jason 1978 sonic boom report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long...

47
( 1 I I '<:!' I CJ) I I ! AQ \ :!) 0 I \ Q1 , <, I j I -- - -- -- -- -- . ./ "-'- = c..> U-I I -' - I u... C'.:> c,:: 'fo ehn ic. 1 Ro po rt JSR_78-OO NO, . rn b< t 1978 JASON 1978 SONIC 800M REPORT " G. Ch oi ''''''' S. Flat "" fl GaM,n F. SRI IMe'n&li"" . 1 1611 North Str .. , Ar lir>Q.on , 2220Y 03 01 067

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Page 1: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

( 1 I

I '<:!' I CJ) I I ~ ! AQ

\

:!)

0 ~

I ~

\ Q1 , < , I

j

I ----- - -----. ./

'~ "-'-= c..>

U-I I -' -I u...

C'.:> c,::

, ~

'foehn ic. 1 Roport

JSR_78-OO

NO, . rnb< t 1978

JASON 1978 SONIC 800M REPORT

" G. M""Oo~.ld , Ch oi'''''''

S. Flat"" fl GaM,n

F. hf~; n'

SRI IMe'n&li"" . 1 1611 North K~nt Str .. , Ar lir>Q.on , V,,~ini. 2220Y

03 01 067

Page 2: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

• •

REPORT DOCUMEJHAT~'O~"~" ~'~A~G~E~---T--::~:F~5~~=~-1

" ,

JASON 1978 SONIC jOaH REPORT . ;t 'i! -= , •.

'" 1611 Nonh K .. nt Arlin8[On. VA

'""'~ Str .. ,n

22209

Projects 1'00 l11hon Boulevard Ar l 1n8t on , VA 2Z209 UNCLAS~l FlED

"

Cl ... r ed !or open publicat ion; distribution unlt~it .. d .

SO~!C BOOHS

'" THtaMOSPIIERE

repor ts <h. 0'

.. of tn.

Co.,t MylteTY led to .. nu~b .. r of lho"-k ... " .. ,;.nto t he thenaoaphen. Thl s r SP"lt

boa. •• ) do not heat t h .. th .. ~.ph.r • • ~b ) andre) do not caulu' cllmic .. l chana •• 1n

• gollOH Of 1 NOV" IS OQOUH

studt", of ,h",,_ t ha t do not <h •

r

Page 3: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

..

DD fO~M 1473(B~CKI , J/I,IO 7J

lDmOH Of , .. Oil 15 " D.D~EH

Page 4: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

EXEClJ'rIV[ SlIMMARY

Press repo,ts ot "[;!l.St Co .. s t !o!ys te:ry Boo",s" have led to a nu",ber of

studi~~ of the propagation of shock ""aves into the cherl!losphHe . The

ene:gv, E 1n a n :-l~ "ave 15 proportional to th .. square of the r"lui" "

overpressure. S

s " t,p/p

"hen! L is the length of the N-w.,'''. p the ;atllb1'!nt pre ssu:e .. "d ~ p

th" Shock overprtssu:re . The fractional oone~gy los . du e to the entropy

jUlDP aCroU the shock travdbg in the x direction is proportional to

the r ellttve overpren"re at every point a long the travel path

IdE ;, S [ dx 21' L

\oIh e r. 'f 1s the r a tio of the heat capacities . Solv ing for the r e l,H ive

overpressure at high al tit"d es , we obtain a limiting strength of th .. Shock

"

'"'He H is the .cale height. Even though t h .. N-",.vt lengthens ;as the

shock propag;ltes upwuds , the incre;lu in sC;lle height insures t hat shock

rema ins " e;lk at ;Ill heilhts. For;l typ1cd Concorde )o!;lch 2 boOtll having

Page 5: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

, '(

an inlt1al Itrlolth of 100 Pa, the turnover hatlht of the boo. Ii 160 ~.

In ruch i nS this alt1;_,,0'; , 90% of the cnugy is 10,1 by 100 lao and 97X of

t he HIers), is l ost 1)y 160 b. As. r"""lt of the shoel< rluining .... k.

the chang ' 'o f ... blent tempe .. lture is · 2 only about 10 OK .nd the vel ocity

l m?l r ted t o the thermcs phere 11 about ) c~/s . fhe boac I •• 1.0 lengthened

to about 4 km, So that the peak fr equency at ground level. is .bout 0.1 Hz.

SST gene tlted hoo .. s thUB

3) do not CIU,,, en"",!".l "hanSI' 1n the thllnoosphe . e.

c..neral considerations of the propaga tion of Inhasound .uggeH •

nUlibe. of r •••• rch pr10rltl •• 1ndudto"

1) eonBt ruc t1 on and op.nUnn of dir.ctio ..... l Inhuound

attays t o monitor natural and artificial infta sound

of anHh:hl source.;

3) toupled infuson!c and !onoapheric oburvaUon to

study thl interaction of near aurtace ,Inluted

diiturbancl. vith ths t ono.phere;

4 ) conatruction and operation of a dirlctional arr lY of

infrasoniC sourCII to s tudy propalation conditione

in the ther.osphlrl.

,.

"

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CONTENTS

EXECUTIVE SllM1ARY

tIST OF FIGURES

1 O\'ER\'IEIi.

II PROPAGA:ION Of SONIC BOO~S IN A STRATIFIED A~OSPHERE

A. Sonic Boo," Chanct"rlstlc5 II. No Lengthening; Isothenul Atmosphere C. L .. ngrhenlng D. At!!los~he"ic )1oo;l d E. Sonic Booll1 Propu'tl",s as .. Function of Altitude F. Ot~n EUIOcts

111 RESEARCH NEEDS.

A. Obs"rvation"l II. theoretic.!

REFERENCES •

• •

APPEl;DIX A: SOSIC IIOOM PROPAGATlO);

A. Fundamental Equations 1\ . R" lat ions within the Sonic S¢OIll

C. R"lations aft .. :- $or.lc Boo", PAilage

• •

'" vi

, 7

8

" D D

" n

13

" • " • • 19

• J>

• J> • " • ;0

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LIST o r nCU1U:s

/

Hlillri 1 Pro?I,lt 1on o f In l~ ell N-~Ive . • ,

Tilure , Temperlt"r ., Pre.l"re I"d Density Dist ribution 1n [he Atmosphe re . "

Hgure ) Varh tion o f Sho ck St ren,th .nth Altit ude 1>

Hgur e , ~cay o f Energy o f a \leak Shoc k Pr opllating \'lnicIUy in the Atmospher e "

Fillure , Lengthen1n, of I \leak Shock 1n the At.osphere 20

Tilure '-1 Cond it ions in a Ptopl,lting \<.'''I k Shock 30

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OVERVlE:'.'

