Propagation of sound above an impedance plane in a downward refracting atmosphere

$Page 1: Propagation of sound above an impedance plane in a downward refracting atmosphere$
Propagation of sound above an impedance plane in a downwardrefracting atmosphere

Kai Ming Lia)

National Center for Physical Acoustics, University of Mississippi, University, Mississippi 38677

~Received 10 April 1995; revised 11 September 1995; accepted 25 September 1995!

An asymptotic analysis has been carried out to study the propagation of sound at long ranges in anarbitrary downward refracting sound-speed profile. It has been shown that the solution can beexpressed in a form of the Weyl–Van der Pol formula and the present theory offers an extension forthe established theory to the case of a vertically stratified medium. The analysis starts from the fullwave equation and the derivation assumes that the ambient properties only depend on the heightabove an impedance ground and that there is no wind or turbulence in the medium. In addition, thetheory also assumes that the sound speed of the atmosphere increases monotonically with height.For such an atmospheric condition, the medium forms a waveguide that enhances the sound levelsat long ranges along the ground surface. It is demonstrated that the acoustical path length derivedby the asymptotic method is identical to that derived by the analytical ray trace model. Furthermore,the asymptotic method also allows a proper inclusion of the surface wave term rigorously. ©1996Acoustical Society of America.

PACS numbers: 43.28.Fp, 43.20.Bi, 43.50.Vt

INTRODUCTION

In this article, we investigate the sound propagation out-doors in a downward refracting atmosphere above a compleximpedance ground. The influence of meteorological condi-tions and the effect of finite ground impedance have longbeen identified as the major factors affecting outdoor soundpropagation.1,2 We are mainly concerned with the meteoro-logical influences on sound propagation in the current inves-tigation. Recently, several theoretical studies3–5 have sug-gested that the wind has significant effects on the predictedsound-pressure level. But theories of sound propagation un-der the influence of wind usually involves a mathematicaldevelopment that rapidly becomes intractable. Neverthelessthe presence of wind only affects the sound field quan-titatively.6 We can still understand the general behavior ofmeteorological influences by ignoring the wind effect and byconsidering of a temperature-stratified medium.

When the speed of sound in a medium increases mono-tonically with height, the medium is downward refractingwhich, together with the reflecting ground, forms a surfacewaveguide and supports sound propagation at long ranges.Raspetet al.7 use the method of normal modes to study theinteraction of the downward refracting sound wave with theimpedance ground. Their investigation is limited to an ideal-ized situation of a linear sound-speed profile. Furthermorethe method of normal modes inevitably leads to a numericalsolution that becomes difficult to interpret in a more compli-cated sound-speed profile.

On the other hand the classical ray trace approach allowsone to visualize the propagation of sound in terms of rays. Itis intuitive to suggest that, in a downward refracting me-dium, some of the rays which launch upward initially will

bend gradually toward the horizontal axis~parallel to the flatground! and they will eventually come down to hit theground. Consequently, the repeated reflection and refractionof some of the rays contribute to give the total field for soundpropagation at long distances. In an earlier study,8 it has beendemonstrated that the distant sound field predicted by the raytrace approach agrees closely with that calculated by the fastfield program~FFP!. We remark here that the FFP is an ef-ficient numerical scheme that is based on the full waveequation.9,10

In this paper, we wish to examine the sound field bystarting from the full wave equation. We also include simul-taneously an arbitrarily sound-speed profile and a compleximpedance plane in our analysis. It is hoped to understandthe validity of the ray trace approach in a downward refract-ing atmosphere and, in particular, the situation for the re-flected waves where there are multiple bounces from theground surface.

We base our analysis on the method of Fourier transfor-mation so that the wave equation is reducible to a one-dimensional Helmholtz equation. The well-known WKBmethod11 provides an asymptotic solution for the one-dimensional Helmholtz equation where the source and re-ceiver are far from the turning point. The sound field canthen be expressed in a Fourier integral which can subse-quently evaluated by the method of steepest descents. Li12

uses this approach to derive an asymptotically expression forthe sound field in a vertically stratified medium at shortranges. However his analysis is invalid at long ranges be-cause the rays will pass through turning points. Here weattempt to extend Li’s work to give the sound field at longranges. The ‘‘local’’ difficulty of the WKB method at turningpoints is overcome by using Airy functions.

In Sec. I, we derive an expression of the sound field interms of inverse Fourier integrals. The analysis starts fromthe full wave equation so that the theory can be applied gen-

a!On leave from Engineering Mechanics Discipline, Faculty of Technology,The Open University, Milton Keynes MK7 6AA, United Kingdom.

746 746J. Acoust. Soc. Am. 99 (2), February 1996 0001-4966/96/99(2)/746/9/$6.00 © 1996 Acoustical Society of America

Downloaded 23 Oct 2012 to 136.159.235.223. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

erally not only in an atmospheric environment but also inother stratified media, such as, an oceanic environment. Wealso show that a closed form analytic solution can be ob-tained for the transformed pressure in the case of a linearsound-speed profile. In Sec. II, we use the method of steepestdescents to estimate the inverse Fourier integral. We alsoidentify the close analogy between the classical ray traceapproach and our present method.

