gilbert (1989a) the parabolic equation applied to outdoor sound propagation (1)
TRANSCRIPT
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Application of the parabolic equation to sound propagation
in a refracting atmosphere
Kenneth E. Gilbert
NationalCenteror Physical4coustics,niversity, ississippi8677
Michael J. White
u.s..4rrnyConstructionngineering esearch aboratory, .O. Box4005,2902NewmarkDrive,
Champaign,llinois61820-1505
(Received12July 1988; cceptedor publication 8October1988)
A wide-angle arabolicequation PE) model s presentedhat is applicable o sound
propagationna steadynonturbulent)tmosphereverlyingflat, oc•,11yeactinground
surface. he numerical ccuracy f the PE model s shownby comparing E calculationso
calculationsrom a "fast-field rogram" FFP). For upward efraction, he PE and FFP
solutions gree o within 1 dB out to rangeswhere he sound-pressureevelsdrop below he
accuracyimitsof bothmodels. or downward efraction,he PE and FFP agree o within 1
dB except t deep nterference inima.Parabolic quation alculationsre alsocomparedo
measured aluesof excess ttenuationor 15 different ombinations f frequenciesnd ranges.
In general, he PE modelgivesgoodagreementwith the average xperimental alues.For
upward efraction t the highest requency 630 Hz), however, he PE predicts strong
shadow zone that is not observed in the data.
PACS numbers: 3.28.Fp
INTRODUCTION
Over 40 yearsago,Leontovich nd Fock introduced he
parabolic quation PE) methodand applied t to electro-
magneticave ropagation..2Sincehat ime, hePEmeth-
od hasbeenused n suchdiverseareasas quantummechan-
ics, plasmaphysics, eismicwave propagation, ptics,and
underwater acoustics. n outdoor sound propagation, a
number f potential pplicationsxist, but, nevertheless,
thePE method asseen nly imiteduse?
The presentarticlediscusses wide-anglePE modelre-
centlydevelopedor soundpropagationhrougha nontur-
bulentatmosphere verlying flat, locally eacting round
surface?hepurposes o show hat hePE algorithm an
giveaccurate umerical olutionsor soundpropagationn
realisticoutdoor environments. n particular, we want to
show that the PE can accuratelyhandle arbitrary sound-
speed rofiles nd the locally eacting mpedance ondition
used n atmospheric cousticso represent he effectof the
ground.Consequently, e limit the numerical omparisons
to situations with no horizontal variation in the environ-
ment. Suchenvironments an alsobe handledby the "fast-
fieldprogram"FFP),6'? owecanmakea detailed om-
parison of the two solutions.To further test the model, we
compare he PE predictions ith measured aluesof excess
attenuation.
Sinceour ultimateobjectives to treat morecomplicated
outdoorenvironments,t will be helpful o brieflydiscusshe
basic deas hat motivate he presentwork.
In the areas of applicationmentionedabove, he pri-
mary motivation or the PE generallyhasbeen o treat wave
propagationn a complicatednhomogeneous edium. n
underwateracoustics,or example, he primary motivation
for the PE hasbeen he need o predictsoundpropagationn
ocean environmentswith significanthorizontal variation
(e.g.,continentallopes,ceanronts, ndeddies).ThePE
can reatsuchcomplicated nvironmentsn a relatively im-
ple way becauset neglects ackscattered avesand uses
"marching" lgorithm o propagate aves utward rom the
source.Given a startingsolutionat the source, he PE ad-
vanceshe solution n range, aking nto accounthorizontal
changesn the environment s the solution s stepped ut.
Statedmathematically,he PE approximates complicated
boundaryvalueproblemwith a simpler nitial valueprob-
lem.
Although we consider only horizontally stratified
("range-independent")nvironmentsn thisarticle, he PE
algorithmtself, iven ange-dependentnputs, an n princi-
ple predictsoundpropagationn a complicatedange-de-
pendentenvironmentsuchas a turbulent atmosphere. he
ability of the PE to treat such nhomogeneousnvironments
is, n fact, the mainreasonor developinghe present tmo-
sphericPE model.Hence, when n this article we state hat
the presentPE model s limited o range-independentnvi-
ronments,one should keep in mind that it is not the PE
algorithm tself hat has he imitations,but rather he pres-
ent environmental nputs being used n the calculations.
