Reflexion of a spherical wave by the plane interface between a perfect fluid and a porous medium

Journal of Sound and Vibration (1978) 56(l), 97-103

REFLEXION INTERFACE

OF A SPHERICAL BETWEEN

WAVE

A PERFECT

POROUS

BY THE PLANE FLUID

AND A

MEDIUM

P. J. T. FILIPPI AND D. HABAULT Laboratoire de Mkcanique et d’dcoustique, Centre National de la Recherche Scientz$que, 13274 Marseille Cedex 2, France

(Received 8 June 1977 and in revisedform 30 August 1977)

By using the Fourier transform of the system of equations and continuity conditions, it is easy to obtain the Fourier transform of the solution. This last function is decomposed into several terms which are identified as Fourier images of known functions. An exact representation of the scattered sound pressure field is obtained as a combination of the radiation of the image source and layer potentials. Approximations are given when the point spherical source is located on the interface.

1. INTRODUCTION

The constant normal impedance boundary condition is not always sufficient for characterizing the effect of an obstacle on a sound wave. It is therefore sometimes necessary to consider the sound propagation within the obstacle. Because of the complexity of such a problem, a simple approach is to assume that, when the reflexion of sound waves is considered, the obstacle behaves as a homogeneous porous medium: this enables one to characterize the obstacle by two complex coefficients (a complex “mass density”, and “sound speed”) instead of one (impedance). Furthermore, it is possible to transform the transmission problem into a boundary value problem (with the non-local boundary conditions taken into account), thus ignoring what actually occurs within the obstacle. The present paper is devoted to the study of the reflexion (or scattering) of a spherical sound wave by a plane interface between a perfect fluid and a porous medium. This simple case leads to explicit formulas and allows the development of a metrology of the two characteristic complex parameters defining a boundary which behaves as a porous medium. The second section is devoted to the transformation of the transmission problem into a boundary value problem. In section 3, the exact solution is given. Section 4 deals with the particular case of a point source located on the interface. In section 5 a possible metrology is suggested. Some of the results here proposed have already been established by several authors [l-3]. 2. BOUNDARY CONDITION AT THE INTERFACE OF A PERFECT FLUID AND A POROUS MEDIUM FOR SIMPLE HARMONIC MOTION Let sZz be a bounded region of space ([w3 or W), limited by an indefinitely differentiable boundary r; and let Q, be the unbounded complement of ?&. The domain Q, is filled with a porous homogeneous isotropic medium which can be characterized [4] by a complex mass 97

P. J. T. FILIPPI AND D. HABAULT

98

density pz and a complex sound speed c, (pz and c2 depending on the frequency). The domain Q, is occupied by a perfect homogeneous isotropic fluid of mass density p1 and sound speed Cl.

The sound pressures p, in Q1 and pz in D, due to a simple harmonic excitation, the radian frequency of which is oO, satisfy the following Helmholtz equations: (A + WP, =f

in .Q,,

(1)

(A + k$p, = 0

in C12,

(2)

kf = o$cf,

i = 1,2,

in which it has been assumed that sources are present in O1 only. On r, it can be shown [4] that pi and pz satisfy the continuity relationships, Trpi = Trp,, Tr(%pJp,)

(3) (4)

=Tr(%p,/pA

(Tr u stands for the value, on r, of u). If the time dependence is assumed to be of the form e-IO’, then, at infinity, pi has the following asymptotic behaviour : ap,/& - ik,p, = o(r(1-n)i2),

p1 = O(r(l-“)‘2),

for r + co,

(5)

where n is the dimension of the space; any other equivalent condition, as for example the limiting amplitude principle, can be used [5]. Let G, be the solution defined in R” by the equation (A + k:) G2 = 6,

(6)

which is bounded. The Green formula applied to p2 and G2 on I’ yields +Trp, + 5 {Trp, a,, G2 - Tr &p2 G,) = 0, I+Tr anp2 + j {Trp, a,, a,, G2 - Tr &p2 8, G,} = 0.

(8)

r

In these equations a,. means that the normal derivative is taken with respect to the integration point; a,, is the normal derivative with respect to the field point. In expression (8), the first integral is defined as a limit [6]. Using the continuity relationships (3) and (4), one obtains two boundary conditions for p1 : fTrp,+j

Trp,&G2--zTr%.p,G,

=O,

r ( Trp,a,a,,.

G2 -zTra..p,&G,

=O.

