Equations for finite-difference, time-domain simulationi

lor00

an

ab

e

ct

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rfor the sound pressure that have been previously used for analytical and numerical studies of soundpropagation in a moving atmosphere. Practical FDTD implementation of the second set of equationsis discussed. Results show good agreement with theoretical predictions of the sound pressure due toa point monochromatic source in a uniform, high Mach number flow and with Fast Field Programcalculations of sound propagation in a stratified moving atmosphere. 2005 Acoustical Society ofAmerica. @DOI: 10.1121/1.1841531#

PACS numbers: 43.20.Bi, 43.28.Js @MO# Pages: 503517

I. INTRODUCTION

Finite-difference, time-domain ~FDTD! techniques havedrawn substantial interest recently due to their ability toreadily handle complicated phenomena in outdoor soundpropagation such as scattering from buildings and trees, dy-namic turbulence fields, complex moving source distribu-tions, and propagation of transient signals.18 These phenom-ena are difficult to handle with frequencydomaintechniques that are currently widely used, such as parabolicequation approximations and the Fast Field Program ~FFP!.FDTD techniques typically solve coupled sets of partial dif-ferential equations that are first order in time. In this regard,they are a departure from methodologies such as the para-bolic approximation, which solve a single equation for thesound pressure that is second order in time. Many suchsingle equations for the sound pressure in a moving inhomo-geneous medium are known in the literature ~see Refs. 914

and references therein!. Although these equations were ob-tained with different assumptions and/or approximations, allcontain second- or higher-order derivatives of the soundpressure with respect to time, and are therefore not amenableto first-order FDTD techniques. Our main goal in the presentpaper is to derive equation sets that are appropriate as start-ing equations in FDTD simulations of sound propagation in amoving inhomogeneous atmosphere and to study the rangeof applicability of these sets.

The most general possible approach to sound propaga-tion in a moving inhomogeneous medium would be based ona direct solution of the complete set of linearized equationsof fluid dynamics,911,15 which are first-order partial differ-ential equations. Although this set could be used as startingequations for FDTD codes, even with modern computers it istoo involved to be practical. Furthermore, this set containsthe ambient pressure and entropy, which are not usually con-sidered in studies of sound propagation in the atmosphere.Therefore, it is worthwhile to find simplified equation setsfor use in FDTD calculations.

In the present paper, the complete set of linearized equa-tions of fluid dynamics in a moving inhomogeneous mediumis reduced to two simpler sets that are first order in time and

a!Portions of this work were presented in V. E. Ostashev, L. Liu, D. K.Wilson, M. L. Moran, D. F. Aldridge, and D. Marlin, Starting equationsfor direct numerical simulation of sound propagation in the atmosphere,Proceedings of the 10th International Symposium on Long Range SoundPropagation, Grenoble, France, Sept. 2002, pp. 7381.

503J. Acoust. Soc. Am. 117 (2), February 2005 0001-4966/2005/117(2)/503/15/$22.50 2005 Acoustical Society of Americaof sound propagation in movingand numerical implementationa)

Vladimir E. OstashevNOAA/Environmental Technology Laboratory, Boulder, CoNew Mexico State University, Las Cruces, New Mexico 88

D. Keith Wilson and Lanbo LiuU.S. Army Engineer Research and Development Center, H

David F. Aldridge and Neill P. SymonsDepartment of Geophysical Technology, Sandia National LDavid MarlinU.S. Army Research Laboratory, White Sands Missile Rang

~Received 24 February 2004; revision received 14 O

Finite-difference, time-domain ~FDTD! calculations aequations that are first order in time. Equation sets apinhomogeneous medium ~with an emphasis on the apaper. Two candidate equation sets, both derived froproposed. The first, which contains three coupled equvelocity, and acoustic density, is obtained without anytwo coupled equations for the sound pressure and veterms proportional to the divergence of the mediumpressure. It is shown that the second set has the same onhomogeneous media

ado 80305, and Department of Physics,3

over, New Hampshire 03755

s., Albuquerque, New Mexico 87185

, New Mexico 88002

ober 2004; accepted 5 November 2004!

typically performed with partial differentialropriate for FDTD calculations in a movingosphere! are derived and discussed in thislinearized equations of fluid dynamics, areions for the sound pressure, vector acousticpproximations. The second, which containsor acoustic velocity, is derived by ignoringvelocity and the gradient of the ambient

a wider range of applicability than equations

amenable to FDTD implementation. The first set contains ]

three coupled equations involving the sound pressure, vectoracoustic ~particle! velocity, and acoustic density. No approxi-mations are made in deriving this set. The second set con-tains two coupled equations for the sound pressure and vec-tor acoustic velocity. Although the second set describessound propagation only approximately, the assumptions in-volved in deriving the second set are quite reasonable inatmospheric acoustics: Terms proportional to the divergenceof the medium velocity and the gradient of the ambient at-mospheric pressure are ignored. To better understand therange of applicability of the second set, we compare the setwith equations for the sound pressure that have been previ-ously used in analytical and numerical studies of soundpropagation in a moving atmosphere. It is shown that thesecond set has the same or a wider range of applicability thanthese equations for the sound pressure.

Furthermore in the present paper, a basic numerical al-gorithm for solving the second set of equations in two-dimensional ~2-D! moving inhomogeneous media is devel-oped. Issues related to the finite-difference approximation ofthe spatial and temporal derivatives are discussed. FDTDsolutions are obtained for a homogenous uniformly movingmedium and for a stratified moving atmosphere. The first ofthese solutions is compared with an analytical formula forthe sound pressure due to a point monochromatic source in auniformly moving medium. The second solution is comparedwith predictions from before FFP.

Although the explicit emphasis of the discussion in thispaper is on sound propagation in a moving inhomogeneousatmosphere, most of the derived equations are also valid fora general case of sound propagation in a moving inhomoge-neous medium with an arbitrary equation of state, e.g., in theocean with currents. Equations presented in the paper arealso compared with those known in aeroacoustics.

The paper is organized as follows. In Sec. II, we con-sider the complete set of equations of fluid dynamics andtheir linearization. In Sec. III, the linearized equations arereduced to the set of three coupled equations for the soundpressure, acoustic velocity, and acoustic density. In Sec. IV,we consider the set of two coupled equations for the acousticpressure and acoustic velocity. Numerical implementation ofthis set is considered in Sec. V.

II. EQUATIONS OF FLUID DYNAMICS AND THEIRLINEARIZATION

Let P (R,t) be the pressure, % (R,t) the density, v(R,t)the velocity vector, and S (R,t) the entropy in a medium.Here, R5(x ,y ,z) are the Cartesian coordinates, and t istime. These functions satisfy a complete set of fluid dynamicequations ~e.g. Ref. 16!:

S ]]t 1 v" D v1 P% 2g5F/% , ~1!S ]]t 1 v" D% 1%"v5% Q , ~2!504 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005S ]t 1 v" D S50, ~3!P 5P ~% ,S !. ~4!

In Eqs. ~1!~4!, 5(]/]x ,]/]y ,]/]z), g5(0,0,g) is the ac-celeration due to gravity, and F and Q characterize a forceacting on the medium and a mass source, respectively. Forsimplicity, we do not consider the case when a passive com-ponent is dissolved in a medium ~e.g., water vapor in the dryair, or salt in water!. This case is considered in detailelsewhere.9,17

If a sound wave propagates in a medium, in Eqs. ~1!~4!P , % , v, and S can be expressed in the following form: P5P1p , %5%1h , v5v1w, and S5S1s . Here, P , %, v,and S are the ambient values ~i.e., the values in the absenceof a sound wave! of the pressure, density, medium velocity,and entropy in a medium, and p , h, w, and s are their fluc-tuations due to a propagating sound wave. In order to obtainequations for a sound wave, Eqs. ~1!~4! are linearized withrespect to p , h, w, and s . Assuming that a sound wave isgenerated by the mass source Q and/or the force F and in-troducing the full derivative with respect to time d/dt5]/]t1v" , we have

dwdt 1~w" !v1

p%

2hP%2

5F/% , ~5!

dhdt 1~w" !%1%"w1h"v5%Q , ~6!

dsdt 1~w" !S50, ~7!

p5hc21hs . ~8!

Here, c5A]P(% ,S)/]% is the adiabatic sound speed, andthe parameter h is given by h5]P(% ,S)/]S . The set of Eqs.~5!~8! provides a most general description of sound propa-gation in a moving inhomogeneous medium with only onecomponent. In order to calculate p , h, w, and s , one needs toknow the ambient quantities c , %, v, P , S , and h . Note thatEqs. ~5!~8! describe the propagation of both acoustic andinternal gravity waves, as well as vorticity and entropywaves ~e.g., Ref. 18!.