With a realistic ange-dependentescription f theenviron-
ment, the PE model presentedhere should be able to treat
complicated ange-dependentffectssuch as atmospheric
turbulence nd irregularground opography.
I. THEORY
A wide-angleparabolic equation can be derived from
the so-called one-waywaveequation,"which s applicable
to soundpropagation n situationswhere backscattered n-
ergy snegligible.Thissectioniscusseshe ormal evelop-
630 J. Acoust.Soc. Am. 85 (2}, February1989 0001-4966/89/020630-08500.80 ¸ 1989 AcousticalSocietyof America 630
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mentof a wide-anglearabolic quationrom heone-way
equationndgiveslthehysical otivationor theapproxi-
mations sed.Webegin yconstructinghe ormaloperator
representationf theone-way ave quation singheusual
two-waywaveequation s a startingpoint.
A,,Operator solution or the one-way wave equation
Considerhe two-way Helmholtz)waveequationor
the acoustic ressure in an environment ith azimuthal
symmetry,
(oa2ea oa2
Here, the wavenumber is a functionof bothrange and
height and s givenby to/c(r,z), where o s the angular
frequencyndc is the sound peed. he associatedarfield
equationor hevariable = x• P is
Oa?+Q=0, (2)
wlhereheoperator is defineds09/•z • q-k 2. Since q.
(2) contains second erivativen r, it permits oth orward
and backwardpropagation f sound.We want to derivean
equation hat treatsonly forwardpropagation. o obtain
such n equation, emomentarilyakek to be ndependent
of' angeandwrite Eq. (2) as he productof two operators,
(i•4-ix[•)(•r-iv•)u0. (3)
The solution or u that is obtainedby solving he one-way
equation,
= + ix/u, (4)
oar
is alsoa solutiono Eq. (2) for range-independent. Note,
however,hat while he two-waywaveequation, q. (2),
permitsboth two-wayand one-waypropagation,he one-
way wave equation,Eq. (4), explicitlyexcludes wo-way
propagation.he one-way quation ropagates aves xclu-
sively n the forwardor backwarddirection, espectively,
depending n whether he ( + ) or ( - ) sign s chosen.
Ilk is ndependentf range, shasbeen ssumedor the
moment,henEq. (4) witha ( + ) sign orrectly ropagates
a forward-going ave.For a realistic tmosphere herek
varieswith ange, owever, q. (4) accurately ropagates
forward-going aveonly if the backward-goingnergy s
smallcomparedo the forward-goingnergy. hat is, Eq.
(4.) applieswhenechoes re negligible.Hereafter,we shall
as:sumehat theacousticield sdominated y forward-going
waves nd that propagations accurately escribedy Eq.
(4).
Sincewearenow imiting hesolutiono forward-going
waves, t is convenient o remove a "carrier wave" by
defining a new, more slowly varying wave •, where
• --=u exp( -- igor).The removal f a carrierwave s not
essentialor the formal derivationgivenhere, but later it will
be important for a numericalsolution.
The equation or •, which hasa carrier waveremoved, s
given by
oa0i(4 - (5)
oar
whereis heunitoperator.For hesake fconciseness,
unit operatorswill besuppressedereafter. The wavenum-
ber for the carrierwavego s chosen o hat it is dose o the
dominanthorizontalwavenumbern the spectral ecompo-
sitionof•. A typicalchoice, or example,s to set% equal o
to/•, where is the average oundspeed. he motivation or
sucha choice s based ssentially n a plane-wave ictureof
propagation. or nearlyhorizontalplane-wave ropagation,
the horizontal wavenumber s between% and gocos0 ...
where0•,• is the maximumangleof propagationwith re-
spect o the horizontal.Although he plane-wave icture s
only approximate, t provides n estimate or % that has
proven ufficientlyccurateor thecalculationsoneso ar.
