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A very-wide-angle acoustic model for underwater soundpropagationRobertA. DalrympleOceanEngineeringGroup,Department f CivilEngineering,Unioersity f Delaware,NewarlLDelaware 19716LevsiriC. MunasingheOcean ngineering roup, epartmentf CivilEngineering, niversityf Delaware, ewark,Delaware 19716David H. WoodDepartmentfMathematicaicience&epartmentfComputernd nformationciences,niversityfDelaware, Newark. Delaware 19716James T. KirbyDepartment f CivilEngineering, niversityf Delaware,Neumrk,De/aware 9716(Received January1990; cceptedor publication 1 May 1990)A parabolicodel,alidorwide nglesout o90* romhe ssumedropagationirectionfor a homogeneousnvironment), s presented. umericalcomputationsor the modelaredonealmostentirely n the Fourierdomain,and the modelcanbe shown o be theoreticallyexact or a homogeneouscean. n addition, he modelcan mplicifiy handle ange-dependentsound-speedrofiles. n erroranalysisndicateshat the model s moreaccuratehan hestandard arabolic quation SPE) and he modified-wide-anglearabolic quation(MWAPE) for constant erturbations f the ndexof refraction. he accuracy f the model sexaminedby comparison f computed olutionswith exactsolutionsor range ndependentcases. everaldealized ange-dependentases re alsoexamined.PACS numbers: 3.30,Bp

INTRODUCTION

The parabolicequationmethod, first introduced nacousticsyTappert ndHardin,hasbeen sedocomputethe acousticpressureield due to a source or long-rangewavepropagationn a sound hannel (seeRef. 2 for a re-cent overviewof parabolicmodels). A disadvantage f themethod s that there s a restrictionon the propagation n-glesof the acousticwaves,although his s not necessarilyseveredrawback,due to the small grazinganglesof thewaves rapped n a waveguide.A numberof recentworkshave focused n extendingtheparabolicmodel o widerangles. he basisor thesemod-els s the Helmholtzequation,whichgovernshe modifiedacoustic ressure , and whichwill be illustrated ereasa2' a2'ko%2,o=O, (1)&2,

whereko is a characteristic avenumber;n(r,z) = Co/C(r,z) = k(r,z)/ko = indexof refraction;Co s the sound peed ssociated ith ko;C(r) is the soundspeed; (2)k(r,z) = to/C(r,z) is the wave number;to is the angular frequency; is the range direction;z is the depth;p(r,z) is the acoustic ressure.

The Helmholtzequation an be rewrittenas

arwhere

-- + koZQ2p 0, (3)

Q: is an operator n: + -- --k (4)Factoring he Helmholtz equation n a range-independentenvironment results in

(r+ikoQ)(r-ikoQ=O. 5)If we are concernedwith wavespropagatingn the positivedirectionand neglect he nfluence f the waves h the oppo-sitedirection the backscattered aves), henour governingequation s, in parabolic orm,

It is the representationf Q that determines ow wellthe parabolicmodel unctions t largepropagationngles.FollowingappertandThomsonndChapman,wecanrepresentQ asQ = x/1+ p + E, (7)

where

= k az (8)e = n2 -- 1. (9)1863 J. Acoust. oc.Am.88 (4), October 990 0001-4966/90/101863-14500.80 1990Acoustical ociety f America 1863


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The standard arabolicmodel arises y approximatingby the inear erms n a series xpansion:QI + 6/2 -I- t/2. (10)

Thecorrespondingarabolicquationsiko i c9 pP=ikop+ (n- 1)p+- -- (11)Or -- 2ko c9z

If we rewrite Q exactly asQ= 41 +it + e = x/1 + e/(1 +it), (12)

then we can approximateQ in a variety of ways. From abinomialexpansion, eglectingermsof order, and higher,andneglectingerms nvolving roducts f and/, we have,correct to order 6,

Q.-x/l+ It + e/2. (13)Tappert suggestshat thismodel s not an improve-mentover the standardparabolic quation.Alternatively,reversinghe rolesof 6 and t in (12), wecan approximateQ asQ/I + e +it/(2/1 + e), (14)

as n Tappert andTappert ndLee.Thomson nd Chapman,utilizing he methodologydevelopedy FeitandFleck in optics,ntroducedhe ol-lowingrepresentation:Qx/l -It + (41 + e- l). (15)They noted hat the error of this approximation o Q ismuchsmallerhan he wo presentedreviously.Another classof wide-anglemodelshasbeendevelopedusingPad6 approximants f the Q operator. n underwateracoustics,reene for example, asused ( l, 1 Pad6ap-proximant, hileKnightlyet alfi extend he methodo a(2,2) approximant,with the numericaldisadvantagef in-troducing mixed fifth-order derivatives n the governingequation.