I. FORMULATION

A. Derivation of the integral solution for thepropagated sound

Consider a point monopole source~with strength of 1Pa m! situated at a height ofzs above a reference plane in anatmosphere with the sound-speed profile depends on the ver-tical heightz only. We choose the reference plane arbitrarilyat z50 for convenience~see Fig. 1!. The propagation ofacoustic pressurep from a point monopole source with anangular frequencyv in the vertically stratified atmosphere isdescribed by the wave equation13

r“–@~1/r!“p#1k2p52d~r2r s!, ~1!

whered~•! is the Dirac delta function,r(z) is the mean den-sity of air, k(z)5v/c(z) is the wave number of the acousticdisturbances, andc(z) is the speed of sound in the atmo-sphere. The field point and source point are denoted byr andr s , respectively, in Eq.~1!. The time-dependent factore2 ivt

is understood and suppressed throughout.To simplify the analysis, a new variableP is introduced

to replace the acoustic pressure such that

p5ArP. ~2!

Further the terms involving (d2r/dz2) and (dr/dz)2 are ig-nored in the following analysis because the variation in themean density of air is usually small in a normal atmosphericcondition. Introducing a zero-order Hankel transform pair,P̂andP, where

P̂~kr ,z!52pE0

`

rPJ0~krr !dr, ~3!

P51

2p E0

`

kr P̂J0~krr !dkr ~4!

and J0(krr ) is the zero-order Bessel function of the firstkind, we can transform Eq.~1! to a second-order ordinarydifferential equation in terms ofz as

d2P̂

dz21kz

2~z!P̂52S 1

Ar D d~z2zs!, ~5!

where

kz~z!51Ak22kr2. ~6!

Here, kr and kz may be interpreted as the horizontal andvertical wave number, respectively. The solution of Eq.~5! isrequired to satisfy the Sommerfeld radiation condition; i.e.,the solution contains no incoming waves asz→6`. Thepositive root forkz(z) is chosen in Eq.~6! in order to ensurea finite and bounded solution forp asz→6`.

To bring out the features of ray theory, we limit our-selves to the problem of a downward refracting atmosphereand there is only one simple zero ofkz(z) at the pointzp ,i.e., kz(zp)50. The zero of the vertical wave numberkz cor-responds to the turning point of an acoustic ray. We remarkthat a straightforward application of the WKB method wouldfail at the turning point. However Langer’s method~see, forexample, Ref. 14! may be employed to find a uniform as-ymptotic approximation which is valid for all heights includ-ing the turning point. The uniform asymptotic solution forEq. ~5! is14,15

P̂~kr ,z!5r21/2~z/kz2!1/4@d1 Ai ~2z!1d2 Bi~2z!#, ~7!

where the nondimensionalized variablez(z) is given by

z~z!5F32 Ez

zpkz dz8G2/3, ~8!

for kz2>0, i.e.,zp.z>0, or

z~z!52F32 Ezp

z

~2kz!1/2 dz8G2/3, ~9!

for kz2,0, i.e.,zp,z. The dummy variablez8 is used in the

integrals of Eq.~8! and ~9!. The functions, Ai(z) and Bi(z),are Airy functions of the first and second kinds. The arbitraryconstantsd1 and d2 with units m1/2 are to be determined.Both z andkz vanish at the turning point but the ratio,z/kz

2

remains bounded and henceP̂ is finite in all the region. Fora complex impedance plane, the solution of Eq.~5! mustsatisfy

]p

]z1 ik0bp50 at z50, ~10!

whereb is the specific normalized admittance of the ground.By satisfying the continuity of pressure atz5zs , the

discontinuity of the particle velocity across the source, theSommerfeld radiation condition at infinity, and the imped-ance boundary condition, the approximate solution of Eq.~5!for P̂ is obtained as

P̂~kr ,z!52p~rs!21/[email protected], /~k.k,!2#1/4 Ai ~2z.!

3$D Ai ~2z,!2Bi~2z,!%, ~11!

where

FIG. 1. The schematic diagram of the source/receiver geometry.

747 747J. Acoust. Soc. Am., Vol. 99, No. 2, February 1996 Kai Ming Li: Sound in a downward refracting atmosphere


D5~k0

22kr2!1/2z0

21/2 Bi8~2z0!1 ik0b Bi~2z0!

~k022kr

2!1/2z021/2 Ai 8~2z0!1 ik0b Ai ~2z0!

, ~12!

k.[kz(z.), z.[z~z.!, etc.,z0 is the evaluation ofz at z50,and rs is the mean density of air at the source height. Thevertical distancesz. andz, are defined as

z.5max~zs ,z! and z.5min~zs ,z!.

The functions, Ai8 and Bi8, are the derivatives~with respectto their arguments! of the Airy functions of the first andsecond kind. In Eq.~12!, the expression forD is derived bynoting

dz

dzUz50

52~k022kr

2!1/2z021/2.

Substituting Eqs.~4! and~11! into Eq. ~2!, we can writethe integral representation of the total sound field above animpedance plane in a downward refracting atmosphere as

p~x,y,z!521

2 S r

rsD 1/2E

0

`

[email protected], /~k.k,!2#1/4

3J0~krr !Ai ~2z.!$D Ai ~2z,!

2Bi~2z,!%dkr . ~13!

Here, the ratior/rsmay be taken as unity in Eq.~13! becausethe variation of mean density of air is small in normal atmo-spheric conditions.