We advancehe solution n rangeby ntegratingEq. (5).
For shortenough angesteps,we can ake Q to be ndepen-
dentof range.The formaloperator olution o Eq. (5) then
is
o6(r Ar) = exp[iAr(,•- -- go)]06(r), (6)
wherehr = r -- ro.Ultimately,we want an mplicitsolution
schemef the Crank-NicolsonypefiHence,wewriteEq.
(6) as an implicit equation,
exp[ - i(Ar/2)(x/• -- go) •(r + Ar)
= exp[i(Ar/2)(x/Q-- too)•(r). (7)
Sincehespectralaluesf theoperatorAr/2) (x/•- - %)
aregenerallymuch ess han 1,wemakea linearexpansion f
the exponential peratorsn Eq. 7). Using hisapproxima-
tion, the solutionor •bcanbe writtenas
[ 1- (i Ar/2)(x• - too)]•(r+ Ar)
= [1+ (iAr/2)(x•--•Co)]06(r). (8)
To actuallysolve or 6, we needan approximationor the
square ootof the operatorQ. The approachaken s to use
the"rationalpproximation"orx•- discussedelow.
B. Rational pproximationor the operator /•
The dea fa rationalpproximationorx/0-was rigin-
ally introduced y Claerbout or extrapolation f seismic
waves.To useClaerbout's ethod, e firstwritex/Q- s
tcox/1q, where = (Q/• - 1 . Theassumptions that
thespectrumf Q does otdepartmuch rom so hat he
spectral aluesof q are alwaysmuch ess han I in magni-
tude. Hence,we might considermakinga linear expansion
andsetx/-1 -q equal o 1 q-q/2. This expansion,n fact,
leads o the narrow-angle arabolic quation riginally n-
troducedntounderwatercousticsyTappert.o n order o
obtaina moreaccurate pproximation, e couldkeepmore
terms n the expansion. owever,suchan approachwould
lead o a morenumerically omplicatedlgorithm. he ap-
proach taken by Claerbout s to write
x/• + q • (A + Bq)/(C + Dq), (9)
where A, B, C, and D are real constants A = 1, B = 4,
C----l, D= I) and I/(C+Dq) means he inverseof
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(C + Dq). Expandedn a powerseriesn q, the rationalap-
proximationbove grees ith heexpansionf x/1+ q
throughhequadraticerm.Thusweobtainquadratic ccu-
racywithanapproximationhatweshall eeater snomore
complicatedumericallyhana linearexpansion.
C. Operator equation for wide-angle propagation
Substitutionf he ationalpproximationorx•- into
Eq. (8) giveshe ollowing perator quationor advancing
4 from r to r + Ar:
M-&(r + At) = M+0•(r), (10)
where
M ñ = (C+Dq) + (i%Ar/2)[(A -C) + (B-D)q].
(ll)
Using q= (Q/•o - 1) = (•2/•z2+k2)/•o - I gives
M ñ in termsmore familiar operators,
M ñ =M l __+m2+ (M3 __+ 4) --
whereMi' m2, m3, andM4 are
Ml = C + D( n2-- 1),
•2
(12)
•'
M2 = (i% Ar/2)[ (A - C) + (B- D)(n 2-- 1)1,
(13)
(14)
(15)
3 --D/•,
and
M4 = i( Ar/2•Co)B -- D). (16)
In Eqs. 13) and (14), the ndexof refraction isdefined s
•/c, where and c are, respectively, reference oundspeed
and the actualsoundspeed.
As will be discussedn Sec. I, the numerical solutionof
Eq. (10) using he definitionsor M ñ in Eqs. ( 11 -(16)
requireshata largebutsimple ystem f equationsesolved
on each angestep.