Equations ~5!~8! were derived for the first time byBlokhintzev in 1946.17 Since then, these equations have beenwidely used in studies of sound propagation ~e.g., Refs.911!. In the general case of a moving inhomogeneous me-dium, Eqs. ~5!~8! cannot be exactly reduced to a singleequation for the sound pressure p . In the literature, Eqs.~5!~8! have been reduced to equations for p , making use ofdifferent approximations or assumptions about the ambientmedium. These equations for p were subsequently used foranalytical and numerical studies of sound propagation. Theyare discussed in Sec. IV. Note that the equations for p knownin the literature contain the following ambient quantities: c ,%, and v. On the other hand, the linearized equations of fluiddynamics, Eqs. ~5!~8!, contain not only c , %, and v, butOstashev et al.: Moving media finite difference time domain equations

also P , S , and h . This fact indicates that the effect of P , S , ]P ~% ,S ! ]

and h on sound propagation is probably small for most ofproblems considered so far in the literature.

The effect of medium motion on sound propagation isalso studied in aeroacoustics, e.g., see Refs. 12, 1924 andreferences therein. In aeroacoustics, the starting equationscoincide with Eqs. ~1!~4! but might also include terms de-scribing viscosity and thermal conductivity in a medium. Us-ing these equations of fluid dynamics, equations for soundwaves are derived which have some similarities with Eqs.~5!~8!. For example, Eqs. ~5!~8! are equivalent to Eqs.~1.11! from Ref. 12, and Eq. ~6! can be found in Refs. 19, 22,23. The main difference between Eqs. ~5!~8! and those inaeroacoustics are sound sources. In atmospheric acoustics, inEqs. ~5!~8! the sources F and Q are assumed to be knownand are loudspeakers, car engines, etc. In aeroacoustics, thesesources have to be calculated and are those due to ambientflow. Furthermore in some formulations in aeroacoustics, theleft-hand side of Eq. ~5! contains nonlinear terms.2123 Notethat FDTD calculations are nowadays widely used in aeroa-coustics, e.g., Refs. 19, 20, 24.

Also note that in aeroacoustics it is sometimes assumedthat the ambient medium is incompressible and/or isentropic,i.e., S5const. Generally, these assumptions are inappropriatefor atmospheric acoustics. Indeed, sound waves can be sig-nificantly scattered by density fluctuations, e.g., see Sec.6.1.4 from Ref. 9. Furthermore, in a stratified atmosphere Sdepends on the height above the ground. The range of appli-cability of the assumption S5const ~which is equivalent tos50 or p5c2h) is studied in Sec. 2.2.4 from Ref. 9. For astratified medium, this assumption is not applicable if thescale of the ambient density variations is smaller than thesound wave length or if the ambient density noticeablychanges with height.

III. SET OF THREE COUPLED EQUATIONSA. Moving medium with an arbitrary equation of state

Applying the operator (]/]t1 v") to both sides of Eq.~4! and using Eq. ~3!, we have

S ]]t 1 v" D P 5 c2S ]]t 1 v" D% , ~9!where c25]P (% ,S )/]% differs from the square of the adia-batic sound speed c25]P(% ,S)/]% . Using Eq. ~2!, Eq. ~9!can be written as

S ]]t 1 v" D P 1 c2%"v5 c2% Q . ~10!The next step is to linearize Eq. ~10! to obtain an equationfor acoustic quantities. To do so we need to calculate thevalue of c25]P (% ,S )/]% to the first order in acoustic per-turbations. In this formula, we express % and S as the sums% 5%1h and S5S1s , decompose the function P into Tay-lor series, and keep the terms of the first order in h and s:J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashec25]%

5]%

P~%1h ,S1s !

5]

]% F P~% ,S !1 ]P~% ,S !]% h1 ]P~% ,S !]S sG5

]P~% ,S !]%

1]2P~% ,S !

]%2h1

]2P~% ,S !]%]S s . ~11!

The first term in the last line of this equation is equal to c2.Denoting b5]2P(% ,S)/]%2 and a5]2P(% ,S)/]%]S , wehave c25c21bh1as5c21(c2)8. Here, (c2)85bh1asare fluctuations in the squared sound speed due to a propa-gating sound wave. In this formula, s can be replaced by itsvalue from Eq. ~8!: s5(p2c2h)/h . As a result, we obtainthe desired formula for fluctuations in the squared soundspeed: (c2)85(b2ac2/h)h1ap/h .

Now we can linearize Eq. ~10!. In this equation, weexpress P , % , v, and c2 as the sums: P 5P1p , % 5%1h ,v5v1w, and c25c21(c2)8. Linearizing the resulting equa-tion with respect to acoustic quantities, we have

dpdt 1%c

2"w1w"P1c2h1%~c2!8"v5%c2Q .~12!

In this equation, (c2)8 is replaced by its value obtainedabove. As a result, we arrive at the following equation fordp/dt:

dpdt 1%c

2"w1w"P1$@%b1c2~12a%/h !#h1~a%/h !p%"v5%c2Q . ~13!

Equations ~5!, ~6!, and ~13! comprise a desired set ofthree coupled equations for p , w, and h. This set was ob-tained from linearized equations of fluid dynamics, Eqs. ~5!~8!, without any approximations. The set can be used as start-ing equations for FDTD simulations. In this set, one needs toknow the following ambient quantities: c , %, v, P , a, b, andh .

B. Set of three equations for an ideal gasIn most applications, the atmosphere can be considered

as an ideal gas. In this case, the equation of state reads ~e.g.,Refs. 9, 17! as

P5P0~%/%0!g exp@~g21 !~S2S0!/Ra# , ~14!

where g51.4 is the ratio of specific heats at constant pres-sure and constant volume, Ra is the gas constant for the air,and the subscript 0 indicates reference values of P , %, and S .Using Eq. ~14!, the sound speed c and the coefficients a, b,and h appearing in Eq. ~13! can be calculated: c25gP/% ,a5g(g21)P/(%Ra), b5g(g21)P/%2, and h5(g21)P/Ra . Substituting these values into Eq. ~13!, we have

dpdt 1%c

2"w1w"P1gp"v5%c2Q . ~15!A set of Eqs. ~5!, ~6!, and ~15! is a closed set of three

coupled equations for p , w, and h for the case of an ideal505v et al.: Moving media finite difference time domain equations

gas. To solve these equations, one needs to know the follow- Equations ~17! and ~18! were derived in Ref. 25 @see

ing ambient quantities: c , %, v, and P .

Let us compare Eqs. ~5!, ~6!, and ~15! with a closed setof equations for p and w from Ref. 1; see Eqs. ~12! and ~13!from that reference. The latter set was used in Refs. 1, 2 asstarting equations for FDTD simulations of outdoor soundpropagation. If Q50, Eq. ~15! in the present paper is essen-tially the same as Eq. ~13! from Ref. 1. @Note that Eq. ~15! isalso used in aeroacoustics, e.g., Ref. 19.# Furthermore for thecase of a nonabsorbing medium, Eq. ~12! from Ref. 1 isgiven by

dwdt 1~w" !v1

p%

2pPgP% 50. ~16!

Let us show that this equation is an approximate version ofEq. ~5! in the present paper. Indeed, in Eq. ~5! we replace hby its value from Eq. ~8!: h5(p2hs)/c2, and assume thats50. If F50, the resulting equation coincides with Eq. ~16!.Thus, for an ideal gas and F50 and Q50, Eqs. ~12! and~13! from Ref. 1 are equivalent to Eqs. ~5! and ~15! in thepresent paper if s can be set to 0. The range of applicabilityof the approximation s50 is considered above.

IV. SET OF TWO COUPLED EQUATIONSA. Set of equations for p and w

In atmospheric acoustics, Eqs. ~5! and ~13! can be sim-plified since v is always much less than c . First, using Ref.16, it can be shown that "v;v3/(c2L), where L is thelength scale of variations in the density %. Therefore, in Eq.~13! the term proportional to "v can be ignored to orderv2/c2. Second, in Eqs. ~5! and ~13! the terms proportional toP can also be ignored. Indeed, in a moving inhomoge-neous atmosphere P is of the order v2/c2 so that theseterms can be ignored to order v/c . Furthermore, in a strati-fied atmosphere, P52g% , where g is the acceleration dueto gravity. It is known that, in linearized equations of fluiddynamics, terms proportional to g are important for internalgravity waves and can be omitted for acoustic waves.