(10)

It can be shown [7] that the two boundary value problems, of expressions (l), (5) and (9), or (l), (5) and (lo), are well posed : the solution p1 exists and is unique; furthermore, the knowledge of p1 enables one to construct pz through its Green formula representation. Condition (9), being the simplest one, will be used in the following. Note that the result remains true if both media are conservative or dissipative ones, and if both Sz, and D2 are unbounded.

REFLEXION

3. REFLEXION

OF A SPHERICAL

99

WAVE

OF A SPHERICAL SOUND WAVE AT A PLANE INTERFACE ;

GENERAL EXPRESSION OF THE SOLUTION Let Q1 be the three-dimensional half-space z > 0, Q, being the half-space z < 0. A point spherical source, of unit strength, is located at S (0, 0, s). It will be useful to replace the Sommerfeld condition (5) by the limit amplitude principle: the frequency o, is first replaced by 0 = o0 + ia,

E > 0;

the solution p1 of the boundary value problem (l), (5) and (9) is then the limit, for E + 0, of the bounded solution of equations (1) and (9) in which o. is changed into w; k, and k, stand for w/cl and w/cz, or we/cl and wo/cz. 3.1.

FOURIER

TRANSFORM

OF THE SOLUTION

Using the two-dimensional Fourier transform with respect to the cylindrical co-ordinates p and 0, and the cylindrical symmetry of the problem, transforms equation (1) into (d2/dz2 + K:)p, = 6,, m

The Fourier transform of the boundary condition (9) is ~2

1

d

+M(s; 590) -----Pl(s;

5,z)

Pldz

I=0 2(4rc2t2 - k;)“2

(l-2)

= 0,

because %TrG,

1

=-%Trg=

1

=-

2(47c2t2- k$112

2iK2

(r is the distance to the co-ordinate origin). This equality is proved in reference [8] for complex k,, but it remains valid when k, tends to a real number.

The solution of equations (11) and (12) is given by 1 zh=2iK,

elK,lr-sl +

(13)

a(t)=.

Here Ki is chosen with positive imaginary part to ensure that@, be bounded; A(c) is given by A(0 =

(~2 KI

-

PI K2Mp2

KI

(14)

+ ~1 K2).

For the same reason, the imaginary part of K- is chosen to be > 0. 3.2.

INVERSION

FORMULA

To obtain an easy way of inverting expression (13), the function A is first written as A= =p

(~2 KI

-

PI K2Mpf

P: + P: +

G

-

P: Kt)

P: pt(k: - k:)

2

(p:

P3 - P: _2w:(G-k:)

1

(p'z -

d)'

1

_2

b2 - 4n212

- p:)’

K,

b2 - 4n2r2 E ’

with

b2= (PI k: - p: k:)/(p: - p:).

~-Pl P2

K1

P: - P:

K2

-

(15)

P. J. T. FILIPPI AND D. HABAULT

100 It is well known [8] that

%-I (l/K,) = -2 i(eik+/4np),

(16) Im(b) > 0

%-I{ l/(b2 - 4~~4~)) = -(i/4) H&p),

(17)

(H,, is the Hankel function of the first kind), and so 1

1

- -’ Ho(bpQ, ‘; b2 - 47~~4~ z2 2

p-1

7

(18)

where(r,stands for the convolution product in the plane. The first term of A gives the field radiated by the image source S’(0,0, -s) with the amplitude (P: + P:)l(P: - P3. The second term corresponds obviously to a simple layer potential, the layer support being the z = --s plane. The two last terms to be inverted are of the form eiK,(s+z) 1 J”W’~

=Tf(r);fg 1

1

(19)

I’=-s

Such a term is the Fourier image of a double layer potential, the layer support being the z = --s plane; the Fourier transform of the half layer density is obtained by letting z + s = 0 in expression (19) : indeed, because the layer support is a plane, the value, on the plane, of the double layer potential is equal to the half layer density, the integral being zero. Finally, the sound pressure p1 can be represented by elk,“S.

p&X)=-

p;

X)

47cr(S, X)

-Y)

eik,r(X,

’ 4nr(X, Y)

+ p;

eik,r(S:

2’)

P?pt(ki - k3

i

pt - p: 47rr(S’, X) + 2

(P4 - P3’

H&( s L,__-S

VI x

s aeIk,r(X, Y) a

elk,t(Y)

dx’dy’-&%

-_

e’k,r’X,

P: - P: I,=_-s 4nt( Y) 8~’4MX,

Y)

dx’dy’+;

x

Y)

1

( Y) G 4,r(X y> dx’dy’, 3

where X = (x, y, z),

s =

(0,0, s),

b2 = (P: k: - P: k:)/(p:

S’ = (O,O,--s),

(20)

Y = (x’, y’, z’),

- P:),

t(Y) = distance (0, Y) and(:)= convolution in the plane.