, Public attention ~as drawn t o unexplained atmospheric sonic phenomena

by a 5," r 1"5 o~ press r epOTU of ''I1yite rious East Coas t 1>o01n" fr o," e arly

Oecember 1977 through ~arch 19781 •2 , Numerous Citi zen reports of int ense

sounds in Ne~ Jerse y , in the vicinity of Charleston, South Carol in. and

011 the southwest ti~ of Nova Scotia prompted a number o f independe nt

In add i tion to the dtiu_I'I reports, certain of the

events were r ecorded by acoustic arrays of t he L~ont -Doherty Obser va t ory6

and by fei6~1c arrays of ~ont-Doherty Observatory , ~es tern Obs ervatory

and Baptist College ir. ehu-l". t'", ",h".t acoustic event s are dist1nguish,.d

by vertical motion witr. no horhontal dls)'lacement. Citizens also not1!d

l1ghts 1n th .. Il<y. ~"","t1me~ a~so~hted with the no i.". at tim"s wh"n

th ... re ""' no thunderstoT!' activity. Sound ev ... ntS have also b ... ... n repo n ... d

in oth ... r IIrell 5 of th ... IUH coast auch as t he Tide" a t "r r ... gion of \'lrginia.

but almost all r ... ports have come from Nova Scotia. Ne~ J ... rsey and South

Carol ina . Pre~~ r ... pOr~1 of e Ve nts in New Je rs ... y a nd Sou th Carolina c .... sed

in l ate March. but citizen r a poru of booms in Nova Scotia contir.ued

t hrough Au&ust 1978.

Public concern led Sena t o r Harriaon Wi l liam. (N .J .) t o reqU"' lt th ...

Whit ... Houee to c onduct an investigation. Frank PreiS. Dir ... cto r o f the

Offica of Sci ... n e. Ind Tachno l ogy Po11cy, refe rred the ... tt e r to the

1

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nepart~t of o.fan •• which in turn d •• tln.t.d th~ N.val k •••• reh Lahor­

.tory (NRL) II the leld ag~ney. The r~lulting NRL Icud;3 conc l uded thlt

I .Ijotlty of the New J~r.~y .nd South C.rol ln. booms could be ~xpllined

b)' the s"puson!c fli&ht of 1II1lHuy drtnft In th~ .... "nill' arus" off

the ~ast COI. t. The NRL .["dy not~d thlt unusuII [~perat"re Ind wind

conditione could prab.bly Iccount for thl Implifleltlon Ind propagation

aircraft ... eu r .. ponl1hl~ . Th~ NRL stud y Ilia bri~fly noted thu tht

boo •• i n Soua Scotil could be .ttributed t o Concorde fli,ht l.

In it l study, NRL ~xI .. in~d .. I ny Ilt~rnltive expllnltion, for the hooms,

including: nuelelr explosions, lIIilitary r.lelr ch Ind d~veloPlllent aetiviti~I,

military Or civilian Ulf of hl&h eli:plo.ivel, .hip dtltlC.rl , USSR ,hi p oplr-

Itions , ,.ophy.tel l ~xplorl[ion , antipodal events , mi •• il~ launches Ind re-

.nt ri.s, 10 .. Iltitudl ,"tellites, explolive freezing of ,uper eool.d ... t.r

vapor , .. ~t~orttl' , .. inter lightninl, direct '~l'lIIic ,"norltlon and th~

explosion of biog~nic or abiogenic IIIlthane. NRL conclud~d that all .. an-

.. d~ loure.s could be ~xcluded with the exc~ption of .t11tlry a i rerlft off

tIova Scoti.. NRL conddued tlMt natural sou r"~s "~n hl&h l y ilIIpr obabl~;

h~n.ive analYlll of .bout 600 I~plrlt.

Notin& t hlt t he reports of the .a.t coalt boo=s Ipprox1aat.ly coin-

elded with the initiation of Concor dl .~rvice into New York'i J.T. Kennedy

Airport in lat. Nove-ber 1911 , Jer~y Ston~ of t~ rederation of ~rlcan

Sc ienti st. su,.e.t~d th.t Concorde VI. r~.pon.1bl~ for the New J~r,ey and

2

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o

Charle.ton eVf-nta •• veIl Ill; for the Nova Scot1& b001U. If Coneo:rde lor"!'"

the source of the South Carolina boo=a. then the shock would have to propa-

gate 1200 km, the dhtance hom the lut s upersonic sepent of the flight.

Liszka] has observed at KiTuna, Sweden, Concorde flights at distances of

3000 to 4S00 '<m wHh Ii nilS amplitude of about 0.1 Pa in the fre~uene~

bAnd .. round 2 Hz. Liszka interprets the observed signal as a focused

boo," i:I ",hteh the signals from '" finite Sf-glIl" nt of the flight path arrive

simultaneou lly at a ?articula r point on the earth'~ s urface .

Stimulated by Stone's ,

speculation, Carvin carried out", pre l iminary

an"lys is o~ the propagation of a lin u r, lossleu loIave into the [ " !luouS

, and hot the rmo s ph" re. A Concord ., sonic bo01ll has a leading _hocl< ",tth ~n

OVetp~eisure ~ ? of ~bout 100 p~ nur ground level; that is, the st~ength

(relative overpressure) which we designate by s - !p/p -J

is about 10 .

'.'1th C:arwin' s a pproximation s . such a shock become s very strong at 180 K::-_

altitude, since:

"'here I' is the ilmbtent pressure and "'here the energy E tn the shock

i _ proportional to (~p/p)2. The corres ponding t""'ptnturt rise inside

th'" shock "'Ould be 104

OK with a resulting kinetic t~perature correspon-

ding to a psnicle velocity ",ell above the escape v .. locity. C:~rwin ttnt~ -

tively concluded that the Concord e generated shocks could significantly

perturb the thermosphere by depositing energy end aoment~ lesding to

theraeepheric .scape and high thermosph"'tic w1nd s . Further, the shock

J

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would propagate throu,h the thermosphere at an average velocity ,lightly

less than the velocity of the aircraft at the point fr~ which the .hock

waS launched. thus the acous tic disturbance could travel through the

~tmosphere at meSn velocities well above the velocity of sound at s ea

l evel . The booms r efracted by the thermosphere , called hyperbooms by

Carwin, could trave l horl~ont.lly for dist anc es of hundreds of thousanc s

of kll~eters and throug~ fOCUSing might produce the observed east coast

booms.