I. THE WIDE-ANGLE MODELIn thispaper, he problem s Fourier ransformedn thecross-rangeirectionprior to obtaining he parabolic qua-tion form. This will permitan exact epresentationf Q for

isovelocity ceans, ndveryaccurate esults singstep-wiseintegrations or each of the Fourier modes,which are theninverse ransformed o find the sound-pressurerofile.Thistechnique asused y Dairymple ndKirby to successful-ly model surface water wave propagation over a uniformslope ndbyDalrymple tal. to modelwaterwavepropa-gation over irregular bathymetry.A cylindrical coordinate systemwill be used o describethe domain,with r in the rangedirection,z pointingdown-ward and 0 as the azimuthal angle.We will assumehat theazimuthal ariationsveryslight nd hat hedependencefpressurep n theazimuthalangle0 canbe neglected. t thesurface,we will assume pressureelease oundary ondi-tion, p(t:,0) = 0, and, at the constantdepth bottom, a no-flow boundary condition,

0p=0 onz=h. (16)The equation overninghe pressureield s givenby theHelmholtz equation,o2p 1ar r 0 -- +k2p=0' (17)

where k(r,z) is the wave number governing he acousticalwavemotion.A representative avenumber, o r), is ntro-ducedvia the index of refractionn and a parameter,v(r,z),which is a measure of the deviation of the index of refractionfrom unity:

=kon =k(l--). (18)Now, our equation s

O:'p + (k -kgd)p0. (9)3r r 3rFinally, he reduc fore 0f the Helmholtz uation is ob-tained by removing he radial spreadingby assuming hat

p(r,z) =p(r)/. (20)Substituting q. (20) into . (19) andneglectinghe ewith 1/(4), i.e.,a far-field pproximation,eobtain

OrWe now introduce the Fourier sine transform, which is

definedsymbolicallyasP(r,v,, ) = (P). (22)

Explicitly, the Fourier transfore paim areI (r,z) in zdz, (23)

(r,z) p(r,y,,sin z, (24)where

y=(m+)(,/h), m = 0,1,2....m. (25)A Fourier sine transform of the ruced Heimholtz

equation, . (21), transforms he equation o one solelydependent n the r erdinate;dP (kg ffm)P-(kgdP)=0. (26)dr

To obtaina parabolicmel from thissecond-orderqua-tion, we split the uation, assumingn Fouder space hatthe pressure,(r,y ) can be split nto a forward-propagat-ing wave and a backscattered wave, p(r,y)= p + (r, y,, ) + p- (r,y), which satisfy he following pairof equations:dP+ ikg - + + G(r,r, (27)drdp- i4k P- - G(r,r, (28)dr

whereG(r,y ) is unknown priori. [If the ast erm n s.(27) and (28) were zero, .e., for a homogeneousnviron-ment, hen he parabolic plittingwouldbe exact. Substi-1864 J. Acoust. oc.Am.,VoL88, No.4, October 990 DalrympletaL:Wide-angiecoustic odel 1864


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tuting these equations nto (26) and assuming[P+1>>P- ] resultsn anexpressionor G(r,y, ):a(r,r,.) a(,o2r,.)/ar2xo- Ym 2o2 - /2rn

that

(29)For range-independentavenumbers, nly the right-handterm of G(r,7., ) exists:

6(r,rm = is( - t,vP)/2x/kg r. (30)Equation 27) with either (29) or (30) serves sa para-bolicmodel. f p(r,y,. is defined uch hatp+ (r,v, ) - p(r,y,.) xp(i]k- y.r), (31)

thenEq. (27) canbe rewritten sOeXp(ix0-T,,,)___t9(o2 -,. /Or24ko _ 7m (32)

Equation (32) is better suited or numerical solutionsthan Eq. (27) becausehe representationf + as in Eq.(31) removeshemain scillatoryerm,k - y,, +fromnumerical omputationso hat fewer ange amplesare needed.