With these approximations, Eqs. ~13! and ~5! become

S ]]t 1v" D p1%c2"w5%c2Q , ~17!S ]]t 1v" Dw1~w" !v1 p% 5F/% . ~18!

Equations ~17! and ~18! comprise the desired closed set oftwo coupled equations for p and w. This set can also be usedin FDTD simulations of sound propagation in the atmo-sphere. In order to solve this set, one needs to know thefollowing ambient quantities: c , %, and v. These ambientquantities appear in equations for the sound pressure p thathave been most often used for analytical and numerical stud-ies of sound propagation in moving media. The set of Eqs.~17! and ~18! is simpler than the set of three coupled equa-tions, Eqs. ~5!, ~6!, and ~13!, and does not contain the ambi-ent quantities P , a, b, and h . It can be shown that Eqs. ~17!and ~18! describe the propagation of acoustic and vorticitywaves but do not describe entropy or internal gravity waves.506 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005also Eqs. ~2.68! and ~2.69! from Ref. 9# using a differentapproach. In these references, Eqs. ~17! and ~18! were de-rived for the case of a moving inhomogeneous medium withmore than one component ~e.g., humid air or salt water!.Equations ~17! and ~18! are somewhat similar to the startingequations in FDTD simulations used in Ref. 3; see Eqs. ~10!and ~12! from that reference. The last of these equationscoincides with Eq. ~17! while the first is given by

]w

]t2w~v!1 p

%01@w"v#50. ~19!

Using vector algebra, the left-hand side of this equation canbe written as a left-hand side of Eq. ~18! plus an extra termv(w). Equations ~10! and ~12! from Ref. 3 were ob-tained using several assumptions that were not employed inthe present paper when deriving Eqs. ~17! and ~18!: ]v/]t5]%/]t5%50, c is constant, and ]w/]t@v(w). Itfollows from the last inequality that the extra termv(w) in Eq. ~19! can actually be omitted. Note that inRef. 4 different starting equations were used in simulationsof sound propagation in a muffler with a low Mach numberflow. The use of Eq. ~19! resulted in increase of stability insuch simulations.

Also note that equations for p and w similar to Eqs. ~17!and ~18! are used in aeroacoustics, e.g. Refs. 20, 24. Theleft-hand sides of Eqs. ~7! in Ref. 20 contain several extraterms in comparison with the left-hand sides of Eqs. ~17! and~18! which, however, vanish if P50 and "v50. The left-hand sides of Eqs. ~75! and ~76! in Ref. 24 also contain extraterms in comparison with the left-hand sides of Eqs. ~17! and~18!, e.g., terms proportional to the gradients of c and %. Theright-hand sides of the equations in Refs. 20, 24 describeaeroacoustic sources and differ from those in Eqs. ~17! and~18!.

At the beginning of this section, we provided sufficientconditions for the applicability of Eqs. ~17! and ~18!. Actu-ally, the range of applicability of these equations can bemuch wider. Note that it is quite difficult to estimate withwhat accuracy one can ignore certain terms in differentialequations. We will study the range of applicability of Eqs.~17! and ~18! by comparing them with equations for thesound pressure p presented in Secs. IV BIV F, which havebeen most often used for analytical and numerical studies ofsound propagation in moving media and whose ranges ofapplicability are well known. This will allow us to show thatEqs. ~17! and ~18! have the same of a wider range of appli-cability than these equations for p and, in many cases, de-scribe sound propagation to any order in v/c . For simplicity,in the rest of this section, we assume that F50, Q50, andthe medium velocity is subsonic.

B. Nonmoving mediumConsider the case of a nonmoving medium when v50.

In this case, the set of linearized equations of fluid dynamics,Eqs. ~5!~8!, can be exactly ~without any approximations!reduced to a single equation for sound pressure p ~e.g., seeEq. ~1.11! from Ref. 11!:Ostashev et al.: Moving media finite difference time domain equations

] 1 ]p p

]t S %c2 ]t D2S % D50. ~20!For the considered case of a nonmoving medium, Eqs.

~17! and ~18! can also be reduced to a single equation for p .This equation coincides with Eq. ~20!. Therefore, Eqs. ~17!and ~18! describe sound propagation exactly if v50.

C. Homogeneous uniformly moving mediumA medium is homogeneous and uniformly moving if the

ambient quantities c , v, etc. do not depend on R and t . Forsuch a medium, the linearized equations of fluid dynamics,Eqs. ~5!~8! can also be exactly reduced to a single equationfor p ~see Sec. 2.3.6 from Ref. 9 and references therein!:

S ]]t 1v" D2

p2c22p50. ~21!

For the case of a homogeneous uniformly moving me-dium, Eqs. ~17! and ~18! can be reduced to the equation forp that coincides with Eq. ~21!. Therefore, Eqs. ~17! and ~18!describe sound propagation exactly in a homogeneous uni-formly moving medium. In particular, they correctly accountfor terms of any order in v/c .

D. Stratified moving mediumNow let us consider the case of a stratified medium

when the ambient quantities c , %, v, etc. depend only on thevertical coordinate z . We will assume that the vertical com-ponent of v is zero: v5(v,0), where v is a horizontalcomponent of the medium velocity vector. In this subsection,we reduce Eqs. ~17! and ~18! to a single equation for thespectral density of the sound pressure and show that thisequation coincides with the equation for the spectral densitythat can be derived from Eqs. ~5!~8!.

For a stratified moving medium, Eq. ~17! can be writtenas

S ]]t 1vD p1%c2Sw1 ]wz]z D50. ~22!Here, 5(]/]x ,]/]y), and w and wz are the horizontaland vertical components of the vector w5(w ,wz). Equa-tion ~18! can be written as two equations:

S ]]t 1vDwz1 1% ]p]z 50, ~23!S ]]t 1vDw1wzv8 1 p% 50. ~24!

Here, v8 5dv /dz . Let p , w , and wz be expressed as Fou-rier integrals:

p~r,z ,t !5E E daE dv exp~ ia"r2ivt ! p~a,z ,v!,~25!

wz~r,z ,t !5E E daE dv exp~ ia"r2ivt !wz~a,z ,v!,~26!J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashew~r,z ,t !5E E daE dv exp~ ia"r2ivt !w~a,z ,v!.~27!

Here, r5(x ,y) are the horizontal coordinates, a is the hori-zontal component of the wave vector, v is the frequency of asound wave, and p , wz , and w are the spectral densities ofp , wz , and w . We substitute Eqs. ~25!~27! into Eqs. ~22!~24!. As a result, we obtain a set of equations for p , wz , andw :

2i~v2a"v! p1i%c2a"w1%c2]wz]z

50, ~28!

2i~v2a"v!wz11%

] p]z

50, ~29!

2i~v2a"v!w1v8 wz1iap%

50. ~30!

After some algebra, this set of equations can be reduced to asingle equation for p:

]2 p]z2

1S 2a"v8v2a"v 2 %8% D ] p]z 1S ~v2a"v!2

c22a2D p50,

~31!where %85d%/dz .

For the considered case of a stratified moving medium, asingle equation for p can also be derived from Eqs. ~5!~8!without any approximations. This equation for p is given byEq. ~2.61! from Ref. 9. Setting g50 in this equation ~i.e.ignoring internal gravity waves! one obtains Eq. ~31!. There-fore, Eqs. ~17! and ~18! describe sound propagation exactlyin a stratified moving medium, and, hence, correctly accountfor terms of any order in v/c .

E. Turbulent medium

Probably the most general of the equations describingthe propagation of a monochromatic sound wave in turbulentmedia with temperature and velocity fluctuations is given byEq. ~6.1! from Ref. 9:

FD1k02~11!2S ln %%0D 2 2iv ]v i]x j ]2

]xi]x j

12ik0c0

v"Gp~R!50. ~32!Here, D5]2/]x21]2/]y21]2/]z2; 5c0

2/c221; k0 , c0 ,and %0 are the reference values of the wave number, adia-batic sound speed, and density; x1 , x2 , x3 stand for x , y , z;v15vx , v25vy , v35vz are the components of the mediumvelocity vector v; and repeated subscripts are summed from1 to 3. Furthermore, the dependence of the sound pressure onthe time factor exp(2ivt) is omitted.