Gardner and Roger s 9 have carried out a detailed analysts of the propa­

ga tion of • weak shock in the atmosphere taking explici t account of the

non-linear character of even a weak shock, the density and ther~l atruc­

ture of the atmosphere and attenuation due to linear dissipation. Cardner

and Rogers find Chat the non-linearities of a weak . hock lead to • substan­

tial energy 10s5 as the shock propagates upward 1n the thermally strltified

atmosphere . They conclude tha t,

1. tp/p never exceeds 0.2 in the thermospher e so that th e

ahock remaina weak throughout it5 path. A weak shock

could ndther lignificantly pertur b the tltmpeuture or

wind field of the t hermosphere.

2. A Concorde boom generated at 17.5 b height ... ith the

airpllne traveling at Mlch 2 will reach the ground a t

a horizontal distance of 320 km fr~ the airplane.

3 . The Bound rltceived at this distance will have I sull

over pr easure of le5s thin 0.5 PI and the anergy viII

peak It about 0.2 Hz.

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• \

The analysis presented 1n this repon carried out i"dependently of

the NRL wo~k u.e5 a much simpler =odel of .~ic boOG prop.g.tion to ar -

r1'>'" at conclusions 5imilar to those of Gardner and R0iten. In this

an.lys i s the fra c tional energy loss due to the ennopy jump for a shook

pro?aga ting !n the x direction 1s

,

"hne L is the 50nic be"", length and y the r .. tic of the hut ca?aci-

ties . The fractional energy 10$5 is thus pro"o:nional to the overpressur e

~t every point along th~ trav e l path. The ,elative overpressure ~p/?

stays "ell below the lilrlit of at all altit u~es , ... here , • "

the scale height of the atmospher" . This limit nays ",d l belo,", 0.5 .t

.11 altitudes (even though the sonic boom lengt hens as i t travels to high

altitude s) be cause the Ical" height of the atmosphe~e also inc~ea se s a t

high dtitudes. thus, the large scale height in the the.",osphere insures

tha t the shock will r~in 5 .... 11.

The re sults obtained here and by Gardner .. nd Rogers are con s 1ste~t

with ." Grover.. observations o f Con~orde gener"ted ,ho~ki.

sonic array, Grover measured at .. horizontal distance of 300 km from the

Concorde flight path a pressure pulse of O . ~ to 0.8 Pa in the frequency

r ange of 0.3 to 0.5 H:.

,

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\

of 4S00 b and at • frlq"en':), of about 2 Hz can alSD be inurpreud in

ta"1lI o f the calculationl pr" • .,"nd her. . Aft .... the flrat bounc .. off the

ther.o.phe .. e . the ground pre •• ure (~p/p) at a horizontal dt ltance of

300 kto is about 1 0.5

. The around - re fl ec t ,,<l s hock s tr ength 11 n.l ll u by

• r .ctor of 100 compl,ed t o the i nitlal shock havins • pr .... ut .. aMpli t ude

o! 100 Pa . \l l :'h the blall over- pre"urI, non-linen _Ikening ,,11 1 be

sma l l and the infr. sound " .ve v iii propasa t e a distance R ",lth only

geometrical reduction in s trength pr oport ional to

ba low 100 ~ is l ... s than about .001 db /kto U ItlHz.

provided the

Fo r <lilt .nc ... of

~ 500 b, the geometrical . preading redue •• the pre.sure ampl itude by 11

10 ..... above 100~. Thul. m .... ur .. d rm. amplitude of 0.1 ,. at 4500 k=

<li ltanca i. consist eo t wit h .n aQ~litud~ of 0. 4 t o 0.8 , •• t )00 ~ ~ven

,

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-

11 PROPAGATION OF SONIC BOOHS IN A STUTIfIED Al'KOSl'HERE

~e shov here that up.ard -going Gonic booms viII not significantly

affect the~05?heric t~peratures or w1nds. and that, although these

upward-going boo~s ~ay ~e ref~acted downward by the the rmosphere, they

lose so !tueh ene rgy dur!.ng the ir up"a rd travel that t hey cannot result in

large sonic boolts on the ground .

For simplicity of illustration We con s ider vertically traveling

planar sonic boo"". lie thus avoid the ""-""-5sHy fnt con Sidering either , geo(>etrical ipru.ding or the change 1n directi" .. due tc rehaetion that

• non-vertical w.vefrant ""\,lld u:p .. rience. Ife can dralol all Significant

conclus ions froc this limple case. Our hastc problem 1s to solve for the

Ionic be"", strengt h as a function of altitude . We \<".,0,,· thn rather close

to the airc~a!t altitude, the strength (relativ" overpressu~e) is about

10. 3 . If the boo," ~eache5 a strength of unity, • nWllbet of l,"po~tant

effects 'w'Ould occu~; bu" if our vertically traveling Ionic boo," a l"'''Y5

has a strength well below unity .5 it trave ls tnto the rarefied upp e r

at,"csphere, then no lonlc booll' It Iny angle " ill ever bec"",e strong

(igr.oring cusps Ind caunie s ), and "e can easUy calculate I onic boo,"

effeetl.

,

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A, Sonic 1100111 Characteristics

A s upe rsonic aircra ft c rute~ I compliclted I hock--"Ive &truccur .. that

propigatc l a,,'ay from the aircra ft's flight pith, At lIKld erate distances

fro~ the aircraft, the s tructur e is reasonably well approxima ted oy a

s~etric N-wave a s 5ho~ in Fig. 1.

'.1-----,

f i,.,,. 1 PRoPAGATION OF AN IDEA L N-WAV£

OJ,,,,,tl,,,, of ""';,: bcom tr ••• 1

Th" true sonic 000::1 will differ fr"" this ide ali zed form, but th"

dHfu"ncu will not affect our result s (s"e Appcndi>< A).