II. ERROR ESTIMATESIt is possibleo estimate he accuracy f Eq. (32) asanapproximateolutiono Eq. (21), the Helmholtzequation,for a range-independentound-speedrofile.A perturbation

s(z) fromn2(z) -- 1 of ordere canbedefined s ollows:2 = __ n2(z) _ 1] = -- es(z). (33)To analyze he effectof perturbinghe indexof refrac-tion by a term of order e, consider he casewheres(z) = 1and e is a smallconstant.Then, an exactsolution o Eq. (32)can be formally statedas follows:b boxp ikoer/2o r )' (34)The solution o Eq. (27) is then)+ =bo xp ix[o-- mr+ (ikoer/2]k- y,.) .

(35)The inverseFourier transformof Eq. (35) yields he

result

P* = o in(y,z)Xexp[ko - 72,,+ (ikoer/2Gk, ) ],

(36)where + s the acoustic ressuref the forward-propagat-ing wave. (37)

Equation 36) is an approximationo the solution fEq. (21), which, n thiscase, sp+ = boin(y.z)xp(ix/k-- r + koer)(38)

The fact hat the arguments f the exponentialsn Eqs.(36) and ( 38 arenot hesame eads o phase rrors, uchasare oftenobservedn modelingwavepropagation y usingEq. (27), which has only first derivativeswith respect orange.An estimate f the phase rror inherent n our methodcan be determined nd compared o similar errors n twoother ypesof parabolicwaveequations,he standard ara-bolicequation SPE), and the modifiedwide-angle arabol-ic equation (MWAPE). The phaseerror in our method isdue to the use of

k ='(ko2 72r) koe/(2x[kom (39)asan approximationor

kexact4(ko q-ek&-'2m). (40)The approximationo k, t of the standard arabolicequationsgiven yMcDaniel andRobertsonisk, = ko+ ko/2- ,/(2ko). (41)

Similarly, the corresponding pproximationused in themodified-wide-angle arabolic equation as given by St.Mary et al? andRobertson is

/,oe r r4 er42 2ko 8k 03 4ko 8 (42)

It is possibleo rewritekt as ollows: .... =4(-r)+ 2x/(o_ rm)

-- + h.o.t. (43)8(,o- )4( g )and he ollowing rrorestimatesor a constt perturbation

mn then be made:

)...--kw=ko 1 +h''t'o(), )+ )...-&= ok8 h.o.t.

+6k(+h.o.t.)o(), (45)k...-k,=koff3ff+h.o.t.)+o,16k 32ko (46)

Curves f 6 vs sin- (v/k o for s. {)-(46) aregiven n Fig. 1 as a functionof the aolute vues of(k ... -- k, )/k o, (k,, -- k, )/k o, and-- k )/ko, restively. Since/k o is thesine f theangleassociat with the mth mode,note hat, for anglesassociat with small values of m, the MWAPE can handle alarger range of 6 for the same value of the errorsl(k,,,--k)/ko[ and [(k ... -k)/kol than theVWA model. However, as the angle (or m) incre,es, theerror ](k ... -- kmp )/kol of theMWAPE is ger than

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usingKaiser-windowechniques,nda high-angleEsourceunction y Greene wereconsidered.ll three ni-tial fields avemodel esultsor the estcaseshat differedveryittle rom ach ther n hegraphicalcale f he esul-tant transmission losses.The referenceound peedor the nitial ieldwas akentobe1510m/s,while hemodeleferenceave umberoandhencehe model eferenceound peedwasobtainedfromk = -L k2 z, (54)500 ao

wherehe ntegral as esignedo minimizehemagnitudeofkovinG(r,y,,. This ave odeleferenceoundpeedsof 1520, 547, nd1574m/s orcases , B, andC, respec-tively. t should enoted hat he ntegral oes ot ake ntoaccount he higher soundspeed n the false bottom and,hence,he actualmodel eferenceound peedsouldbehigher.UsingThomson nd Chapman'snitial field, he com-putationalesultsf he ransmissionossdB)versusange(km)are hownnFigs. 4. Theoverallgreementsgood.An nterestingeature f themodelsshownn Fig.2 (caseA), wherehe esultant etailgenerallygrees ellwith heFFP resultn the 10- o 20-kin egionhanwithFishman ndMeCoy'shigh-frequency odel esult.The modelwassome-what ensitiveo hechoice fko, sshownn Fig.3(a) and(b) for model eferenceound peedsf 1547 nd 1600m/s,respectivelyorcaseB. The higher ound peedesultedn asmall mprovementf detailof thehigh ransmissionossesat 11, 15.5, and 17 km. Results for caseC were obtained forthesoundpeedf 1574m/s,asshownnFig.4 a) aswellas