The total sound field may now be obtained either byevaluating the integral of Eq.~13! numerically or by approxi-mating it asymptotically. In the following sections, we willaddress the later method for a deeper understanding of theproblem and its solution. We remark that the analytical solu-tion may also be obtained by the method of normal mode7

but, in this paper, we restrict our analysis to the asymptoticevaluation of the integral.

B. Analytical solution for a linear sound-speed profile

To evaluate the integral of Eq.~13!, we need first todeterminez from Eqs.~8! and~9!. For a sound velocity pro-file given in the form of7,16

c~z!5c0~122az!21/2, ~14!

the integrals of Eqs.~8! and ~9! can be evaluated explicitly.The solution is written in a compact form as

z5 l 2@k02~122az!2kr

2#, ~15!

where l5(1/2ak02)1/3, which is known as the wave layer

thickness.7 The constanta in Eqs. ~14! and ~15! may beinterpreted as the normalized sound velocity gradient17 be-cause for sufficiently smalluazu, the profile can be approxi-mated by

c~z!5c0~11az!.

We point out that the value fora has an order of 131025

m21 ~see Ref. 18! in a typical windless outdoor environment.Using Eqs.~6! and ~15!, we can determine the ratio of

z/kz2 to give

~z/kz2!1/45Al . ~16!

Similarly, the termD in Eq. ~12! can be simplified to

D5Bi8~t!1q Bi~t!

Ai 8~t!1q Ai ~t!, ~17!

where t5(kr22k0

2) l 2 and q5 ik0b l . Substitution of Eqs.~15!–~17! into Eq. ~11! leads to

P̂~kr ,z!52p l

ArsAi ~2z.!SAi ~2z,!

3Bi8~t!1q Bi~t!

Ai 8~t!1q Ai ~t!2Bi~2z,! D , ~18!

wherez. andz, denote the variables evaluated according toEq. ~15! at z. andz, , respectively.

We emphasize that other pairs of independent solutions,such as Ai~2z! and Ai(2ze2ip/3), may be used as a solutionin Eq. ~7!. Nevertheless using the relations between theseindependent solutions~see Eqs. 10.4.6–10.4.9 in Ref. 19!,we can rewrite Eq.~18! in terms of Ai~2z! and Ai(2ze2ip/3)as

P̂~kr ,z!522p l

Arseip/6 Ai ~2z.!SAi ~2z,e

2ip/3!

2Ai ~2z,!e2ip/3Ai 8~te2ip/3!1q Ai ~te2ip/3!

Ai 8~t!1q Ai ~t! D .~19!

It is worth pointing out that Eq.~19! is the exact wave solu-tion for the sound-speed profile given in Eq.~14!.7,20 Notehere that the monopole source strength used in this paper is1.0 Pa m while it is 4p Pa m in Ref. 20. It is reassuring tofind that the transform pressure developed in this sectionagrees with the previous works in the case of a linear sound-speed profile. Other more realistic sound-speed profiles, forexample. a logarithmic profile, may be used at the expense ofmore computational time in order to evaluate Eqs.~8! and~9!numerically.

In his analysis, Rasmussen20 ignores the mean densityvariations and computes the sound field by direct numericalintegration of Eq.~13! for a downward refracting atmo-sphere. However, the sound field can be computed more ef-ficiently along a number of sampling points simultaneouslyif the fast Fourier transform method is used instead. The fastfield program~FFP! implements this idea. There are numer-ous publications in this area9,10 and the details and its imple-mentation will not be repeated here. We will concentrate onthe asymptotic method in the following sections.

II. ASYMPTOTIC REPRESENTATION OF THESOLUTION

A. Further approximation of the transformed pressure

In this paper, we endeavor to find a closed form solutionfor the high-frequency sound field for long-range soundpropagation in a downward refracting medium. We shall usethe method of steepest descents11,12 in order to evaluate the



inverse transform integral of Eq.~13! asymptotically. To aidthe process, we need to approximate the Airy functions andtheir derivatives. For a practical situation of atmosphericsound propagation at long ranges, we havezp.z. , wherezpis the height of the turning point in whichkz(zp)50. In thiscase,z0, z, , andz. are all positive and real, see Eq.~8!. Inan idealized situation of a linearized sound-speed profile, wefind from Eq.~15! that z is proportional to (k0/2a)

2/3 whichis about 300 for a normalized sound velocity gradient of theorder 1024 m21 and a frequency of about 50 Hz. This nu-merical example demonstrates thatz0@1 except for a smallregion near the turning point in whichkz(z)'0. Thus we canuse the asymptotic representation to approximate the Airyfunctions and their derivatives in Eq.~13!. For z.0, the as-ymptotic expansions for the Airy functions of the first andsecond kind and their derivatives are given by19

Ai ~2z!'p21/2z21/4 sinS Ez

zpkz dz1

p

4 D , ~20a!

Ai 8~2z!'2p21/2z1/4 cosS Ez

zpkz dz1

p

4 D , ~20b!

Bi~2z!'p21/2z21/4 cosS Ez

zpkz dz1

p

4 D , ~20c!

Bi8~2z!'p21/2z1/4 sinS Ez

zpkz dz1

p

4 D . ~20d!

The forms used here for Airy functions are accurate tothe first order and the higher-order terms will be ignored inthe subsequent analysis. These approximations for Airy func-tions are not only valid forz0 but are also applicable forz,

andz. becausez..z,.z0 in a downward refracting atmo-sphere. Using Eq.~20a–d!, we can simplify Eq.~12! for D to

D5~2 i !311V exp$2ik0~Lp2L0!2 ip/2%

12V exp$2ik0~Lp2L0!2 ip/2%, ~21!

where

V5N02b

N01b, ~22!