One" w" hav" esta blished the pr"ssur e waveform, we may ca lculat e

vsrious useful qUlnt i ti .. . from elamentsry .hock-wave theory (se e

Appendix A):

8

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h - altitud~ (distance along direction of sonic boom travel)

u - speed of first shock discontinuity

c - speed of sound - (Y'1lT)1j

Ta - ambient tempe rature before sonic boom passage

Pa - ambient pressure befor e and after sonic boom passage

~p - overpressur e just behind the fiu e shock discontinuity

s - relative overpressur e ~p/pa (called the sonic boom strength)

Y - ratio of specific hea ts for the atDosphere as an ideal gas

bI - increa s e in te~perature just behind the fir s t shock discontinuity

increase in ambient temperature the entire sonic bo~

after the passage of

E - total energy (per unit area) in the son1c boom wave front .

Within the boo~ t he t emperature ri se is first order in the s trength

while the heating of the air behind the boom is third order:

The energy 1n the sonic boom is given by

y+ l 2 '-;;TLP s . , .

,

(2)

'"

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Aa t he sonic boo. travela. cner lY t . 10at In he~ t lnl the a t.oa phc r c. The

en . ... ,! 10 •• r e l ation is

ld'

"" • • -2yL " )

or twice at l a r ge if the ideal N .. ave a hape i . main t ained (lee App. A) .

The elr behind the sonic boO!ll .,111 not o nly b", hut." (by liT.)

but wtll a lao be glvan ~.ntum. Thi s momentum tranafer will r .ault

in .t~.pher ic wi nds. with .peeda given by

(5)

The fi r a t .hock travel. Slightly f •• te r than an acoult i t .. ave:

,OJ

ra.u l ting I n , lengthaning o f the boom while it travel s

Th. above siapl, r alat l onl, plus a ~odel for the atma'pha ... a, are

an t hilt Is nuded to esubU.h c,lta l"uul u of this ae:t1on.

10

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B, No Lengthening; Isothermal At.olpheu

A simple examp h w111 il lustrate th~ phy s i~s of the situation ,

L~t '-IS iKr,o~e the ~ hanKu in L .. i th distanc e , taking L a, a constant

Lo fu rthermo~e , l e ~ us consid e ~ an iso t he noal a tmo sphe r e , Th e sound

spee': c Is a co"stant in such an ,)t"", s,her " , , \.'t need t h" pr"s suu ..

a function of al titud e i n orde r to ",ake us ~ o f 0) . \I" obtain this

t;,~oUllh the sound - speed relation

and the hydrostatic r e latl~n

r e$ulting in the solution

p - p e . ,

c" • yp/o

_h/ H •

Tha t is, :he "e l l -kno~'TI u~onent ial atl!lospher e .

(8)

(9)

DO)

Let p , s be the , , ambient pressur e and s onic -boom I trenllt h at the airc r a ft a lti tud e ,

The n ( 3), (4), and (10) yield a d1f:erential "quation for t he energy E

as a function o f alti tude:

, " ' " -- (11 )

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The solution to th 1s equa tion ~an be easily found, and yield. immediately

the ~ne~iY and It~enKth a. a luncti on of Iltitude:

, . , °

••

'0

/ h/2E

'i '

h/21! • "-----.,."O,'--lr,,",----, H h/ 21! , • ___ '_. e s - l

lTlo 0

(12)

(L )

The solution for t he s t r e ngth s ho~ ! a strong sat uration phenoaenon,

lince at h1gh a l titude, rega rdl lSI of t ht Initia l itrlnlth at ai rcraf t

a l titude, ~e have

(14)

Typical valuu fo t Lo (-300 .. eten) and Hs (-10 kill) wi ll yield a

.ar uration yalue I .. a" _ 0 . 1.

Io'e ~ould It t his point, eaaUy cll cul l t e all the Itl1o.phe ri~ effect l

o f interut w1thin this a1,.ple .adel. lie would f1l'ld that a I trength of

<0 . 1 neva r l .adl t o . iinificant heating or winds in the upper atl10sphere

£qa. (I ) , (2) and (~), I nd t hat

o f its enerlY b y the t1me it h .. rea~h.d a height of 160 b Eq . (12 ).

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(1

(.'" ) 0 <IOl 00\< II ' -, 011\ O~-4

(~l)

011\ 001> 4 ou,'" OOE ~ 0 , -,

: apnlnT" 10 uoll~un. " n p"~d$ punos ,IOn Bulzp .. H ..... ~"d .{q

~n;u~u, 5,41 Ta pa'" iO.'I '''''I 001 lnoq~ 10 "pnl;1r1l Ull' lW ·\lUa ?pns J " 41U

~s,,~~"u, 01 su,~"q ( g z ' ~Li) ;u . 'ld~oOJl! "IP ul ;u ,nu"d":>l " 41

l~PO" 1,~.u.dso:lll\' "(1

' BuellS OmOl a'l lOU saop woo,! " 41 ' pas n s t

"~"4d~o~"ql alll $apnrlU; "41 3J "qd s~l r alll 1'" l a pom ","~nll" ,, ~om " ,> : "41 pu.; " l'o · . ~ " '1dS O""" 1""'-' '' '110 $1 p"",,,sn "41 u, .Jns saJ d lu~,q.'" u1

;)sea J' .. ? p,d,, ; ~tn JO H r. e ,,,« 5J"1 10 'PO,!' BU0.11S ! 01 .. S ... " 11U1 " \,\ 1

'''p r' ~;l1'' =>\ OLI

~~ (! - S) _,1 0 4 5 ~'JCJ1S " "0101 ;)<:; ptne", <:ooq "41 H 'P pun ",' S.1.:i.IlI"J" ~

l " P ,U; l"'rj·{l '11, '" :"na ; :)Ul .( lT1'nU'lU01 .'''''' 41'Ua J lS 0100'1 " 41 11"53;

"sy · S . S~"J1U' Oil" '1 1'ilu" J1S 3\(1 JO a"t"" UO~l" J n."s "'P S"5 ~"J1U 1

"pn1l l tY ~IP ~Y u~" . ' SUOl"l"~t') ~"O U1 ualll!lua t 01 awO,! ~11l ""'11~ ~"

JI "al'!l!lnhu lOU Sl lL) "bJ 1II00q all,. 10 !lulU311,1'llual all, l • .. aA~""1!.

lIUfu,Hp 9'ua, "J

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aD that t he .o~nd .pe~d tner •••• s linearly to 2eo It In altiTude of

IbO k=. In the c ••• of • lontc boom fro. , Ma ch 2 aircraft , the upp er

t u rn1n& point of the IPl>rOll.'t l r ay trac. ~11 1 therefor l b. It an

artitud. of 160 km.