D: Vertical density dependence
Including erticaldensity ariationsn Eq. ( 1 yieldsa
wave equation hat contains he operator (c•/c•z)(1/p)
X(c•/•9z) instead f o•2/o•z. By replacing2/•z • with
p(•/•z) (l/p) (•/•z) in Eq. ( 1 , such ariations anbeac-
countedor. Making this eplacementnddividingbyp, we
obtainslightlymodifiedorms or M e. In •s. ( 10)-(14),
we have
•z •z•p/• (17)
and
pendencef the acousticield.Then the operator-function
equations ecomematrix-vector quations. he matrices
are tridiagonal nd the vectors re of dimension , whereN
is the numberof pointsused o discretizehe field. n the
discussion elow, vectors are denoted by boldface char-
acters.No special otation s used or matrices, ince hey
can be easily dentified rom the context n which they are
used.
A. Finite element discretization
In parabolic quationmodels, commonmethod or
discretizing he vertical dependence f • is finite differ-
ences. Here, insteadof finite differences,we use a closely
relatedmethod,inear initeelements.l'l: This approach
makest easyo ncorporatemall-scaleertical ariationsn
bothdensity nd sound peed.n addition, t is straightfor-
ward o ncorporateonuniformertical ointspacingn the
numericalgrid.
We firstexpand in termsof linear initeelement asis
functions,j Z):
c)(r,z)-- C•hj z) (19)
and
•(r + Ar,z) • D•hj(z), (20)
J
whereheC•andDj are heexpansionoefficients.hebasis
functions• z) are he"hat" unctionshownnFig.1.The
expansionoefficients• and Dj are simply )(r,z•)and
•( r + Ar,zj , respectively.ote hatexpanding)( ,z) and
• (r + Ar,z) in termsof hat functionss equivalento linear
interpolationetweenhegridpoints •. Thebasisunction
formalismgivesa convenientmatrix representationf the
operators e. To obtain hematrixrepresentation,hehat
functionexpansionor • is substitutednto Eq. ( 10):
[MI-- 2 M3--4)(';z)](r+ r,z•)h
.
= M,+M+M3+
Ozp &]l •
(21)
Then, we multiplyby h• (z) and ntegrateoverz. The integra-
tion requires omputation f integrals f the form
m•1) h, h•dz, (22
M,--+M,/p, n = 1,2. (18)
With the numerical method used here to discretize the verti-
cal dependencef 4, the modifiedorm of M ñ is aseasy o
represent s the original orm.
II. NUMERICAL IMPLEMENTATION
This sectionoutlines he numericalapproachused to
solveEq. (10). Statedsimply,we discretizehe verticalde-
• I h• h2 hi. hj hi+ hN. hN
e• 0 , , -' , , , , , : , , ,
I Z] Z Z Zj. Zj Z+ ZN.ZN.ZN
HEIGHT
FIG. 1. Basis unctions ("hat" functions) used to discretize he vertical
dependencef the acousticield n Eqs. (19) and (20).
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f 1
and
H•(3) hi zz•-•zz• z, (24)
wherei=pc: is hebulkmodulusorair. fp-•(z) and
• - I (Z) are epresentedspiecewiseinear unctions, ecan
derivesimpleanalytic unctionsor the matrices, ( 1 ,
H(2), andH( 3 . (The Appendix ives nalytic xpressions
for heH ma[rices.Wenotehere hat,becausehat unc-
tionh• z) isnonzero nlyover • • toz;. •, theH matrices
are ridiagonal.inceheH matricesre ridiagonal,hema-
tricesM e are also ridiagonal. ence,with •(r,zi) and
•(r + Ar,z• defineds he lementsf he ectors(r) and
•(r + At), respectively,eobtain large utsparseystem
of equations f the orm
M-•(r+ ar) = R(r), (25)
w:herehe right-hand ideR(r) is obtained y multiplying
•(r) byM +. Oneach ange tep,heequationsresolved
usingoauss limination.