N~z!5~1/k0!Ak22kr2, ~23!

Lz5L~z!5Ez

N~z8!dz8. ~24!

The subscripts 0 andp denote the parameters evaluated atz50 andz5zp , respectively. We recall that the positive rootis chosen forN(z) in Eq. ~23! to ensure a finite and boundedsolution. The choice of the lower bound in the integral of Eq.~24! is arbitrary because we are mainly concerned with theirrelative magnitudes. For instance, in Eq.~21!, we have twoterms involving (Lp2L0) which can be written as

Lp2L05~1/k0!E0

zpAk22kr2 dz8.

The variable,Lz which characterizes the vertical componentof the acoustical path length, has units ofm. We will call Lzthe vertical path length in this paper. In addition, it will be-

come clear in the next section that the termV in Eq. ~22!may be regarded as the plane wave reflection coefficient.

It is perhaps well understood that there are repeated re-flection and refraction of some of the rays from the source tothe receiver as they propagate in a downward refracting me-dium at long distances. Mathematically speaking, this in-triguing physical phenomenon is due to the presence of theoscillatory term in the denominator of Eq.~21! @see also Eq.~12!#. Thus there may be many rays contribution to give atotal field at the receiver as the sound ‘‘channels’’ in themedium. Each separate ray will have a particular stationaryphase point in the complexkr plane.

To facilitate the analysis, we expand the oscillatory termof the denominator in Eq.~21! into the binomial series togive

$12V exp@2ik0~Lp2L0!2 ip/2#%21

5 (m50

`

Vm exp$2imk0~Lp2L0!2 imp/2%. ~25!

In view of the fact that the positive root is chosen forN0 andthe real part of the normalized admittanceb is always greaterthan zero for absorptive ground surfaces, it is straightforwardto show thatuVu<1. Consequently, the infinite sum in theright side of Eq.~25! converges for all realistic ground sur-faces.

If we now replace the Airy functions with their asymp-totic approximations, combine the sine and cosine functionsand use Eq.~25! to replace the oscillatory term in the de-nominator ofD, then Eq.~11! becomes

P̂5 P̂d1 P̂r , ~26!

where

P̂d5S i ~rs!21/2

2k0AN,N.D 3†exp$ ik0~L.2L,!%

1exp$ ik0@~Lp2L.!1~Lp2L,!#2 ip/2%‡, ~27a!

P̂r5S i ~rs!21/2

2k0AN,N.D (j51

`

Vj$~Glu! j1~Guu! j1~Gll ! j

1~Gul! j%, ~27b!

~Glu! j5exp$ ik0@~L.2L0!1~L,2L0!12~ j21!

3~Lp2L0!#2 i ~ j21!p/2%, ~28a!

~Guu! j5exp$ ik0@2~L.2L,!12 j ~Lp2L0!#2 i ~ jp!/2%,~28b!

~Gll ! j5exp$ ik0@~L.2L,!12 j ~Lp2L0!#2 i ~ jp!/2%,~28c!

~Gul! j5exp$ ik0@~Lp2L.!1~Lp2L,!12 j ~Lp2L0!#

2 i ~ j11!p/2%. ~28d!

In Eqs.~27a!, ~27b!, and~28a!–~28d!, the subscripts. and, represent the parameters evaluated atz5z. and z5z, ,respectively.

Equation ~26! is the asymptotic approximation of thetransform pressure whereP̂d andP̂r correspond to the direct



wave and reflected wave terms, respectively. We can invertthe Hankel transform ofP̂ according to Eq.~4! and thensubstitute it into Eq.~2! to obtain the acoustic pressure. Theinterpretation of each term in Eqs.~27a! and ~27b! will bedeferred until the next section.

B. Asymptotic representation of the total field andthe evaluation of direct waves

To evaluate the inverse transform integral, it is conve-nient to use a polar coordinate system.12 First, we introducethe index of refraction,n(z) of the medium, such that

k~z!5k0n~z! ~29!

and write the horizontal wave numberkr in its polar form,

kr5k~z!sin m5k0 sin m0 , ~30!

wherem is the polar angle measured from the verticalz axis.Then we can express the vertical wave numberkz and thevertical path length@see Eqs.~6!, ~23!, and~24!# as

kz5k0N~z! ~31!

and

Lz5Ez

N~z8!dz8, ~32!

where

N~z!5An22sin2 m0. ~33!

Equations~30! and ~31! are obtained by using Snell’s law,where

n sin m5sin m05constant, ~34!

to relate the polar angles at different heights. We remark thatthe anglem varies along the ray path with 0,m,p becausethe source and receiver are above thez50 plane. It is moreconvenient to choose an anglem0 as the identification ofdifferent rays. Obviously different rays have differentm0.The angle of a particular ray at its turning heightzp isp/2, soaccording to Snell’s law

n~zp!5sin m0 .