The pt'"YU .nd density 1n ou r lKKlel atmolphere .. ay .ga lT. be

found by the uu of ~h . sound-speed uhtloo

and the hydrostat ic relation

lhe'l .~u.tlon. i~.di.tely l.,d t o • !2£!l lell, hl1lh[

-I '. . -1. a II .. , h

(16 )

(18 )

which . ay ba 1ntl,ratad to obtain the pr •• sure I I • func tion of alt1 tudl

(Hg. 2b ).

E. Sonic Booe Properties II • Func tion of Alti tude

Civen our .,dltl atlllOlpheu [0 1. (15 ) and (18 ) and Our propaRation

of • lon ft boOll [qa . (3), (4) and (7) " '" .... y lasily n"lurlcally

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-, -~~ -• ~ < , ,~

• ~ • 0

,00 " )

• • ~ <.

".

------'

i -.oc toO Il:1O '00Cl >:IDO ' _00 ,MIO ~"<n,c TI"~!"UUIII "

CoIiN$,n . ' flO.' '0'" '0'" '0"· ,,- ,,' ,,' ,D" ,,' - ,

~I , , , --, ~~ -

;_l I U , < . , I ,

, ,- 1 ! ,-- , , <Of._TT ,

' 00 ""tau_. -- ---• ... " . ". ' 0" ,,' ,,, ,~ ,,' •• HII.aulli . ...... '

Fitu'. 2 TE~PE""TVRe . 'II.ESSUR( ... ND DENSITY DISTRIBUTION IN THE ... TMO$PHERE

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-, 10 And lengt h Lo ~ 30 .eter~. believlnl thel., to be

ehan,c terlnic of .. supeuonle aircraft. (One effect not ... ntioned

previous ly "'AS taken 1nto account: that the inc r ease of sound apeed

with a lt i t ude also Clue ... a l e n&the ning of the bo~, contri buting to

l""o:.ri n& its 5tUnlt~ .. )

f i s ure 3 shows the ItTeolth o f the boom as .. fun c t ion of

a l t1tude. The boo. reache ... e t!enlth of little more than 0.1 at

160 k.c altitude. Note that an extrapolation of t he 110ther,..1 .~!lo.phne

(valtd below 100 kg) would haue.giuen .. Itrona shock in this reaton.

Thus the thermospherlc properties jUlt manlse to IVOld the produc tion

of atrona shocks.

Fra. this r esult ve ~y calcul ate ther=ospherlc h •• tlng and vinds.

1nlide the ,hock

Behind the shock

6T :l:.l'T i 300 , .

" . •

•• ,!; ) "_/Iee

The" value& are clearly ne,IS,tbl' compared co other fluctuation. in

th. atmolphne.

"

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-

Q,Olf------------I'---------- --- --- - - ----j

'" '00 "" ~ \~m)

fipJ .. 3 VA-fil IATION Of SHOCI( STRENGTH WITH AI..TITUO E

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Figure 4 ahow. the calculated energy in the Ionic bo~ a5 a

func tion of altitude . We See that 901 of the energ y i s lost before

naching 160 Jua altitude.

Figure 5 aho ... . the lengthening of the boO'::1. At 160 kl:L a ltitude

the booll'. r eaches a length of abou t 4 \all.

\I e may describe t he hinory of an "Iward-gobg sonic boo",. follo l.1 ing

Figures 4 and 5 , by approxi:o.ating the rrCle re!ractive pa th ( ... ah an

upper turning point at abou t 160 km) by a v ertic a lly up ... ard path to

160 k~ follo ... ed by a vertically do...n ... ard path to t he ground from 160 km.

In that app rox imation only 31 of the initia l ene rgy is re~ainini at t he

upper turning point, and the boom has lengthened to 4 b. One can sho,",

that in the do...nward-g01ng leg of the path, only about )01 of the

energy at 160 km 1s los t, and the l~ngth does not change significant ly;

the smallnes s of these changes 1s du e to the rapid de crease in the

strength as the boClO'l entera den.e r a tmo sphere. Hence the ligna 1 r each­

ing the ground after refracting off the thenoolpheu 11 app roeximate ly

50 times le'l energetic than the Ionic boom that tr.velled dir ec tly

do"",,,",srd from the .ircraf t , and it 11 about 4 lao in length, livinS

peak frequendlt s in th lt infrasoniC region (-O.l H.z).

The loss in energy and the lengthening results in • strength at

the ground of

18

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' ",~-----'-------r------'------'-------r----~

, '.

"I--------~,__------_1

0.01 '--'-___ -:';: ___ ---' ___ -,:':-___ --'-___ --:":-__ -'

100 ISO

FIgo,o... • DECAY OF ENEI'IGY OF "WE"'':: $HOCK ~RoPAG"TING VERTICALLY IN THE ATMOSPHERE

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,

,~--------------------~------------~

I. Ikm\

" 1------~-------------------1

" ". h lkml

'"

Fi .... i LENGTHENING OF A WEAK SHOCK IN THE " TMOSPH ERE

20

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• •

----------------

I f t~i$ 51gn~1 bounc~$ off t h~ ground ~nd undergoes ~nother thermo-

sperie refract ion, i t s streng t h ~il 1 re~ain 50 s~all th.t no si gnif­~

i cant ene rgv l os I or l ength. nini wi ll occur .

r . Othu Efhcts

A more exten5ive tr ea tment of thls problem would consid er the

cOrrect di rection-changing path o f an anglee sonic boom that s pr eads

in thn." d1'"" 05io05. Such a treatootnr would a150 consider t he caustic

that fortu after the wavefron t has turned over. I n ad~1tlon viscosity

1n the air <:o"ld !>e included. A!ter this work "'as co"plned an

extens ive paper by Ga r dne r and Rogers (~.val Re5tarc~ Labora tory ) was

r eceiv ed that t ook all these effects into a ccollnt. They reached

conclusions that a r e qualitatively and eVe n quaMltatlvel~ si .. tla.r to

thos e we ha.v e derived wit h our si .. plified model.