The range of applicability of Eq. ~32! is considered indetail in Sec. 2.3 from Ref. 9. This equation was used forcalculations of the sound scattering cross section per unitvolume of a sound wave propagating in a turbulent mediumwith temperature and velocity fluctuations. Also it was em-ployed as a starting equation for developing a theory of mul-507v et al.: Moving media finite difference time domain equations

tiple scattering of a sound wave propagating in such a turbu- dQ

lent medium; see Ref. 9 and references therein. Furthermore,starting from Eq. ~32!, parabolic and wide-angle parabolicequations were derived and used in analytical and numericalstudies of sound propagation in a turbulent medium, e.g.,Ref. 26. For example, a parabolic equation deduced from Eq.~32! reads as

2ik0]p]x

1Dp12k02S 11 mov2 D p50. ~33!

Here, the predominant direction of sound propagation coin-cides with the x-axis, D5(]2/]y2,]2/]z2), and mov522vx /c0 .

In Ref. 9, Eq. ~32! was derived starting from the set ofEqs. ~17! and ~18! and using some approximations. There-fore, this set has the same or a wider range of applicabilitythan equations for p that have been used in the literature foranalytical and numerical studies of sound propagation in aturbulent medium with temperature and velocity fluctuations.

F. Geometrical acousticsSound propagation in a moving inhomogeneous medium

is often described in geometrical acoustics approximationwhich is applicable if the sound wavelength is much smallerthan the scale of medium inhomogeneities. In geometricalacoustics, the phase of a sound wave can be obtained as asolution of the eikonal equation, and its amplitude from thetransport equation. In this subsection, starting from Eqs. ~17!and ~18!, we derive eikonal and transport equations andshow that they are in agreement with those deduced fromEqs. ~5!~8!.

Let us express p and w in the following form:

p~R,t !5expik0Q~R,t !pA~R,t !, ~34!w~R,t !5expik0Q~R,t !wA~R,t !. ~35!

Here, Q(R,t) is the phase function, and pA and wA are theamplitudes of p and w. Substituting Eqs. ~34! and ~35! intoEqs. ~17! and ~18!, we have

ik0S %c2wAQ1pA dQdt D52 dpAdt 2%c2"wA , ~36!ik0S wA dQdt 1 pAQ% D52 dwAdt 2~wA !v2 pA% .

~37!

In geometrical acoustics, pA and wA are expressed as a seriesin a small parameter proportional to 1/k0 :

pA5p11p2ik0

1p3

~ ik0!21 . . . , ~38!

wA5w11w2

ik01

w3

~ ik0!21 . . . . ~39!

Substituting Eqs. ~38! and ~39! into Eqs. ~36! and ~37! andequating terms proportional to k0 , we arrive at a set of equa-tions:508 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005%c2w1Q1p1 dt 50, ~40!

w1dQdt 1p1

Q%

50. ~41!

Equating terms proportional to k00, we obtain another set:

%c2w2Q1p2 dQdt 52dp1dt 2%c

2"w1 , ~42!

w2dQdt 1

p2Q%

52dw1dt 2~w1 !v2

p1%

. ~43!

From Eq. ~41!, we have

w152p1%

QdQ/dt . ~44!

Substituting this value of w1 into Eq. ~40!, we obtain

F S dQdt D2

2c2~Q!2Gp150. ~45!From this equation, we obtain an eikonal equation for thephase function:

dQdt 52cuQu. ~46!

Here, a sign in front of uQu is chosen in accordance with thetime convention exp(2ivt). Equation ~46! coincides exactlywith the eikonal equation for sound waves in a moving in-homogeneous medium ~e.g., see Eq. ~3.15! from Ref. 9!which can be derived from Eqs. ~5!~8! in a geometricalacoustics approximation. Thus, in this approximation, Eqs.~17! and ~18! exactly describe the phase of a sound waveand, hence, account for terms of any order in v/c .

Substituting the value of dQ/dt from Eq. ~46! into Eq.~44!, we have

w15p1%

QcuQu 5

p1n%c

, ~47!

where n5 Q/uQu is the unit vector normal to the phasefront. Now we multiply Eq. ~42! by dQ/dt and multiply Eq.~43! by c2%Q . Then, we subtract the latter equation fromthe former. After some algebra and using Eq. ~46!, it can beshown that the sum of all terms proportional to p2 and w2 iszero. The resulting equation reads as

dp1dt 1cn"p11%cn

dw1dt 1%cn~w1 !v

1%c2"w150. ~48!In this equation, w1 is replaced by its value given by Eq.~47!. As a result, we obtain

%n

c ddt S np1%c D1 1c2 dp1dt 1 n"p1c 1%S np1%c D

1p1n~n" !v

c250. ~49!Ostashev et al.: Moving media finite difference time domain equations

In geometrical acoustics, the amplitude pA of the sound pres-

sure is approximated by p1 . Equation ~49! is a closed equa-tion for p1 ; i.e., it is a transport equation.

The second term on the left-hand side of Eq. ~49! can bewritten as

1c2

dp1dt 5

ddt S p1c2 D1 p1c4 dc

2

dt 5ddt S p1c2 D1 p1c4 bd%dt . ~50!

Here, we used the formula dc2/dt5bd%/dt; see Eq. ~2.63!from Ref. 9. According to Eq. ~2!, d%/dt in Eq. ~50! can bereplaced with 2%"v. When deriving Eqs. ~17! and ~18!,terms proportional to "v were ignored. Therefore, the lastterm on the right-hand side of Eq. ~50! should also be ig-nored. In this case, Eq. ~49! can be written as

%n

c ddt S np1%c D1 ddt S p1c2 D1 n"p1c 1%S np1%c D1

p1n~n" !vc2

50. ~51!

This equation coincides with Eq. ~3.18! from Ref. 9 if in thelatter equation terms proportional to "v are ignored. Equa-tion ~3.18! is an exact transport equation for p1 in the geo-metrical acoustics derived from Eqs. ~5!~8!. Thus, if theterms proportional to "v are ignored, Eqs. ~17! and ~18!exactly describe the amplitude of a sound wave in a geo-metrical acoustics approximation, and correctly account forterms of any order in v/c . Note that in Ref. 9 starting fromthe transport equation, Eq. ~3.18!, a law of acoustic energyconservation in geometrical acoustics of moving media isderived; see Eq. ~3.21! from that reference. Since Eq. ~51!coincides with Eq. ~3.18!, the same law @i.e., Eq. ~3.21! fromRef. 9# can be derived from Eq. ~51! provided that the termsproportional to "v are ignored.G. Discussion

Thus, by comparing a set of Eqs. ~17! and ~18! with theequations for p which are widely used in atmospheric acous-tics, we determined that this set has the same or a widerrange of applicability than these equations for p . Note thatthere are other equations for p known in the literature ~seeRefs. 9, 11, 17 and references therein!: Monins equation,Pierces equations, equation for the velocity quasi-potential,the AndreevRusakovBlokhintzev equation, etc. Most ofthese equations have narrower ranges of applicability thanthe equations presented above and have been seldom usedfor calculations of p .

V. NUMERICAL IMPLEMENTATION

In this section, we describe simple algorithms for FDTDsolutions of Eqs. ~17! and ~18! in the two spatial dimensionsx and y . Isolating the partial derivatives with respect to timeon the left side of these equations, we have

]p]t

52S vx ]]x 1vy ]]y D p2kS ]wx]x 1 ]wy]y D1kQ , ~52!

J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashe]wx]t

52S wx ]]x 1wy ]]y D vx2S vx ]]x 1vy ]]y Dwx2b

]p]x

1bFx , ~53!

]wy]t

52S wx ]]x 1wy ]]y D vy2S vx ]]x 1vy ]]y Dwy2b

]p]y 1bFy , ~54!

where b51/r is the mass buoyancy and k5rc2 is the adia-batic bulk modulus. In Eqs. ~52!~54!, the subscripts x and yindicate components along the corresponding coordinateaxes.

The primary numerical issues pertinent to solving theseequations in a moving inhomogeneous medium are summa-rized and addressed in Secs. V AV C. Example calculationsare provided in Secs. V D and V E.

A. Spatial finite-difference approximationsThe spatial finite-difference ~FD! network considered

here stores the pressure and particle velocities on a grid thatis staggered in space, as shown in Fig. 1. The pressure isstored at integer node positions, namely x5i Dx and y5 j Dy , where i and j are integers and Dx and Dy are thegrid intervals in the x- and y-directions. The x-componentsof the acoustic velocity, wx , are staggered ~offset! by Dx/2in the x-direction. The y-components of the acoustic veloc-ity, wy , are staggered by Dy /2 in the y-direction. This stag-gered grid design is widely used for wave propagation cal-culations in nonmoving media.2730 Here we furthermorestore vx and Fx at the wx nodes, and vy and Fy at the wynodes. The quantities b , k, and Q are stored at the pressurenodes.