This ray ~with polar anglem0 at the ground! cannot reachthose points abovezp . We also specify that the source or thefield point may be regarded as lying on the ascending branchof a ray if p/2.m.0 and on the descending branch ifp.m.p/2; see Fig. 1. We can writeN(z), on the ascendingbranch, as

N~z!5An22sin2 m05n cosm, where p/2.m.0,

and, on the descending branch, as

N~z!5An22sin2 m052n cosm, where p.m.p/2.

We now turn our attention to the study of different termsappearing in Eq.~27a! and ~27b! for the direct waves andreflected waves. It is sufficient just to consider the directwave term as the reflected wave terms have a similar fea-tures. There are two terms for the direct waves@see Eq.~27a!#. We can express the difference in the vertical pathlength for the first term as

L.2L,5Ez,

z.An22sin2 m0 dz5Ez,

z.

n cosm dz,

because both the source and receiver are situated on the as-cending branch. In the second term of Eq.~27a!, the pointz,

is on the ascending branch whilez. is on the descendingbranch. Therefore, we can write

~Lp2L,!1~Lp2L.!5Ez,

z.

n cosm dz.

This is an identical form as the result above. Consequently,we can express the direct transformed pressure,P̂d in a morecompact form as

P̂d5(j51

2 S i ~rs!21/2

2k0AN,N.D

3expH i Fk0Ez,

z.

n cosm dz2xd, j G J , ~35!

wherexd,150 andxd,25p/2. The summation sign in Eq.~35!denotes the sum of the two possible rays connecting thesource and receiver; see Fig. 2. With this unified notation, itis important to remember that (L.2L,) is not necessarilyzero if z.5z, unless the horizontal ranger is zero. We notehere that the second term corresponds to the ray initiallylaunched upward but it will eventually bend downward at theturning point. We also emphasize that phase changes onlyoccur when the ray is in contact with a caustic.11,16The am-plitude of the sound pressure remains unchanged but thephase is retarded byp/2 as the ray grazes a caustic.

FIG. 2. The schematic diagram for the direct wave.~a! Both source andreceiver are situated on the ascending branch of a ray.~b! The source,z,,say, is situated at the ascending branch and the receiverz. is situated at thedescending branch of a ray.



Similarly, Eq. ~27b! suggests that the reflected waveconsists of four terms for eachm. Using the polar coordi-nates, we can rewrite Eq.~22! as

V5cosm02b

cosm01b. ~36!

It is clear from Eq.~36! thatV takes a more familiar form asthe plane-wave reflection coefficient.11 We are now in a po-sition to interpret the terms in Eq.~27b! that contribute to thesound field due to reflected waves. Using the same method asfor the direct wave above, we can recognize that the contri-butions from all possible combinations of the source/receivergeometry are given in Eq.~27b!. They are, respectively,Glu ,Guu , Gll , andGul , where the index ‘‘uu’’ ~upper, upper!means thatz, andz. are situated at the ascending branch ofthe ray, the index ‘‘lu ’’ ~lower, upper! for z, situated at thedescending branch whilez. is situated at the ascendingbranch of the ray, and so on~see Fig. 3!. The same notationis also used in Ref. 11.

It is straightforward to show that the transformed pres-sure for the reflected wave,P̂r can be written in a morecompact form as

P̂r5 (m51

`

(j51

4 S i ~rs!21/2

2k0AN,N.D 3Vm3exp$ i @ I m2x r , j #%,

~37!

where

I m5k0S E0

z,

1E0

z.

12~m21!E0

zpD n cosm dz, ~38!

xr , j are the phase shifts due to the ray in contact with caus-tics andm is the number of reflections from the ground. Wehave x1,m5(m21)p/2, x2,m5x3,m5mp/2, and x4,m

5(m11)p/2. There will only be a phase retardation ofp/2each time the ray passes through a caustic. There will be nochange in the phase at the turning point unless it is also acaustic.11,16 The details of the difference between a causticand a turning point are described elsewhere11 and will not berepeated here.

In Eq. ~37!, I m is introduced for the clarity of presenta-tion and the summations are to include all four possible raysthat hit the ground at least once, twice and so on. In the caseof a short range between the source and receiver such thatthere are no multiple ray paths for either waves, i.e.,m51,Eqs.~35! and ~37! can be reduced to the equations given inRef. 12.

Substituting Eqs.~4! and~26! into Eq. ~2! and using themethod as detailed in Ref. 12, we can express the total soundpressure as

p~r ,z!5k02Ar

4p E2p/21 i`

p/22 i`

P̂dH0~1!~k0r sin m0!

3cosm0 sin m0 dm0

1k02Ar

4p E2p/21 i`

p/22 i`

P̂rH0~1!~k0r sin m0!

3cosm0 sin m0 dm0 . ~39!