Page 29: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

· --.------- ----- --_. .. --

22

Page 30: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

• ------- - ----

III !U:StAllCR NttDS

IHdesprud interest in the "East Coast Myscuy Booms" has focused

anention on th .. probllMll of u"d"rstsnding the propaga tion of low fr ~'lu~ncy

sound 1n .. strongl y stratifie d, 1nh<>mogeneous atltOSp;'H e hav ing Winds of

"" r 10us strengths a t differ ""t le", ,, 15. In the Uni t ed State" the subjec :

received conSide rable atten t ion 1n the 1950" and 1960., fi.n bee a" . .. of

an inte ... est in detecting nucle.r e xplosions in the acmos.he re and s e c o~d

heca"u of tonc ern ov~r the effects of flying supersonic tran.?orts, both

beri"." and the Concord... Hote r ecently the subject of 1nhasound has

been la"'l"ly negl"cted. At the start of th .. c.st Coas t events ther e was

only one low frequency acoustic array in operat ion , t he lamont-Doherty

facility.6 At L~ont, Donn op~rat~s tw~ tripartite array' of capacitor

~1crophone~ . The larger array i, On a 600 to 900 m~ter spacing and has

be~n in op~ration linee 1967 while the smal l er array 1s on 60 80 metn

spacing and was put in ope ration in July, 19 77. During Mar ch and April,

1978, in ordu ~o supple:nent ex is t ing o!:>servatlona l faCiliti e s, ~i ~r e oper­

ated three two-microphone correlator. at Bedford, Mas,schusetts, Atla ntic

City, New Jersey and McLean , Virginia. The study of the rut Cou t 800ms

was aided by the exi 5t ence of leveral leismic array~ where atmospheriC

pr essure disturbances we r e recorded only along th~ vertical axis o f the

s eismolraph. Whi le useful, leismometlrl, becau.e of their poor perfor­

mance in the 0.01 to 5 a, region and bICaUK .. of th l ~pldanc .. mismat ch

"

Page 31: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

across th~ air-ground interf.c ~ , .r ~ 1"55 than ideal instrwoenu for

observing anao.ph .. ri< d1,turbancu. I

Th~ r ~cent inv e stigation by the L .~ont6 , 12 ,13 and . 7 14 l~ Knoll" , groups

ha,'e indlcaad tha t nu .... reu s artificial 50,a",,0 can be detected at lar ge

dis tances, for ex . .. pl" r ocke t launches, the o?er.tlon~ of SST .. , paper

mills , hydroelectric plants a~d ofhhon,-oil drilling "qui"", .. nt. The

1n~1cated richness of these sounds s uggest ch,,: a nIX.bu of propentes o f

th e atmosphere, for example str a tospher ic and thermosphe ric ~ind$, coul d

be lIonitored on II. rO<l Une hasi .. using netted acous tic . rrays. Such obser-

v a tions are possible be cause of , fundam ental pro?~rty of the Atmospher e ;

10ft . sonic wa ves can propaga t e over long distanc es bee a" s .. the attenuation

is very small in the 0.01 to 5 Hz region, -a frequ enoy band whe re IUl tuul

atmospheric noise is also very 10~·.

The general con5iderat ions presented above suggest a number of re5earch

prior ities :

A . Obs~rvationa l

1. The cons truction and operation of .ever~1 netted 1nfr~sonic array'

opera ting in the 0.01 to 5 Hz region to IOonitor Concordt and other

artificial lound. of infrasound along the un Coa.t. The ex isting

~ont fa c ility ne eds to be lupplemented by other arrays in order to

understand better the Concorde ,isnsl.. As noted by Ga rd ner a nd

ROger.9 an arrlY On a baae lengt h of one ki lometer ha~ poor directional

-

Page 32: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

" 'C>! l n 'l~nll 51' 10 l'lll~q U" ,.,,;oTli1 ~ .lit"11'1 3 ;tIn UJ 5"Ol •• n~"nu pUll

lithitl puno.ll "" 51"1.111'" ( 1M l) ,,~uos • .llUl '1.0'1 p .. :>np0.ld (ua~I'.la ,(

q''''S) 5.laHIIH "loos.udns 10 fl'lilTI ''''ll lJ.od;u uouTO P= rlIZS1,

" 'uop""Hp -:>nsno-:>1' 'Ill" slu,punos IH4dsoUOl ,0 UOlll',ilJJOJ '(

'an.ll:>"ds :>115no". " '11 ." uc,la.l runoSl'll", a'll "1 ST1'ul1 5 01 Sal.lnOS

TI":>'1,1.11' ~nC;.ll'h 10 uOTlnql.llUO" a'll pUI' 'sapn11111' SnO;,l"" .1' S~lOqS

'1'''''\ JO uo;~enualU al.p ''1'0,"5 '11! a" I' U; s a llUa nb\U l 10 uOlS.l a "UOl

.l1'~Ull·UOU "'11 ';:pnlS 01 ~aplO U1 ? " P""" all' I'l11a~S n .. ,u ""tqI'B "'"

lOU a ,ll' s all,1,l1'1 "lllaTaO.lph" pUI' 51UVId l,,~od ~l' a lS ' sal1JllJI',

l " PH"PUl 10 1'.Illa ds punosI' llu1 a'll oJO 1'11'a · POOlS.lilP"" £110001 5,

aplCllUOJ " 'n I" UO,SiJ.l puno.ellu, '''P "1 IIln.lllads " ".InOS "1.P pUll 1.L""U~

~0\1 s~ ZH 09 n Ola'l. l'"D ~ ''' Hf '["uopua"uo~ 10 "~.~,,ds altl " ! d"" x ,,

~Ol 'pOo15~~pun .('[lood /..~"" 51 puno5"~1\l1 p"~npold .\1 '["'~11Pll' 10

~n, ~ds Pt " lJ ~""u "<U ' Sannos t";~lnlH JO ell~aGS a~lnos .~

' s,(vnll ~ ll'qn; ,0 UOll",Ot .. q ~ tJ" ,d 01 nIt .. , .. S>'

" S10\1 t>'lnl ~u "I{. ;0 u,8110 aq. PU" ,$l"pUn o. "1" ,,~ Jl , " , lU"5S"

~l" "up q,n s ' uo~ a,o l /..eJJe s" 1I "~ s" H 41P" " pue uos"" s '.{ep

,0 ""'11 JO uoq,unJ V U \10.8" • . \,U"nb" 'l ,, \10 5.,;U' " 4. \11 S["""!

as,ou PtJno~!,.pIl'l. .. q. ~U"~la."p 01 P"/..Old .. a "'I. p,noqs S.(,U~" "l'l."l