fora soundpeedf 1600m/s,shownnFig.4 (b). Similarocase , thehigher oundpeedontributedo a slightm-provementndetailn he10- o20-kmegion. benchmarkforcomputationalpeedan eprovidedithhe nowledgethateach f theabove ases as omputedyone housand20-m ange ectionssinghe ourth-orderunge-Kuttascheme, hich ook100s of scalar rocessingnan BM3090.Since achstepof the fourth-order unge-Kuttaschemeequires ne nverse ouriersine ransform ndoneFourier ine ransformsee heAppendix)or the ourth-orderscheme,ight ransformsre required.n addition,one nverse ourier ine ransforms required t theendoftheentirestepgiving totalof nineFouriersine ransformsfor each angesection r a total of 9000 Fouriersine rans-formsor 20 km. n comparison,heSPE mplementedythesplit-stepourierransformlgorithmookust25s forthesame omputationssing n denticalange tep.The modelwasalsoused o compute ound-pressuretransmissionossesn an idealizedange-dependentceanenvironment,sshownn Fig.5.The ange ependencyasprovided y reducinghe sound peedroman otherwisehomogeneous500m/sby0.2m/spermeter oncentricallyalongheradiiof a circle f diameter0.5kin,extendingfrom range f 5-15kmand eachingmaximumepth f500m at a range f 10km.Thisgave sound peed f 1400m/s at theocean urface t a rangeof 10 km, anda verticalsectionhroughhis ingof reducedound peedsshownnFig. 5. Similar to casesA-C, a false bottom from 1500 mdown o 2560m withanexponentialttenuationrofilewasusedo minimizeeflectionsrom hebottom f thegrid.Anormal-mode olution or a homogeneousceanwith asource t a depthof 500m wasutilized o generatehestart-

zI-

7C

8O

9C

lOCRANGE (kin)

FIG. 2. Computational esults orcase A with a model reference soundspeed of 1520 m/s: ---, VWAmodel result;- - -, FFP result romFishman ndMcCoy;s---,Fishmanand McCoy'shigh-frequency odelresult.

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(i-

7O

9O

7

!oc

RANGE (kin)

RANGE (km)

FIG. 4. (a) Computational esultsfor case C with a model referencesound speedof 1574 m/s: ,VWA model result;- - -, FFP re-sult romFishman ndMcCoys --,Fishman and McCoy's high-fre-quencymodel sult. (b) Computa-tional results for case C with a modelreferenceound peed f 1600m/s.

match hoseof Fig. 6 after a rangeof 10 km and are spatiallyshifted, uggestivef accumulated hase rrors,whichare tobe expected rom the application. f the split-stepFouriertransformalgorithm to a range-dependent nvironment.As a control,Fig. 8 gives comparison f transmission

loss (dB) contours of a homogeneousocean with a soundspeedof 1500 m/s and with an exponentially ttenuatingfalsebottom for a normal-mode nitial field. It is interestingto note that evena weakly range-dependentceanenviron-ment suchas the ring shown n Fig. 5 considerably ffects

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oo

RANGE ( kl ,

0.2 5 10 15 20

//

FIG. 5. Idealizedocean or a range-dependent sound-speed environ-ment. Sound-speed ontour ntervalis 10 m/s.

underwatersoundpropagation.A more severe est was providedby applying the modelto the range-dependentceanenvironment, sshown n Fig.9. The contoursare similar to those n Fig. 5 except hat thesoundspeed aries rom 1500m/s by 0.6 m/s/m concentri-

eally along the radii from 5 to 15 km and givinga soundspeedof 1200m/s at the oceansurfaceat a rangeof 10 kin. Afalsebottom,similar o that of the earlierexample,wasusedto minimize reflections from the bottom. A normal-modesolution or a homogeneousceanwith a source f frequency

0.02

oo

oo

5 lO 15RANSE (k)

20

FIG. 6. Computational esults orthe very-wide-angle odel appliedto the idealized ange-dependentn-vironment with contours of thetransmission loss (dB). Contour in-terval is 5 dB.