For simplicity, we consider in this article only a second-order accurate, spatially centered FD scheme. A centered so-lution of Eqs. ~52!~54! requires an evaluation of each of theterms of the right-hand sides of these equations at the gridnodes where the field variable on the left-hand side is stored.One of the main motivations for using the spatially staggered

FIG. 1. Spatially staggered finite-difference grid used for the calculations inthis article.509v et al.: Moving media finite difference time domain equations

grid is that it conveniently provides compact, centered spatial ]p~ i Dx , j Dy ,t !

differences for many of the derivatives in Eqs. ~52!~54!.For example, ]wx /]x in Eq. ~52! is

]wx~ i Dx , j Dy ,t !/]x.$wx@~ i11/2!Dx , j Dy ,t#2wx@~ i21/2!Dx , j Dy ,t#%/Dx ~55!

and ]p/]y in Eq. ~54! is

]p@ i Dx ,~ j11/2!Dy ,t#/]y.$p@ i Dx ,~ j11 !Dy ,t#2p@ i Dx , j Dy ,t#%/Dy . ~56!

The derivatives ]p/]x and ]wy /]y follow similarly. Thebody source terms can all be evaluated directly, since theyare already stored at the grid nodes where the FD approxi-mations are centered. The same is true of k, which is storedat the pressure grid nodes and needed in Eq. ~52!. RegardingEqs. ~53! and ~54!, the values for b can be determined at theneeded locations by averaging neighboring grid points.

The implementation of the remaining terms, particular tothe moving medium, is somewhat more complicated. Forexample, the derivatives of the pressure field in Eq. ~52!,]p/]x and ]p/]y , cannot be centered at x5i Dx and y5 j Dy from approximations across a single grid interval.Centered approximations can be formed across two grid in-tervals, however, as suggested in Ref. 2. For example,

]p~ i Dx , j Dy ,t !/]x.$p@~ i11 !Dx , j Dy ,t#2p@~ i21 !Dx , j Dy ,t#%/2Dx .

~57!Neighboring grid points can be averaged to find the windvelocity components vx and vy at x5i Dx and y5 j Dy ,which multiply the derivatives ]p/]x and ]p/]y , respec-tively, in Eq. ~52!. Similarly, the spatial derivatives of theparticle velocities in Eqs. ~53! and ~54! can be approximatedover two grid intervals. In Eq. ~53!, the quantities wy and vy~multiplying the derivatives ]vx /]y and ]wx /]y , respec-tively! are needed at the grid point x5(i11/2)Dx and y5 j Dy . Referring to Fig. 1, a reasonable way to obtain thesequantities would be to average the four closest grid nodes:

wy@~ i11/2!Dx , j Dy ,t#

.14 $wy@~ i11 !Dx ,~ j11/2!Dy ,t#

1wy@ i Dx ,~ j11/2!Dy ,t#1wy@~ i11 !Dx ,~ j21/2!Dy ,t#1wy@ i Dx ,~ j21/2!Dy ,t#%, ~58!

and likewise for vy . The quantities wx and vx , multiplyingthe derivatives ]vy /]x and ]wy /]x in Eq. ~54!, can be ob-tained similarly.

B. Advancing the solution in timeLet us define the functions f p , f x , and f y as the right-

hand sides of Eqs. ~52!, ~53!, and ~54!, respectively. Forexample, we write510 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005]t

5 f p@ i Dx , j Dy ,p~ t !,wx~ t !,wy~ t !,s~ t !# , ~59!where p(t), wx(t), and wy(t) are matrices containing thepressures and acoustic velocities at all available grid nodes.For convenience, s(t) is used here as short hand for the com-bined source and medium properties (b , k, vx , vy , Q , Fx ,and Fy) at all available grid nodes. ~Note thatf p@ i Dx , j Dy ,p(t),wx(t),wy(t),s(t)# in actuality dependsonly on the fields at a small number of neighboring gridpoints of (i Dx , j Dy) when second-order spatial differencingis used. The notation here is general enough, though, to ac-commodate spatial differencing of an arbitrarily high order.!

For a nonmoving medium, the solution is typically ad-vanced in time using a staggered temporal grid, in which thepressures are stored at the integer time steps t5l Dt and theparticle velocities at the half-integer time steps t5(l11/2)Dt .2730 The acoustic velocities and pressures are up-dated in an alternating leap-frog fashion, with the fieldsfrom the previous time step being overwritten in place. Con-sidering the moving media equations, approximation of thetime derivative in Eq. ~59! with a finite difference centeredon t5(l11/2)Dt ~that is, ]p@ i Dx , j Dy ,(l11/2)Dt#/]t.$p@ i Dx , j Dy ,(l11)Dt#2p@ i Dx , j Dy ,l Dt#%/Dt) resultsin the following equation for updating the pressure field:

p@ i Dx , j Dy ,~ l11 !Dt#5p@ i Dx , j Dy ,l Dt#1Dt f pi Dx , j Dy ,p@~ l11/2!Dt# ,

wx@~ l11/2!Dt# ,wy@~ l11/2!Dt# ,s@~ l11/2!Dt#.~60!

Note that this equation requires the pressure field at the half-integer time steps, i.e., t5(l11/2)Dt . In the staggered leap-frog scheme, however, the pressure is unavailable at the half-integer time steps. A similar centered approximation for theacoustic velocities indicates that they are needed on the in-teger time steps in order to advance the solution, which isagain problematic. If one attempts to address this problem bylinearly interpolating between adjacent time steps ~i.e., bysetting p@(l11/2)Dt#.$p@ l Dt#1p@(l11)Dt#%/2 in Eq.~60!!, explicit updating equations ~a solution of Eq. ~60! forp@ i Dx , j Dy ,(l11)Dt# that does not require the pressurefield at nearby grid points at the time step t5(l11)Dt) can-not be obtained. Hence the customary staggered leap-frogapproach does not lead to an explicit updating scheme for theacoustic fields in a moving medium. The staggered leap-frogscheme can be rigorously implemented only when the termsparticular to the moving medium ~those involving vx and vy)are removed from Eqs. ~52!~54!.

A possible work-around would be to use the pressurefield p(l Dt) in place of p@(l11/2)Dt# when evaluating f p ,and wx@(l21/2)Dt# and wy@(l21/2)Dt# in place ofwx(l Dt) and wy(l Dt) when evaluating f x and f y . This non-rigorous procedure uses the Euler ~forward difference!method to evaluate the moving-media terms while maintain-ing the leap-frog approach for the remaining terms. From aprogramming standpoint, the algorithm proceeds in essen-Ostashev et al.: Moving media finite difference time domain equations

tially the same manner as the staggered leap-frog method for of FDTD techniques for simulating sound propagation in a

a nonmoving medium. The calculations in Ref. 2 appear touse such a procedure. But the stability and accuracy of thisalgorithm are unclear. An alternative is provided in Ref. 4,which uses a perturbative solution based on the assumptionthat the flow velocity is small.

Here we would like to develop a general technique thatis applicable to high Mach numbers. The simplest way toaccomplish this is to abandon the staggered temporal gridand form centered finite differences over two time steps.The pressure updating equation, based on the approximation]p(i Dx , j Dy , l Dt)/]t.$p@ i Dx , j Dy ,(l11)Dt]2p@ i Dx , jDy ,(l21)Dt]%/2 Dt , isp@ i Dx , j Dy ,~ l11 !Dt#

5p@ i Dx , j Dy ,~ l21 !Dt#12 Dt f p@ i Dx , j Dy ,p~ l Dt !,wx~ l Dt !,

wy~ l Dt !,s~ l Dt !# . ~61!

Similarly, we derive

wx@~ i11/2!Dx , j Dy ,~ l11 !Dt#5wx@~ i11/2!Dx , j Dy ,~ l21 !Dt#12 Dt f x@~ i

11/2!Dx , j Dy ,p~ l Dt !,wx~ l Dt !,wy~ l Dt !,s~ l Dt !# ,~62!

wy@ i Dx ,~ j11/2!Dy ,~ l11 !Dt#5wy@ i Dx ,~ j11/2!Dy ,~ l21 !Dt#

12 Dt f y@ i Dx ,~ j11/2!Dy ,p~ l Dt !,wx~ l Dt !,wy~ l Dt !,s~ l Dt !# . ~63!

Somewhat confusingly, this general temporal updatingscheme has also been called the leap-frog scheme in theliterature,31 since it involves alternately overwriting thewavefield variables at even and odd integer time steps basedon calculations with the fields at the intervening time step.We call this scheme here the nonstaggered leap-frog. Theprimary disadvantage, in comparison to the staggered leap-frog scheme, is that the fields must be stored over two timesteps, rather than just one. Additionally, the numerical dis-persion and instability characteristics are inferior to those ofthe conventional staggered scheme due to the advancementof the wavefield variables over two time steps instead of one.On the other hand, the nonstaggered leap-frog does provide asimple and rigorous centered finite-difference scheme that isnot specialized to low Mach number flows. Other commonnumerical integration methods, such as the RungeKuttafamily, can also be readily applied to the nonstaggered-in-time grid. Some of the calculations following later in thissection use a fourth-order RungeKutta method, which isdescribed in Ref. 32 and many other texts. We have alsodeveloped a staggered-in-time method that is valid for highMach numbers but requires the fields to be stored over twotime levels. This method was briefly discussed in Ref. 6.