_lod pu" p<IX<J lpOg ''''~ I ~OJd 'P"4dso!:l'l" aql 01 pandde .\tlpeu ~~

p'[no~ (',lSnO'" u" a~o '~ep"l) sp131J J"q10 u1 p ~dol""" P s"nb,u~,,,,

itJ,S5a,oJd IIlllP pa l ll,,15,qdos pUll ' s l"lloP pa,punq ~al .. '''''lsu"d

_xaUl l..,[a"'lI1"J a l" ,

sauoqdol',~ 101l,"d,,~ a1q"ll.". I..I'[."'~J

' . SUII' n~"H II . JO Slpua. ~al aq. UI sa" . .. punos'O uo,1n10sa1

~

Page 33: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

T~e in~eractinn o f weak . hock. vit~ t he vaakly ioniz.d per tion , of the ion05ph.re .hould be inve. t i,a t ed both to uDd era t and the

flature of t he coupling, the pouible contr i but i on o f t he neutral~

ion inter act i on to dislipation It ht&h IltttuCes and the po.l i ble

ule of ionospheric ob •• tvation to detec t velk 5hDCk~ l aunched a:

ground lev el Or in the aelOosphere . l.1nka and Olison I ls o obu rved

• I hort-dur ation E-laye r echo a t 130 bi 50 lee beloa the I nivil

of th . infrasor. i c .... u." I.,ggestin& that the • .,per sonic lirC Ti,ft ,..y

have gener ated .noth.r lIIOC:e of pro".,uion, poss 1":>l y gra\' I : y .... ,·u.

4. Artificta l nol ae,entrati"s arrly • . An an .. y of in!r.soune

.ource. pr oducing • narr~ beam of hilh intensi t y could pr~ idt I

poverful tool for probing the ph~'sicll conditions in the hi&h I UDO-

.phere. Such . f.ciUty would be particularly usdul in I nvI.Ugutng

t he di . sipation .echani~s in t he therlllO.phere. Recent calculations

of th l d i ssip.tion "I~e t hat el.,.le.l vi,co,ity .nd helt conduct i on

9 .. r e t he pr inCipal mechanisms f or d ialipatio~. For the t hermosph.r e ,

inha.glnhtiu 1n thl density and tftlpn.t"r e field . on tnpora l ,u',d

.p.tlal Icalel .~ller th .. n tho~e of t hl tranlmitrld vave c.n .ct .s

.n .fflctive eddy vi,co,i t y wh i c h could be much l~rler t h.n thl

clan ical v1lcosl t y. A I""nd .rray coupled t o .n ionolonde fatUity

would be .1 PO~lrful t oo l f or i nvel t li.ting the neutr l l-ion lnter~c-

t ionl no t ed above.

Page 34: JASON 1978 Sonic Boom Report · 2016-10-21 · 10ft. sonic wa ves can propagat e over long distances beea" s .. the attenuation is very small in the 0.01 to 5 Hz region, -a frequenoy

"

,. Thr.oreti"al

1. Ori&in of background noiu in the infrUOl.Ind r"sion. Natural

infrasoni c noise 1 5 g~n"r.lly attributed to turbul ent fluctuati ons

in t h .. wind Held. The relations between th ue flu,,-~uat~o!,5 and

.lI\~len t we athu conditions have not bee n ad equ.t.t"ly u:plored. An

unde rstanding of t he natur e and orig1n of t h e natural background

n o i s e is " sHnti al to . e bcting qll1 H . ites fOT iT.hase nlc un.}' •.

2. Atter.uatlon of weak shocks in the high a tm".pher e . The

" onc ept s of ch s sica l vis=osity anc he n conduction are

probably no longer approprill.t e when :he collis i on :r e quenc:,.' in th~

ambien t atmosphere is of the ~.lI\e order a s the inft.sound f r"quel'cle • .

The phy s ica l nature of the a ttenuation 1n the atmosphere above about

100 ~ neeos to b~ und ~rs tood in ord ~r to int~r?r~t long_~ange pro­

~a gltion of infra,.oun~.

3. Coupling of wui< shocks with 10ni:~d particles. The o bs ervation

of E l ayer height variations aS 50c1at~ with the up"ard ~ropagation

of a "eak shock 5ugge5t a strong coupling. Th~ detailed physics of

the coupling n ~ ~d, ~xa~ination as ~oe5 the possible contribu tion of

weak plas~a effects to the attenuation.

4. Artificial generation of gravity wavaa in the atmosphere. rh~

preaent report has examined the infraaound plrt of the apectrulII.

Artificial sources of gravity " a v'" ' may exist, ,for ,"xa .. pl~ Concorde,

"

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I

and these &rtifi~hl ",aves ~ould pert"rb the ionospher e . A v.ri~ty

of pot entia l artificial 5""r~" s sh""ld be exllDlined in t " .,.5 of souree

str ength 50 as to de termine ",heth"r observations bel ow 0.01 Hz could

yield information about a rtiflcial sources and a bout the pr opagat ion

of gravity wase$ in th .. a t,"osphH" .

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REFERENCES

l. Scie nce. 199, 1416 (1978).

2 . PI", " !" . T(>d"". 61-63 (June 1978).

) .

4. Cha lto:!, S. C. and C. J . !'tacOonald, Sound and light Pheno",ena A Study of Histo~lcal an~ ~ode~n Occurrences , Mitre Technical Re port 7927 (1978).

5 . Public Interest Report , Feduation of Amer1c;a..'1 Scientists, 1l; No . 5 (~ay 19 78) .

6. Donn, W. L., Report on Atlllospheric ACOUHtc Ev .. n t $ , \ll ~ t e r 1977 -1978, prepared for Naval Ru .. .. rch laboratories, C,,\uOlb1a Univusity, ~ew York (1978).

7 . liszka , L .• Long- Distanc e Focasing o f Concord .. Sonic Booms, KG! Repr int 77:303 , Kiruna Geo~hyslc .. l Institute, Ki r una , Sweden , (November 1977).

8. GaNtll , R. L., Specuh.cion on Long-Range Effects of S<JF"rsonic Flight, unpublished note (1976) .

9. J. H. and P. H. Rogers,

10. Grover, f. H •• Geophysica l Effects of Concor de Sonic 300m5. Quart J. ROy . Astron. Soc . • .!.!, 141-160 (1973 ) .

11. Procunier. R. W. and G. W. Sharp, Optill>\lIII Frequency for Detec:tio" of M:.oustic Sounds in the Upper Atll>oilph"re, J. Acoustica l Soc. M., ,!!, 622-62~ (1971).