4.THE PARTICULAR CASE OF A POINT SOURCE AND OBSERVATION POINT BOTH ON THE z = 0 PLANE When both the source and observation point are on the plane, formula (20) is much simplified : indeed, the double layer potentials will reduce to the half layer density only. One obtains eik,r(S.X)

2p: f-Q%

X)

=

-

p:

_

-

2

4nr(S, X)

p:

i

2

~:p:(kl (pz

p2 e*k2r(s* x, + I

p1 p:

+

-

p:

4nr(S, X)

x= (X,Y,O), Two kinds of approximations

p1

2

-

- k:)

eik,r(X,

5

p:)’

z,=.

H”[bt(Y)l

pz(k: - G) (p$

-

p:)’

s = GAO,Oh

&r(X,

Y’)

Y) e’k2”X,

H”[bt(Y)l

dx’

dy’

-

Y)

4rcr(X, Y)

dx’dy’,

Y = (x’,y’, z’).

can be developed for the near field and the far field.

(21)

REFLEXION OF A SPHERICAL WAVE

101

4.1. THE NEAR FIELD APPROXIMATION

It is first to be remarked that an expression of the form e18rW.Y) z=

Y)l

HO&(

1 z,=cl

4rcr(X, Y)

dx’dy’

can be expressed by I= 3l-Wr) 1 J&p) eiSpdp + +Jo(ar) 1 H,(ap) eiSPdp I

0

x=m , x=,

(22)

which is obtained by using the well-known development of the HO function in terms of cylindrical harmonics, and where Je and He = Je + iYe are the Schwarz functions [9]. It is shown in reference [9] that Je(~,il~)=iei8r~(~)[e-i~r-~~](~~,

(23)

~Ye(~,~~)-log(~)Je(~,~~)+iei~r[log(~)J~~a~)-~Yo~a~)]+

+ i elBr @ 2n x( + ieiBri

[$Qn + 1) - $(n + l)] (_E~[e-i@-~L!_!$Z]+

1 (+)($r[

‘“-l (-i/jr) +

e+[Ci@r)

1)),

c a;-ti(Y+ 4=0 .

- log fir + i Si(/?r)] +

ti(d=~;+~(L),

(24)

$(1)=-y.

Ci and Si are the cosine and sine integral functions. These expressions are obviously rapidly convergent: indeed, in expression (23), the nth term is the difference between the exponential function emiBrand the (2n + 1) first terms of its representative series; the same remark applies to formula (24). The last result needed is 02

s

Ho(ar)ei8' dr = 1 In

0

1

log (a’ - /P)l” - ifl

(a’ - /3’)112

(25)

a

Using formulas (23), (24) and (25) one obtains an exact convergent series representation :

2P: e%r

p,(S,X)=--------p: - p: 47rr k,+im log

b

2p, p2 eikzr I i Pi P%G - G) A-p<

4nr

4

M-P:)’

Jo&) ; d&2

1

x

+[H,(hr)-(l+~log~)Jo(~r)][~eiklr~(~)(&~x

x(e-W_zy)]

+f F

Jo@)

[log~J,(br) - MY,

+zo2n x

102

P. J. T. FILIPPI AND D. HABAULT

X

2n n [+(2n +

0

x e-‘klr(Ci(k,r) [ +

x e-“2’(Ci(k,r) ]

1)

_

~l/(~

+

2n-1 (-ik2r)P$(p + i Si (k2r)) + 2 p!

2p, p2 eikzr

p:-p:4nr+Zrr

4m

-k:

+ 1)

b

1 p:p:(k: -k:) (P:-py

(26) :

1 d-X k2 f id-

1

(pz'-p:y

+si

II) ,

near field approximation

1 PI&:-~:)

which is obtained by taking into account functions appearing in expression (26).

1%

m

b

the first terms of the ascending

’

(27)

series of the different

THE FAR FIELD APPROXIMATION

has been shown in a previous

paper [lo] that ei4r

-!Ho(ar)Je

B

:tpr

i B

1

I Upon using expressions

1

E -____ 7rr a(P - cx) [ -‘-2(/3-a)r

WJH,(ap) dp

iJo(w)Je

I

+ O(r-*)

1

1 ”

f-

nr a(/?+ a)

(28), the asymptotic pl(S,X+

]

i+

2(P+cc)r

behaviour

k,

p;

ofp,

1 I

+ O(r-*)

.