, B. Boundary conditions
Incorporatinggeneralboundaryconditionsnto the
parabolicquationsdifficult. ortunately,heusual ound-
ary conditionmposed t theEarth'ssurfacen atmospheric
acousticsssimple noughhat t canbeeasily ncorporated
into the PE. In atmosphericcoustics,neusually ssumes
that hegroundsa ocally eacting urface.hisassumption
resultsn an mpedancehat s ndependentf anglebut de-
pendent n requency.ence, t a given requency,hecon-
dition to be satisfied at the Earth's surface is
•9•+ i•_•= O, (26)
0z z
whereo s hewavenumbernairatz = 0 and is he
complexroundmpedance),videdya referencealueor
pc orair. Note:Hereafter, willbe eferredo simply s
thegroundmpedance.With inear inite lements,varies
linearly etweenridpoints. quation26) thus anbewrit-
ten as
( 3,-Cboe)3 køcb-•-9-ø-O, (27)
2
whereAz = z• -- zo, o = •b(zo), nd b•= •b(z•).The point
zo s on the ground,andz• is the first pointabove he ground.
Equation 27) yields n additional quation hat is ncluded
in the setof equations olvedwhen the field is advanced. t
shouM enoted hat heapproximationor)/& isone-sided
(ii.e.,not symmetric).Because f this,z• - zomustbe taken
quite .smalla tenthof a wavelengthn mostcases).
IX[earhe top of the numericalgrid, we want to havean
oatgoingwave, .e., a radiationboundarycondition.We can
alpproximate ucha boundaryconditionby addingartificial
attenuation o the top part of the sound-speedrofile.The
attenttation bsorbs ound eaching he top of the numerical
grid so hat the amplitudes f waves eflecting ff the top end
of the grid are greatly reduced.
C. Starting field
Since he PE algorithm olves n initialvalueproblem,
the initial field •o(z) is neededat the startingrangero.
(Typically, e = 0 is used.)The present E modeluseshe
standard aussiantartingieldproposedy Brock.3 n all
cases xamined, he Gaussianstarter gavegood agreement
with known solutions.
D. Numerical accuracy
To judge he accuracy f the parabolic quationmodel,
we compare he PE solutionwith the solution rom a fast-
fieldprogramdeveloped t the U.S. Army Construction n-
gineeringesearchaboratoryfi'*hecomparisonsre or
upwardrefractionand downward efractionabovea finite
impedance urface.The source requencys 40 Hz with
source nd receiver eightsof 2 and 1 m, respectively. he
ground mpedances computed rom the empiricalequa-
tionsof DelanyandBazley,4using flow esistivityf 200
cgsrayIs. The resultingground mpedancen pc units is
31.4 + i38.5. The soundspeed s linear initially, and is
capped y a homogeneousalf-spacebove, ccordingo
c(z){cogz'orh, (28)
whereg is thesound-speedradient, isheight, o s330m/s,
and h is 100 m.
A comparisonetweenhe PE andFFP for upward e-
fraction g= --0.12 s •) is shownn Fig. 2 asa plotof
excessttenuationersusorizontalange..sOut o 2.5km
from the source,he PE and FFP calculationsgree o with-
in I dB. Beyond2.5 km, numerical naccuracybecomes
more evident n both models,with the accuracydegrading
first n the PE. For the present alculation,t appearshat
the PE and FFP solutions re both numericallyaccurateout
lO
o•• PE
to4
•0-
-30
-50-
-60- •.
O 1.25 2.50 3.75
RANGE [kml
5.00
FIG. 2. Comparison f parabolicequationand fast-fieldprogramcalcula-
tions or upward refraction.
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to 2.5-3.0 km. Beyond his distance, purious scillations
(numerical "noise") dominate the result, and the curves
cease o decay with range. Note, however, hat when nu-
mericalnoise akesover, he sound-pressureevel s already
120dB below he ree-field alueat 1m. In a realapplication,
such a signalwould probablybe below the ambient noise
levelso he numerical imitations n Fig. 2 wouldnot matter.
Figure 3 showsa model comparison or downwardre-
fraction g = + 0.12 s-•). With downwardefraction, n-
ergy is trapped n the surfaceduct formed by the sound-
speed profile and ground surface. As a result, the
sound-pressureevels all off slowlywith increasingange.