Note that our present numerical modeling efforts are di-rected toward demonstrating the applicability and feasibilityJ. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashemoving atmosphere. We have not undertaken a comprehen-sive comparative analysis of the many alternative numericalstrategies available for the solution of Eqs. ~17! and ~18!.However, several of these approaches ~including the pseu-dospectral method, higher-order spatial and/or temporalfinite-difference operators, and the dispersion relation pre-serving ~DRP! technique! yield accurate simulations ofsound propagation with fewer grid intervals per wavelengthcompared with our numerical examples. In particular, theDRP method, involving optimized numerical values of thefinite-difference operator coefficients ~e.g., Ref. 18!, can bereadily introduced into our FDTD algorithmic framework.

C. Dependence of grid increments on Mach numberFor numerical stability of the 2-D FDTD calculation, the

time step Dt and grid spacing Dr must be chosen to satisfythe Courant condition, C,1/& ~e.g., see Ref. 33!, where theCourant number is defined as

C5u Dt

Dr. ~64!

Here, u is the speed at which the sound energy propagates.@For a nonuniform grid, Dr51/A(Dx)221(Dy)22.] Sincethe grid spacing must generally be a small fraction of awavelength for good numerical accuracy, the Courant condi-tion in practice imposes a limitation on the maximum timestep possible for stable calculations. An even smaller timestep may be necessary for good accuracy, however.

Let us consider the implications of the Courant condi-tion for propagation in a uniform flow. In this case, u isdetermined by a combination of the sound speed and windvelocity. In the downwind direction, we have u5u15c1v . In the upwind direction, u5u25c2v . The wave-lengths in these two directions are l15(c1v)/ f and l25(c2v)/ f , respectively, where f is the frequency. Since thewavelength is shortest in the upwind direction, the value ofl2 dictates the grid spacing. We set

Dr5l2N 5

l

N ~12M !, ~65!

where N is the number of grid points per wavelength in theupwind direction, M5v/c is the Mach number, and l5c/ fis the wavelength for the medium at rest. If N is to be fixedat a constant value, a finer grid is required as M increases.Regarding the time step, the Courant condition implies

Dt,l2Nu . ~66!

This condition is most difficult to meet when u is largest,which is the case in the downwind direction. Therefore wemust use u1 in the preceding inequality if we are to haveaccurate results throughout the domain; specifically, we mustset

Dt,l2

Nu15

1N f

12M11M . ~67!511v et al.: Moving media finite difference time domain equations

Here, k5v/c , H0(1)

, and H1(1) are the Hankel functions, jTherefore the time step must also be shortened as M in-creases. For example, the time step at M51/3 must be 1/2the value necessary at M50. At M52/3, the time step mustbe 1/5 the value at M50. The reduction of the required timestep and grid spacing combine to make calculations at largeMach numbers computationally expensive.

D. Example calculationsIn this subsection, we use the developed algorithm for

FDTD solutions of Eqs. ~52!~54! to compute the soundfield p in a 2-D homogeneous uniformly moving medium.The geometry of the problem is shown in Fig. 2. A pointmonochromatic source is located at the origin of the Carte-sian coordinate system x ,y . The medium velocity v is paral-lel to the x-axis. We will first obtain an analytical formula forp for this geometry.

In a homogeneous uniformly moving medium, c , %, andv are constant so that "v50 and P50. Therefore, Eqs.~17! and ~18! describe sound propagation exactly for thiscase and are valid for an arbitrary value of the Mach numberM . They can be written as

S ]]t 1v" D p1%c2"w5%c2Q , ~68!S ]]t 1v" Dw1 p% 50. ~69!

Here, p and w are functions of the coordinates x , y and timet , 5(]/]x ,]/]y), and the function Q is given by

Q5 2iA%v

e2ivtd~x !d~y !, ~70!

where d is the delta function and the factor A characterizesthe source amplitude. In Eqs. ~68! and ~69!, for simplicity, itis assumed that F50.

Assuming that v,c , the following solution of Eqs. ~68!and ~69! is obtained in the Appendix:

p~r ,a ,M !

5iA

2~12M 2!3/2 S H0(1)~j!2 iM cos aA12M 2 sin2 a H1(1)~j!D3expF2 ikMr cos a12M 2 G . ~71!

FIG. 2. The geometry of the problem.512 J. Acoust. Soc. Am., Vol. 117, No. 2, February 20055krA12M 2 sin2 a/(12M2), and r and a are the polar coor-dinates shown in Fig. 2. For kr@1, the Hankel functions canbe approximated by their asymptotics. This results in thedesired formula for the sound pressure:

p~r ,a ,M !5A~A12M 2 sin2 a2M cos a!

A2pkr~12M 2!~12M 2 sin2 a!3/4

3expF i~A12M 2 sin2 a2M cos a!kr12M 2 1 ip4 G .~72!

Note that a sound field due to a point monochromatic sourcein a 2-D homogeneous uniformly moving medium was alsostudied in Ref. 18 by a different approach. The phase factorobtained in that reference is essentially the same as that inEq. ~72!. Only a general expression for the amplitude factorwas presented in Ref. 18 which does not allow a detailedcomparison with the amplitude factor in Eq. ~72!.

Let us now consider the FDTD calculations of the soundfield for the geometry in Fig. 2. In these calculations, thesource consists of a finite-duration harmonic signal with acosine taper function applied at the beginning and the end.The tapering alleviates numerical dispersion of high frequen-cies, which becomes evident when there is an abrupt changein the source emission. The tapered source equation is

Q ~ t !55~1/2!@12cos~pt/T1!#cos~2p f 1f!,

0

FIG. 4. Normalized sound pressure amplitudeup(r ,a ,M )/p(r ,0,0)u versus the azimuthal angle a forM50.3 and kr520. The staggered and nonstaggeredleap-frog methods and the fourth-order RungeKuttaare compared to the theoretical solution. The nonstag-gered leap-frog and RungeKutta methods are graphi-cally indistinguishable.The methods include the staggered ~with forward-differencing of the moving medium terms mentioned in Sec.V B! and nonstaggered leap-frog approaches and the fourth-order RungeKutta. The time step for the leap-frog methodswas 0.036 ms ~1/4 that used for the RungeKutta!, so thatthe computational times of all calculations are roughly equal.The RungeKutta and nonstaggered leap-frog providegraphically indistinguishable results. The staggered leap-frog, however, systematically underpredicts the amplitude inthe downwind direction and overpredicts in the upwind di-rection. The actual sound pressure signals at t50.11 s, cal-culated from the staggered and nonstaggered leap-frog ap-proaches, are overlaid in Fig. 5. In the downwind direction,the staggered leap-frog method provides a smooth predictionat distances greater than about 22 m. The noisy appearance atshorter distances is due to numerical instability, which wasclear from the rapid temporal growth of these features weJ. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostasheobserved as the calculation progressed. We conclude that thestaggered leap-frog approach, when applied to a moving me-dium, is less accurate and more prone to numerical instabil-ity. This is likely due to the nonsymmetric temporal finitedifference approximations for the moving medium terms.

Figure 6 shows the azimuthal dependence ofup(r ,a ,M )/p(r ,0,0)u for M50, 0.3, and 0.6. All FDTD cal-culations for this figure use the fourth-order RungeKuttamethod. Two calculated curves are shown: one for a low-resolution run with 8003800 grid points and a time step of0.145 ms, and the other for a high-resolution run with 160031600 grid points and a time step of 0.0362 ms. For M50.3, both grid resolutions yield nearly exact agreementwith Eq. ~72!. At M50.6, the low-resolution run has 11spatial grid nodes per wavelength in the upwind directionand a downwind Courant number of 0.64. The high-resolution grid has 22 spatial grid nodes per wavelength inFIG. 3. Wavefronts of the sound pressure due to a pointsource located at the point x50 and y50 for M50.3. The medium velocity is in the direction of thex-axis.513v et al.: Moving media finite difference time domain equations

the upwind direction and a downwind Courant number of0.32. Agreement with theory at M50.6 is very good for thehigh-resolution run. The low-resolution run substantially un-derpredicts the upwind amplitude.