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0.02 10 15RANGE (km)

2O

FIG. 7. Computational esults orthe split-step ourier ransform l-gorithm applied to the idealizedrange-dependentnvironment ithcontours of the transmission loss(dB). Contour interval is 5 dB.

25 Hz at a depth f 500m wasusedo generatestartingfield. hemodelange nddepthncrementsere10and20m, espectively.igure 0showshe esultora modelefer-ence ound peed btainedromEq. (54), whileFig. 11showsheresult or a model eferenceound peed btainedfrom

1 f7k2dz (55):5-JoThe results reshown scontourmaps f the ransmis-sionossdB).Thesensitivityf hemodelo heselectionf

ko s llustratedy hedifferencendetailn he angef10-

0.02 5 10 15

RANGE (kin)20

oo FIG. 8. Computationalesults orthe very-wide-angleodel nd thesplit-step ourier ransform lgo-rithm applied to a homogeneousoceanwith a soundspeedof 1500m/s. Contours are of the transmis-sion oss (dB), and the contour n-terval is 5 dB.

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0.2 5 lo 15 20

FIG. 9. Idealizedocean or a strong-ly range-dependentound-speedn-vironment. ound-speedontour n-terval is 50 m/s.

20 km between igs. 10and 11.However,asexpected,herewasa strongsimilarity n the overallstructure. f a receiverwasplaced t a rangeof 20 km andat a depthof less han 500m, it wouldbe important o evaluate he modelreferencesoundspeed rom an integralwith limits similar to those nEq. (55) and not from Eq. (54). However, if the receiver

wasplaced t a muchgreater epth,Eq. (54) should henbeconsidered.For comparison urposes,he same est casewas alsomodeledby the wide-angle plit-step ourier transformal-gorithmwith a reference ound peed f 1500m/s. The con-tours of transmissionoss (riB) are shown n Fig. 12. A1-

0.02 5 lO 15 2O

FIG. 10. Computational esults orthe very-wide-anglemodel appliedto the idealizedstrongly ange-de-pendent nvironment sing modelreference sound speed computedthrougha depthof 1500m with con-tours of the transmission oss (dB).Contour interval is 5 dB.

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0.02o

o

oo

5 10 15 20

FIG. 11. Computationalesults or thevery-wide-anglemodel applied to theidealizedstrongly ange-dependentn-vironment using a model referencesound peed omputedhrough depthof 700 m with contours of the transmis-sion loss (dB). Contour interval is 5 dB.

though hereare similarities etweenFig. 12 and Figs. I0and 11, n the 10-20-km egion, heyarenot as eadilyvisibleas or thepreviousestcasewherea sound-speedariationofonly 100m/s wasused. his s to beexpectedrom the split-stepalgorithmwith largedeviations fk(r,z) from the refer-enceko r), in addition to its inability to handlestronglyrange-dependentnvironments.The phenomenonf wave ocusingor cases f waves

propagatinghrough refractiveocuss well known.Accu-rate reproduction f experimental esults or this casehasbeenwidely chievedorwaterwaves ymany uthors.7-19Oneof the majordifferencesetween he modeling f waterwavepropagation nd underwater oundpropagations theinitial field used to start off the computations.For waterwaves, n order to be realistic, he initial condition s usuallya monochromatic aveof constant mplitude.However, or

0.02 5 10 15 20

FIG. 12. Computationalesultsor thesplit-step ourier transform lgorithmapplied o the idealized trongly ange-dependentenvironmentwith contoursof the transmission loss (dB). Contourinterval is 5 riB.

1873 J. Acoust.Sec. Am., Vol. 88, No. 4, October 1990 Dalrympleeta/.: Wide-angleacousticmodel 1873


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5 250

o

g

RANG; (m)500

FIG. 13. Idealized ocean or a range-dependentsound-speed nvironmentto illustrate ffects f focusing. ound-speed ontour nterval s 50 m/s.

underwateround ropagation,quitedifferentnitialcon-dition s usedcorrespondingo the form of a depth-depen-dent delta function. We were interested to find an idealizedocean est casecombinedwith an initial sound-pressure on-dition that exhibitedstrong ocusing haracteristics.