Hence,numerical oise s not a problem s t waswith up-
ward refraction.Surface-ductnergys propagated y a lim-
ited numberof "leaky modes"with different ndividualhori-
zontal wavenumbers.The modes add coherently,and,
because of their different individual horizontal wavenum-
bers,produce he characteristicmodal nterference attern
seen n Fig. 3. The overallgoodagreementn the peaks ndi-
cates hat the phase errors in the PE are small. Accurate
phases re necessaryor goodnumericalaccuracy n long-
rangedeterministic redictions.
III. COMPARISON WITH EXPERIMENT
In this section,we compare he predictions f the para-
bolic equationmodel to experimentalmeasurements ade
by Raspet t al. 6Beforemakinghecomparison,owever,
wegivea briefdescription f the proceduressed or thedata
collection and model calculations.
The data reported here were collectedduring an 18-
monthperiod,overopen armland,undera varietyof envi-
ronmentalconditions. oneswerebroadcastrom a height
of 30.5 m and averaged ver 15-min ntervals.Microphones
20.
2.50
RANGE Ikrnl
II
---- FFP
3.•5 ' 5.00
FIG. 3. Comparison f parabolic quation nd fast-field rogramcalcula-
tions for downward refraction.
were ocated1.2 m above he groundat horizontaldistances
ranging rom 305 to 1550 m from the source.The acoustic
data are reportedas excess ttenuation. Atmosphericab-
sorption nd spherical preading ereremoved.
The temperature nd wind speedweremeasured t four
heights 1, 3, 10, and 33 m), and converted o an effective
soundspeed.Once he effective ound peedwasdetermined
at each height, a refraction parameter a was obtained by
fitting he soundspeedwith a logarithmicprofileof the form
c(z) [cø+aln(z/zø)'or >z ,
tCo, for
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40 Hz 160Hz 630Hz
20 1•.
o
-lO
305m
- -10{ -20
-•or -•o -•o ,•
1o
-1o
ß --tO
-1• /''. 3 -60
-20 , s , -80
610m
-1o -20
- -40 884m
-15 -30 -80
20 -40 -100
1o
-10]
15
-20
-2 2, 5 i •
-20 -40
-40 -60
-80
-60 -100
-2 2• 5 i 2 -2 -'• 5 i 2
1150m
-10
-20 -40
-20 -60
40
-30 -80
-60 -100
-40 -80 -120
-2 2• 5 i 2 -2 2• 5 i 2 -2 2• 5 i 2
t550m
REFRACTION PARAMETER, a (rids)
FIG. .Comparisonf arabolicqualionalculationsith easuredaluesf xcessttenuation.n achraph,hebscissas heefractionarameter
andhe rdinates he egativef xcessttenuation.herequencyndangerendicatedlongheopndide,espectively.achataoints 15-min
average;hesolidine s heparabolicquationalculation.
635 d.Acoust.ec. m.,oL5,No. ,February989 K.E. ilbertnd . .White:pplication[ he arabolicquation 635
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facehasbeendeveloped. he modelwasshown o givegood
agreementwith calculationsrom a fast-fieldprogram.
The parabolic quationmodelalsowascomparedwith
excess ttenuationmeasurements nd found to adequately
predict he average xcess ttenuation or neutralor down-
ward-refracting tmospheres ut not for upward-refracting
atmospheres. or upward refraction, he PE predicteda
deepshadow oneat the receiver t 630 Hz, while the data
showedno definiteshadowzone. Possible xplanationsor
the high evels n the regionof a predicted hadow oneare
scattering rom turbulence, ough surfacescattering, r
large-scaleluctuationsn the sound-speedrofile.The PE
model usedhere was imited to deterministic, ange-inde-
pendent, ound-speedrofiles vera smooth round urface.
Hence, the presentmodel did not take into accountany
mechanisms that could weaken the shadow zone. Future ef-
forts will be directed oward including suchmechanismsn
the model.
We conclude hat the parabolicequationmethodcan
accurately reat soundpropagationn a realisticoutdooren-
vironment. n particular,arbitrary sound-speedrofiles nd
a locally eacting round urface anbe used.Because f the
potentialof the PE algorithm or handlingcomplicated f-
fectssuchas turbulenceand irregular ground opography,
the methodshouldproveto be a powerfulcomputational
tool for studies f atmospheric oundpropagation.