The integration path is shown in Fig. 4. Here,H0(1)(u) is the

Hankel function defined by

H0~1!~u!5

4eiu

p E0

`

~4iu2Y2!21/2 expS 2Y2

2 DdY.To simplify the presentation, the summations ofj ~i.e., allpossible rays! for the direct and reflected wave are dropped.It is understood that the total sound field is the sum of all

FIG. 3. The schematic diagram for the reflected wave that hits the groundmtimes wherem51,2,3,... .~a! Glu : z, at descending branch andz. at as-cending branch,~b! Guu : z, at ascending branch andz. at ascendingbranch,~c! Gul : z, at ascending branch andz. at descending branch, and~d! Gll : z, at descending branch andz. at descending branch. FIG. 4. The integration path for the inverse Fourier integral. The original

path of integration and the steepest descent path are shown in the diagram.



possible rays with due consideration of the phase shift whenthe rays pass through caustics. The symbolsxd andxr

(m) areused to represent the phase shift for the direct wave and forthe reflected wave withm reflections from the ground, re-spectively. Substitution of Eq.~35! into the first term of Eq.~39! leads to an expression for the direct wave. Similarly,using Eq.~37! and the second term of Eq.~39!, we obtain anexpression for the reflected wave. They can be written sepa-rately as

pd5ik04p S r

rsD 1/2E

2p/21 i`

p/22 i` 1

2F~m0!g~k0r sin m0!

3expH ik0Ez,

z. n cos~m2f!

cosfdz2 ixdJ dm0 ~40!

and

pr5ik04p S r

rsD 1/2(

m51

` E2p/21 i`

p/22 i` 1

2VmF~m0!g~k0r sin m0!

3expH ik0F E0

z,

1E0

z.

12~m21!E0

zpGEz,

z. n cos~m2u~m!!

cosu~m! dz2 ix r~m!J dm0 , ~41!

where

F~m0!5cosm0 sin m0

@~n.2 2sin2 m0!~n,

2 2sin2 m0!#1/4, ~42!

g~u!5H0~1!~u!exp~2 iu !, ~43a!

or g(u) in its asymptotic form as

g~u!'A2/ipu@11O~u21!#, ~43b!

f and u(m) are the polar angles for the direct and reflectedwaves, respectively. The superscript (m) represents the pa-rameters for the reflected waves that hit the groundm times.A close examination of the exponential function in Eq.~41!suggests that for a short source/receiver separation, there isonly a single term in the sum which has a stationary phasecontribution.12 However, for a longer source/receiver separa-tion, there are a finite number of terms which have stationaryphase contributions.

The next step is to evaluate the integral expressions ofEqs. ~40! and ~41! asymptotically. We will tackle the directwave first. The stationary point is determined by setting

]

]m0Ez,

z. n cos~m2f!

cosfdz50,

which implies that the saddle point is given bym5f. Em-ploying the method of steepest descent,12 one may expandthe integral in Eq.~40! asymptotically to yield

pd~r ,z!5S r

rsD 1/2 exp$ i @k0Rd82xd#%

4pRd, ~44!

where

Rd5F ~n, cosf,!~n. cosf.!S Ez,

z. dz

n cos3 f D3S E

z,

z. dz

n cosf D G1/2, ~45!

Rd85Ez,

z. ndz8

cosf. ~46!

We note that the direct wave, see Eq.~44!, is consistentwith that given by Ostashev4 and Li5 except that, here, weare concerned with a stationary medium but they consider amoving stratified medium. One may interpretRd as an ap-parent geometrical path length andRd8 as an acoustical pathlength in a stratified medium. It is also worth pointing outthat the apparent geometrical path lengthRd is different fromthat given by Ref. 12@see his Eq.~36!#. To be specific, theterm *z,

z.dz/(n cos3 f) in Eq. ~45! is replaced by

(1/cos2 f0)*z,z.n dz/cosf. The discrepancy is due to the ap-

proximation used in the evaluation of Eq.~40! wherecos~m2f! is replaced by cos~m02f0! in Ref. 12.

We end this section by comparing the phase and ampli-tude given in Eq.~44! with that obtained by the classical raytheory. It is obvious that the acoustical path length agreeswith that determined by the classical method~see, for ex-ample, Refs. 11 and 16!, except a phase shift ofp/2 for eachcaustic the ray has passed. The amplitudeB derived by thepresent method is

B5Ar/rs4pRd

, ~47!

with Rd given by Eq.~45!. On the other hand, the amplitudeof the sound wave can be determined by considering theintensity of the wave as detailed in Sec. 43.2 of Ref. 11. Theamplitude of the sound pressure can be derived to give

B5S 1

4p D F ~r/rs!n. sin f.

r ~n, cosf,!u]r /]f.uG1/2

. ~48!

Equations~47! and ~48! suggest the seemingly different ex-pressions for the amplitude of the sound pressure derived bythese two methods. However, using Snell’s law and notingthat

r5Ez,

z.

tanf dz,

we can verify the following identities:

]r

]f.5n. cosf.

sin f0Ez,

z.

sec2 f tanf dz, ~49a!

r

sin f05E

z,

z. dz

n cosf, ~49b!

Ez,

z. dz

n cos3 f5

1

sin f0Ez,

z.

tanf sec2 f dz. ~49c!

With the use of Eq.~49a!–~49c!, we can show that the am-plitude derived by the present approach@see Eq.~47!# isidentical to that given in Ref. 11@see Eq.~48!#.



C. Reflected waves and the total sound field

The reflected wave can be obtained by evaluating theintegral in Eq.~41! asymptotically. We start by consideringthe wave that hits the groundm times wherem51,2,3,... .The sound pressurepr

(m) is given by

pr~m!5

ik04p S r

rsD 1/2E

2p/21 i`

p/22 i` 1

2VmF~m0!g~k0r sin m0!

3expH ik0F E0

z,

1E0

z.