Finally note that it follows from Figs. 4 and 6 that thesound pressure is largest for a5180, i.e., in the upwinddirection. This dependence is also evident upon close exami-nation in Fig. 3.

E. Comparison of FDTD and FFP calculationsThe computational examples so far in this paper have

been for uniform flows. However, the numerical methodsand equations upon which they are based apply to nonuni-form flows as well. In this section, we consider an examplecalculation for a flow with constant shear. The point sourceand receiver are both located at a height of 20 m and the

frequency is 100 Hz. The computational domain is 200 m by100 m and has 600 by 300 grid points. The time step is7.7331024 s and the fourth-order RungeKutta method isused. A rigid boundary condition is applied at the groundsurface (y50 m). An absorbing layer in the upper one-fifthof the simulation domain removes unwanted numerical re-flections. ~The implementation of the rigid ground boundarycondition and the absorbing layer is described in Ref. 34.Realistic ground boundary conditions in a FDTD simulationof sound propagation in the atmosphere are considered inRef. 35.!

Calculated transmission loss ~sound level relative to freespace at 1 m from the source! results are shown in Figs. 7~a!and 7~b!. The first of these figures is for a zero-wind condi-tion and the second is for a horizontal (x-direction! windspeed of v(y)5my , where the gradient m is 1 s21. For Fig.

FIG. 5. Sound pressure traces for ~a! downwind and ~b!upwind propagation. Calculations from the staggeredand nonstaggered leap-frog methods are shown ~dashedand solid lines, respectively!.514 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashev et al.: Moving media finite difference time domain equations

In the present paper, we have considered starting equa-

tions for FDTD simulations of sound propagation in a mov-ing inhomogeneous atmosphere. FDTD techniques can pro-vide a very accurate description of sound propagation incomplex environments.

A most general description of sound propagation in amoving inhomogeneous medium is based on the completeset of linearized equations of fluid dynamics, Eqs. ~5!~8!.However, this set is too involved to be effectively employedin FDTD simulations of outdoor sound propagation. In thispaper, the linearized equations of fluid dynamics were re-duced to two simpler sets of equations which can be used asstarting equations for FDTD simulations.

The first set of equations contains three coupled equa-tions, Eqs. ~5!, ~6!, and ~13!, for the sound pressure p , acous-tic velocity w, and acoustic density h. This set is an exactconsequence of the linearized equations of fluid dynamics,Eqs. ~5!~8!. To solve the first set of equations, one needs toknow the following ambient quantities: the adiabatic sound

pressure p which have been most often used for analyticaland numerical studies of sound propagation in a moving in-homogeneous medium. It was shown that the second set hasthe same or wider range of applicability than these equationsfor p . Thus, a relatively simple set of Eqs. ~17! and ~18!,which is however rather general, seems very attractive asstarting equations for FDTD simulations.

The numerical algorithms for FDTD solutions of thesecond set of equations were developed for the case of a 2-Dinhomogeneous moving medium. It was shown that thestaggered-in-time grid approach commonly applied to non-moving media cannot be applied directly for the movingcase. However, fairly simple alternatives based onnonstaggered-in-time grids are available. We used the result-ing algorithms to calculate the sound pressure due to a pointsource in a homogeneous uniformly moving medium. Theresults obtained were found in excellent agreement with ana-lytical predictions even for a Mach number as high as 0.6.7~a!, the FDTD results are compared with both the exactsolution for a point source above the rigid boundary andcalculations from the FFP developed in Ref. 36. The FDTDresults are nearly indistinguishable from the exact solution.The FFP is also in good agreement, although there is somesystematic underprediction of the interference minima, par-ticularly so near the source. This is likely due to the far-fieldapproximation inherent to the FFP. For the case with con-stant shear, Fig. 7~b!, the interference pattern is shifted. TheFDTD and FFP continue to show very similar small discrep-ancies near the source. On the basis of the results shown inFig. 7~a!, it is highly likely that the FDTD is more accurate.The FDTD calculations required about 100 times as long tocomplete as the FFP on a single-processor computer. Aswould be expected, the FFP is more efficient for calculationsat a limited number of frequencies in a horizontally stratifiedmedium.

VI. CONCLUSIONSJ. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashespeed c , density %, medium velocity v, pressure P , and theparameters a, b, and h . The atmosphere can be modeled asan ideal gas to a very good accuracy. In this case, the first setof equations simplifies and is given by Eqs. ~5!, ~6!, and ~15!.Now it contains the following ambient quantities: c , %, v,and P .

The second set of starting equations for FDTD simula-tions contains two coupled equations for the sound pressurep and acoustic velocity w, Eqs. ~17! and ~18!. In order tosolve this set one needs to know a fewer number of theambient quantities: c , %, and v. Note that namely these am-bient quantities appeared in most of equations for the soundpressure p which have been previously used for analyticaland numerical studies of outdoor sound propagation. Thesecond set was derived from Eqs. ~5!~8! assuming thatterms proportional to the divergence of the medium velocityand the gradient of the ambient pressure can be ignored.Both these assumptions are reasonable in atmospheric acous-tics. To better understand the range of applicability of thesecond set, it was compared with equations for the sound

FIG. 6. Normalized sound pressure amplitudeup(r ,a ,M )/p(r ,0,0)u versus the azimuthal angle a forM50, 0.3, and 0.6. The fourth-order RungeKuttamethod was used. The calculation with 8003800 gridpoints had a spatial resolution of 0.125 m and time step0.145 ms, whereas the 160031600 calculation had aspatial resolution of 0.0625 m and time step 0.0362 ms.515v et al.: Moving media finite difference time domain equations

Propagation in Atmospheric Environments and the U.S.Furthermore, using the algorithm developed, we calculatedthe sound field due to a point source in a stratified movingatmosphere. The results obtained are in a good agreementwith the FFP solution.

Finally note that Eqs. ~17! and ~18! have already beenused as starting equations in FDTD simulations of soundpropagation in 3-D moving media with realistic velocityfields. The results obtained were published in proceedings ofconferences.58 These realistic velocity fields include the fol-lowing: kinematic turbulence generated by quasi-wavelets,5,63-D stratified moving atmosphere,6 and atmospheric turbu-lence generated by large-eddy simulation.7 In Ref. 8, FDTDsimulations were used to numerically study infrasoundpropagation in a moving atmosphere over distances of sev-eral hundred km. The largest run to date incorporated over1.5 billion nodes and took about 100 hours on 500 CompaqEV6 parallel processors.8

ACKNOWLEDGMENTS

This article is partly based upon work supported by theDoD High-Performance Computing Modernization Officeproject High-Resolution Modeling of Acoustic Wave

FIG. 7. Comparisons between the transmission loss calculated with differentmethods. ~a! Homogeneous atmosphere without wind. ~b! Atmosphere withlinearly increasing wind velocity.516 J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005Army Research Office Grant No. DAAG19-01-1-0640.

APPENDIX: SOUND FIELD DUE TO A POINTMONOCHROMATIC SOURCE IN A HOMOGENEOUSUNIFORMLY MOVING MEDIUM

In this appendix, we derive a formula for the sound pres-sure due to a point monochromatic source located in a 2-Dhomogeneous uniformly moving medium ~see Fig. 2!.

For this geometry, Eqs. ~68! and ~69! can be reduced toa single equation for the sound pressure:

S ]]t 1v" D2

p2c22p5%c2S ]]t 1v" DQ . ~A1!Here, the source function Q is given by Eq. ~70! and containsthe time factor exp(2ivt). In what follows, this time factor isomitted. Furthermore, taking into account that the mediumvelocity is parallel to the x-axis, Eq. ~A1! can be written as

S ]2]x2 1 ]2

]y2 1k212ikM

]

]x2M 2

]2

]x2D p~x ,y !5

2iAv S iv2v ]]x D d~x !d~y !. ~A2!

Let

p~x ,y !52iAv S iv2v ]]x DF~x ,y !. ~A3!

Substituting this formula into Eq. ~A2!, we obtain the follow-ing equation for the function F(x ,y):

F ]2]x2 1 ]2

]y2 2S 2ik1M ]]x D2GF~x ,y !5d~x !d~y !.