(28)

takes the form

eiklr

co) E 2i-------ki - kS pi 4nr2

which shows that, as in the case of an impedance re2.

+

I)]

(26) reduces to the following

k, + idb*

It

b

p=O

eik,r

x log

+

- logk,r

with r = r(S, X). Expression

4.2.

log

+ III

+~10e3~o(~r)][~eikzr~(~)(~~x

[$(zn

2P: ---pi-p:

P!

kz+im

Jdbr$ q&

(P: - Pi)”

$(p + 1)

+ 2

p=O

+ [H,(br)-(1

p,(s,x)E

2n-1(-ik,r)p

k,r + i Si(k,r))

-log

1 PI P:(k:- k3 4

[e-w_~qq+~ (-q(qx

1) - $(n + l>l

absorbing

plane, the pressure

(29) decreases

as

REFLEXIONOF A SPHERICALWAVE 5.APPLICATION

TO THE METROLOGY

OF GROUND

103 PROPERTIES

Formula (29) shows that the measurement of the far field gives one complex parameter characterizing the porous medium. A second complex parameter can be obtained by measuring the reflexion of a plane normally incident wave; in such a case, the ratio of the pressure to its normal gradient is given by (30)

P~/%P~ = - i(pzlpl)(l/&).

Other measurements can be performed, subject to some approximations. Indeed, as shown in reference [4], the ratio Ip21/p1 is much greater than unity at relatively low frequency. In fact, let I be the characteristic length of a pore, and p/p1 the dynamic viscosity of air which is known to be @lo-’ cm2/s); pz is related to the dimensionless Darcy’s law constant K*, which is O(l), by pz = - p/ioK*

Therefore

I’.

Ip21/p1 is 01 10-1/oZ21. Under this hypothesis,

expression

(27) can be approximated

by eik,r(S. p1(S,

X)

X)

‘v -2p:

p; - p: 4lTr(S, X)’

(31)

Hence a measurement of the near field pressure will provide an approximation of p2/p1. Consequently, it is possible to perform three different measurements to determine the two characteristic constants of the porous medium, and so check the validity of such a model of the ground. 6. CONCLUDING

REMARKS

The main result of this paper is the exact representation of the total pressure field given by expression (20). This representation allows easy numerical computation : indeed, the integrals involved contain exponentially decreasing functions. It is possible to obtain far-field approximations, too: the kernel eikI’(“*‘)/4nr(x,~) can be approximated by eiklr(X,S’)/4rrr(X,S’), which yields a modified amplitude of the image source. Another important result is that when the characteristic parameters of a porous medium (or a ground having the same reflecting properties) are known in the plane configuration they provide the non-local boundary conditions for any geometrical situation. REFERENCES I. I. RUDNICK 1947 Journal of the Acoustical

2. 3. 4. 5.

Society of America 19, 348-356. Propagation of an acoustic wave along a boundary. D. I. PAUL 1959 Journal of Mathematics andPhysics 38( I), 2-15. Wave propagation in acoustics using the saddle point method. R. B. LAWHEADand I. RUDNICK 195 1 Journal of the Acoustical Society of America 23, 546-549. Acoustic wave propagation along a constant normal impedance boundary. TH. LEVY and E. SANCHEZ-PALENCIA (to appear) Journal of Mathematical Analysis and Applications. Equations and interface conditions for acoustic phenomena in porous media. C. H. WILCOX 1975 in Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-

Verlag. Scattering

theory for the d’Alembert equation. and Vibration 54, 473-500.

6. P. .J. T. FILIPPI 1977 Journal ofSound 7.

8. 9. 10.

Layer potentials and acoustic diffraction. P. FILIPPI 1977 Laboratoire de MPcanique et d’dcoustique, Internal Report No. 1485. ProbEme de transmission pour 1’Cquation de Helmboltz scalaire, et problemes aux limites Cquivalents: application g la transmission gaz parfait-milieu poreux. G. N. WATSON1962 Theory of Bessel Functions. Cambridge University Press. Y. LUKE 1961 Integrals of Bessel Functions. New York: McGraw-Hill Book Company, Inc. M. BRIQUETand P. FILIPPI 1977 Journal of the Acoustical Society of America 61, 640-646.

Diffraction of a spherical wave by an absorbing plane.