12~m21!E0

zpG3E

z,

z. n cos~m2u~m!!

cosu~m! dz2 ix r~m!J dm0 ~50!

and the term involving the reflection coefficient can be writ-ten as

Vm5122mb

cosm01b12m~m21!b2

~cosm01b!21••• . ~51!

Again the superscript (m) denotes the corresponding param-eters for themth bounce ray. In a hard boundary case whereb!1 we can ignore those terms of orderb2 and higher.Hence the contributionpr

(m) can be approximated12,21 to give

pr~m!5H 122miAp

b

b1cosu0~m! wme

2wm2erfc~2 iwm!J

3exp$ i @k0Rr8

~m!2x r~m!#%

4pRr~m! , ~52!

where

Rr~m!5U~n, cosu,

~m!!~n. cosu.~m!!

3S F E0

z,

1E0

z.

12~m21!E0

zpG dz

n cos3 u~m!D U1/23UF E

0

z,

1E0

z.

12~m21!E0

zpG dz

n cosu~m!U1/2, ~53!

Rr8~m!5F E

0

z.

1E0

z,

12~m21!E0

zpG n dz

cosu~m! , ~54!

wm2 5 1

2ik0Rr8~m!~b1cosu0

~m!!2. ~55!

Here, the phase anglexr(m) is used to account for the phase

shift due to themth bounce rays passing through caustics.One may regardRr

(m) andRr8(m) as the apparent geometrical

path length and the acoustical path length of the reflectedwave for the rays that havem reflections from the ground.

We mention here that the third term of Eq.~51! corre-sponds to a second-order pole singularity and subsequenthigher-order terms correspond to the higher-order poles.These multiple pole singularities can be evaluated accordingto the method described in Ref. 22. However, for smallb, itis found that the magnitude is proportional tob(bAk0) j21,where j is the order of the pole. Therefore we can ignoreterms due to these higher-order poles when compared withthat due to the first-order pole forb!1.

The total sound field can be presented more conve-niently by using a spherical wave reflection coefficientQm ,where

Qm5@Gm1~12Gm!F~wm!#m, ~56a!

Gm5cosu0

~m!2b

cosu0~m!1b

, ~56b!

and

F~wm!511 iApwme2wm

2erfc~2 iwm!. ~56c!

Here,Gm ,F(wm),wm are known as the plane-wave reflectioncoefficient, the boundary loss factor, and the numerical dis-tance, respectively. These three terms are used widely intheory of the outdoor sound propagation in a homogeneousatmosphere2 and can be extended to our present situation. Byrewriting the spherical wave reflection coefficient as

Qm5H 122iApb

b1cosu0~m! wme

2wm2erfc~2 iwm!J m

and expanding it in binomial series as

Qm5122miApb

b1cosu0~m! wme

2wm2erfc~2 iwm!

1O~b2!,

we can approximate the curly bracket of Eq.~52! by Qm .The approximation is accurate to the order ofb which isadequate for the hard boundary case whereb!1. Now wecan sum up the contribution of the reflected wave fromm51to ` and write the reflected wave in a more compact form as

pr5S r

rsD 1/2(

m51

` Qm exp$ i @k0Rr8~m!2x r

~m!#%

4pRr~m! . ~57!

The total sound field due to a point source can be calculatedby ignoring the small variation in the mean density and sum-ming the direct and reflected waves given in Eqs.~44! and~57! to give

p5exp$ i @k0Rd82xd#%

4pRd

1(m

Qm exp$ i @k0Rr8~m!2x r

~m!#%

4pRr~m! . ~58!

One can identify that Eq.~58! is in a form of the well-knownWeyl–Van der Pol formula. We stress that our analysis isrestricted to the propagation of sound in a downward refract-ing atmosphere. However there is no restriction on the hori-zontal range but the analysis will not be valid for regions inclose proximity to caustics. The region of caustics for thedirect wave can be found by settingRd50, whereRd is givenin Eq. ~45!. To find caustics for the reflected wave, we canuse Eq.~53! and setRr

(m)50. The total phase shift,xd andxr(m), can then be determined from the knowledge of thepositions of caustics. Hence we can use Eq.~58! to calculatethe total sound field.

The solution in its present form and its variations havebeen used by a number of researchers as a heuristic model to



predict sound propagation outdoors.17,23,24In a recent study,8

it has been demonstrated that the heuristic model performswell at long distances when compared with other numericalmethods on the Benchmark cases.25

III. CONCLUSIONS

This paper has been concerned with the development ofan asymptotic theory for sound propagation in a stratifiedatmosphere above a complex impedance ground. The sourceis assumed to be a point monopole of strength 1 Pa m and thesound-speed profile of the atmosphere is downward refract-ing. To derive the theory, a full wave equation is used as thestarting point in which an arbitrary sound-speed profile andthe impedance boundary condition are included. It has beenshown that the theory extends the Weyl–Van der Pol formulato give the sound field in a downward refracting medium.The interaction of the sound waves with a ground of finiteimpedance is essentially unaffected by the sound-speed pro-file of the medium except that the angle of incidence of theray has been modified due to this atmosphere inhomogeneity.The consequence is that the spherical wave reflection factorcan be calculated as if the atmosphere is homogeneous.

