI

Yf

MÔO[EW8XVP8I J!OGHI *OYJ!Q8W8I$Q1O('zIVPvK nT_R"EHG[W8R *OHIVP.Ggf JLGHjR_Q8Q1JLEHKVI$Ggf IVE,Q1W8XVIR_EHGHRsnR_GHOHIVT"T"I

Yt

A

2 3 54 6 FJLEHPoT"IKNJLP M& T|fhXNT_XNjINE,Q # K JLW8JLKVQ8XNWR"P8XyINPwQDOHEHIpINE,Q1I JLEHE*OHTUJ!R"W8I Ggf JLGHjR_QQ:JLE[KNI

Yf

q W8XVSYJLWQ8R"IOHEHR1pM^W8jXNjINE,Q POHWcTUJ$P8OHW, JLKNI

σ1 = 2πa δzq,T|f JLGHjR_QQ:JLE[KNI PSzXVKNR$ 9O[I

y1

INPwQ(R_EHGHXVSzIVEHGYJLE,Q1IxGHI\T|f JLEH]LT"IθIVQxINT_T"IFIVPQvXNKVW8R_Q8I

y1 =ρ0c0σ1

Yf =ρ0c0

2πaδzYf .

`cfhInSHWINPP8R"MLE 7,iA #rGHMÊ[EHIBJLT"M^WP

< y1ψt′


= ρ0c0Yfψc0ν(a)

∫ 2π

0

ψt′

µ′ν′(a, θ0) dθ0 ;-, A^

# A

"!#$% !'&& (&)! " * +& ,-

".( *!' /0 12

Ggf JLSHWNPT"IVP$InSHWINP8PR"MÊ[PdpX A"HA*LJdIVQ 'YuGHINP pMÊHKQ1R_MÊHP$SHWM^SHW8IVPNqTlf R_E*Q8XN]^W8JLT"IP8OHWθ0INPwQBE9O[T"T"ISzMÔ[W

µ 6= 0IVQ\SMÔHW

t′ = sqµ′ = 0

qHKNI *OHRGHMÊ[EHIBJLT"M^WP.SzMLOHW.T"I$SHWMnG[OHR_Q.PK JLT"JLR"WI

< y1ψt′


= ρ0c0Yfψc0ν(a)ψ

c0ν′(a)δt′cδµ′0 .

-,iA! ,

2 3 54 4 4 76 6 84 FJLEHPuT"IkKNJLPFM&bTlf XVT"XNjINE,QF#dK JLW8JLKVQ8XNW8R_P8IVWFINPwQOHEbW8XVP8I J!O G[I *OYJ!Q1WIkQ1O 'INPuGgfmJ!GHjoRQ8Q1JLEHKNIR"EHG[R_s9R 0

GHO[INT"T_IYtqYW8XN]LOHT"R_NW8IVjoIVE,Q\W8XNSHJLWQ8R"P.INE

θn = nπ/2n 9 # ^qYP8O[W *OYJQ1W8IPOHW JLKVINPxXN],J!T"INP.K:ZHJLKNOHE[I#

a δθδzq T|f JLGHjR_Q8Q1JLEHKVIuP8SXNKVR$ *OHI

y1

INPwQuXVKNW8RQ1I

y1 =ρ0c0a δθδz

Yt

P8Rθ = θn

IVQy1 = 0

P8Rθ 6= θn ,

IQvT"JkP8OHW, JLKNIσ1

sLJ!O[Q4δθa δz

A` f I [S[W8INPP8R_MÊ -,iA$#rG[MÊHEHIJLT"MLW8P

< y1ψt′


= ρ0c0Ytψc0ν(a)

3∑

n=0

ψt′

µ′ν′(a, θn) ,-,iA "*

PM^R_Q< y1ψ

t′


= ρ0c0Ytψc0ν(a)

Jµ′(γµ′ν′)

aµ′bµ′ν′

3∑

n=0

cosµ′θn

PRt′ = c

sinµ′θn

PRt′ = s .

-,iA&^ M^jS[Q8IQ8INE*OyGHINPxW8IVTUJ!Q8R"MÊHPrP8OHRsLJ!E*Q8INP

3∑

n=0

cosµ′θn =(

1 + (−1)µ′

)(

1 + cosµ′π

2

)

=

4P8Rµ′ = 0

jM9GHOHT"M4

0P8R_EHMÊ

,

-,iA-A,

IQ3∑

n=0

sinµ′θn =(

1 + (−1)µ′

)

sinµ′π

2= 0 ,

7,iA *L

IQcGHO JLR_Q *OHI.PINOHT_PcT"INP~joM9GHINP SMÔHW~T"INP *OHINT_Pµ < 3

P8MÊ,QcSHWR"P INE KVM^joSnQ1IpT"INPcJ!O[Q1WINP XVQ1JLE,QcP8OHS[SzM^PXNPJ s^MLR"W.OHEHIBKNMLE*Q8W8R 'HO[Q1R_MÊDEHXV]^T"R_]Î J'[T"Itq[Tlf I [S[W8INPP8R_MÊ 7,iA&LrGHIVs9R"IVE,Q

< y1ψt′


= ρ0c04Ytψc0ν(a)ψ

c0ν′(a)δt′cδµ′0 .

-,iA,

2 ;:=< > 6 f JLSHW8VPFT"IVPvI nSHW8IVP8P8R_MÊHP-, A ,.IVQi7,iA^q T"IBSHW8M9GHOHRQuPK JLT"JLR"WI

< y1ψt′


pX A ,iA #xSINOnQJLT_M^W8P.Q1WIBI nSHW8R_joX$PMÔHP\TUJ pM^WjoIF]LXNEHXVW1JLT_IBPOHR_s!JLE,Q8I

< y1ψt′


= ρ0c0Y1ψc0ν(a)ψ

c0ν′(a)δt′cδµ′0 ,

-,iA,M&

Y1

s!JLO[QYf

PRrT|fhXNT_XNjINE,Q#K JLW8JLKVQ8XNWR"P8IVWkINPwQiOHEHI pINE,Q1IDJ!EHE*OHTUJLR_W8IoGgfmJ!GHjoRQ8Q1JLEHKNIoJLKVMÔHPQ8R *OHIYfIQ

4Yt

P8RuKVIVQDXVT"XVjoIVE*QdINPQdOHE W8XVP8I J!O G[I *OYJ!Q1WIQ1O('zIVPdK9T"R_EHGHWR *OHINP GgfmJ!GHjoRQ8Q1JLEHKNIVPdJLKVMÔHPwQ1R *OHIVPR_EHGHR_s9R_GHOHINT_T"IVP

Yt

A

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