Gilbert (1989a) the Parabolic Equation Applied to Outdoor Sound Propagation (1)

8/9/2019 Gilbert (1989a) the Parabolic Equation Applied to Outdoor Sound Propagation (1)

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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

631 J. Acoust. Sec. Am., Vol. 85, No. 2, February 1989 K.E. Gilbert and M. J. White: Applicationof the parabolic equation 631


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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).

632. J. Acoust. ec.Am.,Vol.85, No.2, February 989 K.E. Gilbert ndM.d. White:Applicationf the parabolic quation 632


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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.

6;;t3 d. Acoust. ec.Am.,Vol.85, No.2, February 989 K.E. Gilbert ndM. J. White:Applicationf theparabolicquation 633


5/8

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


6/8

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


7/8

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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tion of electromagnetic avesalong he earth'ssurfaceby the methodof

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Gilbert (1989a) the Parabolic Equation Applied to Outdoor Sound Propagation (1)

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