ACKNOWLEDGMENTS
The authorswould ike to thank RichardRaspet or
helpfuldiscussionsnd for the data reportedn thisarticle.
Oneof us (MJW) gratefully cknowledgesinancial upport
from the Army ResearchOffice.
0,
z - z,_, )/(z, - z, , ),
h•(z)- 1 z-zi)/(zi+•
L0,
APPENDIX: ANALYTIC EXPRESSIONS FOR THE H
MATRICES
In Sec. I of the mainbodyof the text, thegeneral orms
of the H matriceswere briefly discussed.n this Appendix, ß
we considerheHmatrices n detailandgiveanalyticexpres-
sions.We beginby definingmathematically he basis unc-
tions,whichare used n computinghe H matrices.
As shown n Fig. 1, the basis unctions (z) have he
appearance f a peakedhat (hence, he name "hat" func-
tion). The mathematical definition is
Z•Zi-- I '
Z _ • •Z•Z•,
Z•Zi + I ß
(A1)
The H matricesare constructed rom the basis unctions
the unctions-•(z) and i-l(z), and he operator
Oz)(1/p)(c•/&). For convenience,e define -•(z) as
f• (z) and fi-•(z) as 2(z). Then the matrix elements f
H( 1 and H(2) can be written
(z)-•-•zaz. (A3)
Note that, since he ith basis unction hi (z) overlapsonly
with its nearestneighbor,we have H•(n)= 0 unless
j ----,i + 1. That is, the H matrices re tridiagonal.For the
same eason, he limits of integration re z _ 1 to z•+ • for
H,n), z _ • toz• forH,_ • (n),andz i tozi+ • forH,+ • (n).
To obtain simpleanalytic expressionsor the H matri-
ces,we akep- •(z) and i - • z) to be inear unctionsfz
betweenhe grid points.We furtherassume-l(z) and
fi-l(z) to be continuous t the grid points z . Hence,
between, andz, + l weapproximate,(z) as
f,,(z)=f,,(zi)+(z_zi)(f,,zi+f_,(zi))
i + I -- Zi
(A4)
Inserting he linear approximationorfn (z) into Eq. (A2)
gives, or H(n) for n = 1,2,
H,,(n) =• [(z,--Zi_l)fn(z i 1)
+ 3(zi+, --zi l)fnzi) + (zi+, -z,)f,•(zi+•)],
(AS)
H,ñl (n) = (+•) [f, (zi) +f,,(zi+•)] (ziñ•
(Ar)
In order o compute he third H matrix,H(3), we first nte-
grateby parts o obtain
(f• c•h,(z)•
i;(3) (z)hi(z)•--•z •.
•8hi(z) h•(z)z, (A7
A(z)
wherez• andz• are he imitsof integrationdiscussedarlier.
Sinceheproduct f hi andOh•z)/& is zeroat z• and%,
we have
œ8hi z) Oh•z)
nv(3)=--J-•z A(z) & dz.
(A8)
Inserting Eq. (A4) into Eq. (A8), we obtain
Hii+3= + ) (• zi+-f-L(ziñ(A9
H,(3)= -- [H,+i(3)+H, ,(3)]. (AI0)
Although the computationof H matriceswith linear
finiteelementss slightlymore nvolved han the more amil-
iar difference omputation, here are two main advantages
that ustify he added omputationalffort.First, nonuni-
form grid spacing s easily ncorporated; ence,smallerspac-
ingcanbeusedwheremore ccuracysneeded.hesecond
advantage s that, since he acoustic ield is definedbetween
grid points,small-scale erticalvariations n soundspeed
and densitycan be ntegratedover n the computation f the
H matrices. hus, with linear initeelements,he grid spac-
ing is determinedby the acousticalwavelength nd not by
small-scale variations in the medium.
Hij(n)hi(z)f,,,z)h•(z)z,
The third H matrix is
n = 1,2. (A2)
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