The asymptotic theory developed in this paper is similarto the heuristic model proposed by L’Esperenceet al. for thepredication of sound propagation in the downward refractingatmosphere. It is, perhaps, not surprising to find that, at longranges, their calculated results due to the heuristic modelagreed well with that due to the fast field program. It isbecause both methods are based on the full wave equation.The only difference is that the fast field program evaluatesthe inverse Fourier integral numerically while the presenttheory exploits the technique of approximating the integralasymptotically. Nevertheless the ray trace method providesan attractive alternative to other numerical schemes becauseit gives a better physical understanding of the problem.

ACKNOWLEDGMENTS

The author wishes to thank Henry Bass and James Sa-batier for encouragement and support. He would also like tothank Richard Raspet for discussions and useful commentson the draft of the manuscript. The work was supported inpart by EPSRC, UK.

1J. E. Piercy, T. F. W. Embleton, and L. C. Sutherland, ‘‘Review of noisepropagation in atmosphere,’’ J. Acoust. Soc. Am.61, 1403–1418~1977!.

2K. Attenborough, ‘‘Review of ground effects on outdoor sound propaga-

tion for continuous broadband source,’’ Appl. Acoust.25, 289–319~1988!.

3V. E. Ostashev, ‘‘Theory of sound propagation in an inhomogeneous mov-ing medium~Review!,’’ Izv. Atmos. Oceanic Phys.21, 274–285~1985!.

4V. E. Ostashev, ‘‘High-frequency acoustic field of a point source lyingabove an impedance surface in a stratified medium,’’ Izv. Atmos. OceanicPhys.23, 370–377~1987!.

5K. M. Li, ‘‘A high-frequency approximation of sound propagation in astratified moving atmosphere above a porous ground surface,’’ J. Acoust.Soc. Am.95, 1840–1852~1994!.

6M. J. Lighthill, Waves in Fluids~Cambridge U.P., Cambridge, 1978!.7R. Raspet, G. Baird, and W. Wu, ‘‘Normal mode solution for low-frequency sound propagation in a downward refracting atmosphere abovea complex impedance plane,’’ J. Acoust. Soc. Am.91, 1341–1352~1992!.

8R. Raspet, A. L’Esperance, and G. A. Daigle, ‘‘The effect of realisticground impedance on the accuracy of ray tracing,’’ J. Acoust. Soc. Am.97, 154–158~1995!.

9S. W. Lee, N. Bong, W. F. Richards, and R. Raspet, ‘‘Impedance formu-lation of the fast field program for acoustic wave propagation in the at-mosphere,’’ J. Acoust. Soc. Am.79, 628–634~1986!.

10Y. L. Li and M. J. White, ‘‘A note on using the fast field program,’’ J.Acoust. Soc. Am.95, 3100–3102~1994!.

11L. M. Brekhovskikh,Waves in Layered Media~Academic, London, 1980!.12K. M. Li, ‘‘On the validity of the heuristic ray-trace based modification tothe Weyl–Van der Pol formula,’’ J. Acoust. Soc. Am.93, 1727–1735~1993!.

13P. G. Bergmann, ‘‘The wave equation in a medium with a variable indexof refraction,’’ J. Acoust. Soc. Am.17, 329–333~1946!.

14D. S. Jones,Acoustic and Electromagnetic Waves~Claredon, Oxford,1980!, pp. 344–346.

15F. W. J. Olver,Asymptotic and Special Functions~Academic, New York,1974!.

16A. D. Pierce,Acoustics: An Introduction to Physical Principles and Ap-plications ~Acoust. Soc. Am., New York, 1989!.

17T. Hidaka, K. Kageyama, and S. Masuda, ‘‘Sound propagation in the restatmosphere with linear sound velocity profile,’’ J. Acoust. Soc. Jpn.6,117–125~1985!.

18R. Geiger,The Climate Near the Ground~Harvard U.P., Cambridge,1971!.

19M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions~Dover, New York, 1972!.

20K. B. Rasmussen, ‘‘Outdoor sound propagation under the influence ofwind and temperature gradients,’’ J. Sound Vib.104, 321–335~1986!.

21C. F. Chien and W. W. Soroka, ‘‘Sound propagation along an impedanceplane,’’ J. Sound Vib.43, 9–20~1975!.

22L. B. Felsen and N. Marcuvitz,Radiation and Scattering of Waves~Pren-tice Hall, Englewood Cliffs, NJ, 1973!.

23A. L’Esperance, P. Herzog, G. A. Daigle, and J. R. Nicolas, ‘‘Heuristicmodel for outdoor sound propagation based on an extension of the geo-metrical ray theory in the case of a linear sound speed profile,’’ Appl.Acoust.37, 111–139~1992!.

24K. M. Li, K. Attenborough, and N. W. Heap, ‘‘Source height determina-tion by ground effect inversion,’’ J. Sound Vib.145, 111–128~1991!.

25K. Attenborough, S. Taherzadeh, H. E. Bass, X. Di, R. Raspet, G. R.Daigle, A. L’Esperence, Y. Gabillet, K. E. Gilbert, Y. Li, M. J. White, P.Naz, J. M. Noble, and H. A. J. M. van Hoof, ‘‘Benchmark case for outdoorsound propagation models,’’ J. Acoust. Soc. Am.97, 173–191~1995!.



Documents

Propagation of sound above an impedance plane in a downward refracting atmosphere