12. Co t ten , D. and W. Donn, Soun~ fr om Apollo Rocket . 1n S~ace. Science . lli, 5-65-567 (1971) .

13. tlalachandran, N. K •• W. Donn and D. H. Rind, Concord. Sonic 8 0011>5

as an AtllK).pheric Probe . Science, ill. 47 - 49 (1977 ) .

14. Linl<a, L. and S. Obs.,., . On the Ceneration and Detection of Artifi­chi AtlllO.phuic Wavu , J . Atll>os. ' err . Phys i cs , 11.. 19 33-1939 (1971).

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15. Link., 1., Lon&-dinAn"-" Prop'lnion of lnfr .. ound fro", ArtU1c1,} SOUT,,- U , J . Ac o uSt. Soc . ...... , 2!, 1383-1388 (1974) .

30

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SONIC BOOM P10 PACATION

A. f unda "'~" tal [Sua tl on,

Consider a Ionic boo," eondl t ~"& "f •• 1I0cl< dhcontlnulty f " ll"o: ed

b~ a continuous wave:o"" of lengtll L (FI, . A-I ) . Th~ shock II travellini

into .~bl ~nt . t~o5plle r e a t I peed 1.0:

P, ........

i- , , . , ~ "

" "" " " pix) "

T(x)

Jus t after tll a Illoc k (I .e. It x .~ll but pOlltlve ) we daflne the

pre'lu re, den.it y , and t~pera tur~ " , . , e 1' (0); ~o eT (o) .

The veloc lty of the Ii. ve x ) t. non-zero " I t hin the s1"l0e k , Ind _,_ In ~'e

<I,fine v • v (o ). ,

31

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Th~ full .~t of ~~l.t lons betv~en qu.ntlt1~s On either ,ide of the

shoc k discontinuity ar e the folloving:

Ma ss conurv~tion

Mom en turt con~e:r\'ation

Energy con$ervstlon

Equation o! Stat e

a u -pi,. • 0

" aUVo ' E Po

'"IC(T-• , 0

, ) o

'. T)+!"'I",, a 0 0 0

;" ith in the s onic boom t hese Same cons eTvnion 1'101$ yield:

; Ie (T -. , ) ) • ( 'I + '2 v - a,,(pv)

pix ) - c(x) ~ I(,, )

(ll

0)

(' )

() )

(6 )

(7)

(8)

Equations (1) - (4) yie ld s number of rel . tions by a i m?l. algebra,

whic h we present in t e nnS of the sho c k strength

(9)

.nd the ,mbient sound s peed

(0)

32

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On" c. an ahov that the eho<:k Ipeed 11

(11 )

wh~,~ yields fo r s « 1

( "'-' .J u ;;, I +,1' (IT - b)

T!>.~ temperature juSt b"hind the shoc k is

(1 2 )

which yields f o r 1« 1

(II - l )

The entrop}' c hana" acron t h,. Iho, k 11 /

vhich yia l dl

-+- };1 )-'1 • 1 (1))

33

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5. Relations within the Sonic 5001:1

The exact manne r in ... hich the various thermodynamic variables change

IS I function o f distance behind the shocl 1s a complicated question of

fluid dynam~C5 . How eve r,"oI e ...... y ",ake some gene ra l re ... arks :

• The integr ... l of. the absolute ove rpressure "'ust ulti .... a ly

be close to zero as the gas is assumed to return to velocity

",ueh l e s~ than t ·he s ound speed after the shock has passed.

Thus so<oe regien of negative overpressure is a necess it y .

• In reg i ons of s<oooth pressure behavior. the ther.o.odynam1c

changes ... il1 be is entropic. However, a second shock

behind the fir st ... ill aho have an entropy jUl,..p. Because

of the distortion of the shape of the shock, the entropy

jumps in the s econd shock will be less than that of the

first shock.

We must det ermine the total energy in a unit area of wavefront, and

the energy left behind the w.vefront in the ambie nt atmosphere.

The energy pe r unit volume at a distance x behind the shock is

(14 )

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,

, -

~ ( lC ) • , :' (P.)' . " , -

p(x)p(,,) _ pIx )

'.

,.., I'

(l~)

(16)

(1 7 )

To fi l'l d t~. e t"U I enH BY i n the \lIVe .. . lIust i n t e gr.te t over th.

lena eh of the ", .... d onn : Thus w i ::IUlt kn o... pI x ) . We know that t hll in t ."ral

of p(x) - p(a) ,"uS t be ve ry ~lo •• :0 zero. Let us aSlume t he ulual N-wlv.

On. ean tlle ll ahow t~t

, £. j td >:

, I ( ,<, ,H ,.

• II L X .. 2 (2~ ...

For &HI \Ie find

(18 )

.)1,, < l.r!.!.1 (I · s)Y

(19) ' )(1 + 'I-I )( ' < )l/l !Y' 5 I

"

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C. Relations after Sonic 80011 Pauase

An entropy jump across the first shock folleved by isentropic expan-

aion back to a~bient pressure, will leave the atmosphere heated behind

the Shock. The second shock 1n an ideal N wave will generate as ~uch

entropy as the fir st , since the underpressure in the following shock wav e

equals t he overpressure in the leading wave . In an actual case. the shock

lengthens a. a r esult of ~mentu~ transfer and the s econd shOCK will be

..... aKer. In this r eport we consider the weakening of the shock re sul ting

only from entropy jump at the fi rst shock. r ecognizing this approxi~ation

may underestimate the fr ac tional energy loss by a s ~uch a s a factor of

two but probably much less.

so that

For 5«1 we have

" , •

(20)

(21 )

(1I - 2)

10 that even moderate-strength shocks heat the atmosphere very little.

36

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noetn, that Cv - ~/(y-I) .. e ftnd

the rr.c t ion.l enlflY lOll 1n tr,vIlina • dllt.nee L I,

" , • 2,

H.~c e ~I ~'Y immediately WTlt l the Inef lY 101$ equation

" •

, L

( 22)

(11-4 )

Th, momlntum t r.nsrefred t o the atmosphere i $ rela ted t o t hl .nlrl~

de~o.ltlon. A wave tr.vell i nl .t nearly .coustic s peed I (l. e. 1« 1) would

have t hl . ,latlon

v iii Enern dlPo l1 ud/" n lt ,

"

(23)

- (14)

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(II -5 )

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I

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I

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I I I I

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12