We considered n idealized ange-dependentceanen-vironment imilar o the onediscussedefore xcept hat therangedependency as imited o a semi-ellipsextendingoa depthof 250m andsituated t a rangeof 100 o 350m. Theboundaryof the semi-ellipse asdefined ccordingo

RAE (m)20 500 1000 1500 2000

lO

500

lOOO

15oo

FIG. 14.Computationalesultsorthe very-wide-anglemodelappliedto the idealized range-dependentenvironment singa model refer-ence ound peed f 1500m/s to il-lustrate effectsof focusing.Con-tour interval is :5 dB.

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R 2 + Z 2= 1, (56)where

R = [ (range n meters) 225 /125 (57)Z = depth n meters/250

and the sound peedn the semi-ellipse asdefined ccord-ing tosoundpeed /s= 1500- 300[1 -- x/(R2+ Z ) ].

(58)This idealized ange-dependentceanenvironmentsshownn Fig. 13.The sound peed utsideheellipsewasahomogeneous500m/s, while he nitial condition f a con-tinuoussourceof noisewas represented y an idealizedsquarewave of magnitudeunity and frequency 5 Hz ex-tending rom 100 o 1400m in depth.This initial conditionapproximately epresentshe effectof a great number ofsuperimposedepth-dependentelta functions nd hasanexactanalyticFourier sine ransform.The contoursof soundtransmissionoss dB) for this case s shown n Fig. 14 and

clearlyshows he effectof focusing fter the ellipse.l. SUMMARY

We havedescribed very-wide-angle umericalmethodforsolvingheHelmholtzequation or sound-pressurerop-agation or a constant ensity, ange-dependentcean nvi-

ronment. he algorithmhasbeen uccessfullypplied o ide-alized ocean environments nd is computationallywellsuited or arrayprocessing,articularlywhen he Adams-Bashforth-Mouitonpredictor-corrector cheme s imple-mented,alongwith vectorized omputation f the discretefast Fourier transforms.In the future, the algorithmwill be applied o range-dependentoceanenvironments nd will be extended o in-elude he effects f strongdepth-dependent erturbations fthe index of refractionand the effects f depth-dependentdensityprofiles.

APPENDIXWe rewriteEq. (32) for the range-independentase orthe sakeof brevity:3= {f '[exp(ik02ya,,r)} (AI)ar 2i4/o:- Ymxp(i,q/o-- m )

where = k0. (A2)For the second-order unge-Kutta,(r+ At) = b(r) + 0.5(k, + k2 ), (A3)

where

or

k 2

-- Ar-,( if-/- [ b xp(/x/k2- yz,,,) ] }2ik02 exp(ix/k2

-- Ar-s Vl,f 1{(I//--[ )exp 02 /2rnr+ Ar)]})2io2-- ,. expix/k -- y, r+ Ar)]

[ ( Ar-'[+(r)lexp(ix/k--, --Ar,,ff-,-'+exp(i4k-Ar)- 2i4kNow,

)(r + Ar)=(r+ Ar)exp[i4k- (r + Ar)].Hence, rom s. (A3), (A4), (A6), and (A7),

- ar, [ + (r) ] exp(ik p+ (r+ Ar)=p+ (r)exp(i -- mAr) g -/m

(A4)

(AS)

(A6)

(A7)

(AS)

F. D. TapDeft nd R. H. Hardin,"Computer imulationsf long-rangeocean cousticropagationsingheparabolicquation ethod,"m Pro-ceedingsf the8th nternational ongrexsn 4cousticsGoldcrest, on-don, 1974), Vol. 2, p. 452.W. F. AmesandDing Lee,"Currentdevelopmentsn thenumericalreat-ment of oceanacousticpropagation," ppl. Num. Math. , 2547{ 1987).

F. D. Tappert,Theparabolicpproximationethod,"n Waoe ropaga-tion and UnderwaterAcoustics, ecture Notes n Physics, dited by J. B.Keller andJ. S. Papadakis Springer,New York, 1977), Vol. 10,pp. 224-287.D. J. Thomson ndN. R. Chapman, A wide-angleplit-steplgorithmfor the parabolic quation,"J. Acoust.Soc.Am. 74, 1848-1854 (1983).F. D. Tappertand D. Lee, "A range efraction arabolic quation,".

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