~A4!In this equation, let us make the following transformations:

x5A12M 2X , k5A12M 2K ,

F~x ,y !5exp~2iKMX !C~X ,y !. ~A5!As a result, we obtain the following equation for the fuinc-tion C(X ,y):

F ]2]X2 1 ]2

]y2 1K2GC~X ,y !5 1A12M 2 d~X !d~y !. ~A6!

A solution of this equation is well known:

C~X ,y !52i

4A12M 2H0

(1)~KAX21y2!. ~A7!

Using this expression for C and Eqs. ~A3! and ~A5!, weobtain a desired formula for the sound pressure of a pointmonochromatic source in a 2-D homogeneous uniformlymoving medium:

p~x ,y !5iA

2~12M 2!3/2 FH0(1)~j!2 iMkxj~12M 2! H1(1)~j!G3expS 2 ixkM12M 2D . ~A8!Ostashev et al.: Moving media finite difference time domain equations

Here, j5 (k/A12M 2)Ax2/(12M 2) 1y2. In polar coordi-nates, Eq. ~A8! becomes Eq. ~71!.

1 R. Blumrich and D. Heinmann, A linearized Eulerian sound propagationmodel for studies of complex meteorological effects, J. Acoust. Soc. Am.112, 446455 ~2002!.

2 E. M. Salomons, R. Blumrich, and D. Heinmann, Eulerian time-domainmodel for sound propagation over a finite-impedance ground surface.Comparison with frequency-domain models,Acust. Acta Acust. 88, 483492 ~2002!.

3 T. Van Renterghem and D. Botteldoren, Numerical simulation of theeffect of trees on downwind noise barrier performance, Acust. ActaAcust. 89, 764778 ~2003!.

4 T. Van Renterghem, The finite-difference time-domain method for simu-lation of sound propagation in a moving medium, Doctoral thesis, Uni-versiteit Gent, 2003.

5

18 C. K. W. Tam and Y. C. Webb, Dispersion-relation-preserving finite dif-ference schemes for computational acoustics, J. Comput. Phys. 107,262281 ~1993!.

19 C. Bogey, C. Bailly, and D. Juve, Computation of flow noise usingsource terms in linearized Eulers equations, AIAA J. 40, 235243~2002!.

20 C. Bailly, P. Lafon, and S. Candel, A stochastic approach to computenoise generation and radiation of free turbulent flows, AIAA Paper No.95-092, 1995.

21 J. C. Hardin and D. S. Pope, An acoustic/viscous splitting technique forcomputational aeroacoustics, Theor. Comput. Fluid Dyn. 6, 323340~1994!.

22 W. Z. Shen and J. N. Sorensen, Aeroacoustic modeling of low-speedflows, Theor. Comput. Fluid Dyn. 13, 271289 ~1999!.

23 W. Z. Shen and J. N. Sorensen, Aeroacoustic modeling of turbulentairfoil flows, AIAA J. 39, 10571064 ~2001!.D. K. Wilson, M. L. Moran, L. Liu, V. E. Ostashev, D. F. Aldridge, N. P.Symons, and D. H. Marlin, Development of a high-fidelity simulationcapability for battlefield acoustics, in Proceedings of SPIE AeroSense2003, Orlando, FL, 2003.

6 D. H. Marlin, D. F. Aldridge, N. P. Symons, D. K. Wilson, and V. E.Ostashev, Finite-difference time-domain acoustic wave propagation incomplex atmospheric environments: Second year results, in Proceedingsof the Military Sensing Symposia (MSS) Specialty Group on BattlefieldAcoustic and Seismic Sensing, Magnetic and Electric Field Sensors, Lau-rel, MD, 2003.

7 D. K. Wilson, N. P. Symons, E. G. Patton, P. P. Sullivan, D. H. Marlin, D.F. Aldridge, V. E. Ostashev, S. A. Ketcham, E. L. Andreas, and S. L.Collier, Simulation of sound propagation through high-resolution atmo-spheric boundary layer turbulence fields, in Proceedings of the 16thAmerican Meteorological Society Symposium on Boundary Layers andTurbulence, Portland, ME, 2004.

8 N. P. Symons, D. F. Aldridge, D. H. Marlin, D. K. Wilson, E. G. Patton, P.P. Sullivan, S. L. Collier, V. E. Ostashev and D. P. Drob, 3D staggered-grid finite-difference simulation of sound refraction and scattering in mov-ing media, in Proceedings of the 11th International Symposium on LongRange Sound Propagation, Fairlee, VT, 2004.

9 V. E. Ostashev, Acoustics in Moving Inhomogeneous Media ~E&FNSPON, London, 1997!.

10 A. D. Pierce, AcousticsAn Introduction to its Physical Principles andApplications ~McGraw-Hill, New York, 1989!.

11 L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media~Springer-Verlag, Berlin, 1992!.

12 M. E. Goldstein, Aeroacoustics ~McGraw-Hill, New York, 1976!.13 V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propa-

gation ~Israel Program for Scientific Translation, Jerusalem, 1971!.14 A. D. Pierce, Wave equation for sound in fluids with unsteady inhomo-

geneous flow, J. Acoust. Soc. Am. 87, 22922299 ~1990!.15 D. F. Aldridge, Acoustic wave equations for a linear viscous fluid and an

ideal fluid, SAND Report SAND2002-2060, Sandia National Laborato-ries, Albuquerque, NM, 2002. ~Available from the U.S. Dept. of Com-merce National Technical Information Service, http://www.ntis.gov/ordering.htm.!

16 L. D. Landau and E. M. Lifshitz, Fluid Mechanics ~Pergamon, New York,1987!.

17 D. I. Blokhintzev, Acoustics of an Inhomogeneous Moving Medium~Nauka, Moscow, 1946! ~in Russian!; ~English translation, Department ofPhysics, Brown Univ., Providence, RI, 1956!.J. Acoust. Soc. Am., Vol. 117, No. 2, February 2005 Ostashe24 R. Ewert and W. Schroder, Acoustic perturbation equations based onflow decomposition via source filtering, J. Comput. Phys. 188, 365398~2003!.

25 V. E. Ostashev, On sound wave propagation in a three-dimensional in-homogeneous moving medium, in Diffraction and Wave Propagation inInhomogeneous Media ~MPTI Press, Moscow, 1987!, pp. 4249.

26 Ph. Blanc-Benon, L. Dallois, and D. Juve, Long range sound propagationin a turbulent atmosphere within the parabolic approximation, Acust.Acta Acust. 87, 659669 ~2001!.

27 K. S. Yee, Numerical solution of initial boundary value problems involv-ing Maxwells equations in isotropic media, IEEE Trans. AntennasPropag. 14, 302307 ~1966!.

28 D. Botteldooren, Acoustical finite-difference time-domain simulation ina quasi-Cartesian grid, J. Acoust. Soc. Am. 95, 23132319 ~1994!.

29 R. W. Graves, Simulating seismic wave propagation in 3D elastic mediausing staggered-grid finite differences, Bull. Seismol. Soc. Am. 86,10911106 ~1996!.

30 J. O. A. Robertsson, J. O. Blanch, and W. W. Symnes, Viscoelastic finite-difference modeling, Geophysics 59, 14441456 ~1994!.

31 A. Kasahara, Computational aspects of numerical models for weatherprediction and climate simulation, in Methods in Computational Physics,edited by J. Chang ~Academic, New York, 1977!, pp. 166.

32 E. Kreyszig, Advanced Engineering Mathematics, 6th ed. ~Wiley, NewYork, 1988!.

33 L. R. Lines, R. Slawinski, and R. P. Bording, A recipe for stability offinite-difference wave-equation computations, Geophysics 64, 967969~1999!.

34 D. K. Wilson and L. Liu, Finite-difference, time-domain simulation ofsound propagation in a dynamic atmosphere, Cold Regions Research andEngineering Laboratory Technical Report ERDC-CRREL TR-04-12, Ha-nover, NH, 2004, p. 63, available from the National Technical InformationCenter, http://www.ntis.gov/, Technical Report, ADA423222.

35 S. L. Collier, V. E. Ostashev, D. K. Wilson, and D. H. Marlin, Imple-mentation of ground boundary conditions in a finite-difference time-domain model of acoustic wave propagation, in Proceedings of the Mili-tary Sensing Symposia (MSS) Specialty Group on Battlefield Acoustic andSeismic Sensing, Magnetic and Electric Field Sensors, Laurel, MD, 2003.

36 D. K. Wilson, Sound field computations in a stratified, moving medium,J. Acoust. Soc. Am. 94, 400407 ~1993!.517v et al.: Moving media finite difference time domain equations