-
Equations for finite-difference, time-domain simulationi
lor00
an
ab
e
ct
re
ptmmata
ct
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!.
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