Effective wave numbers for media sustaining the
propagation of three types of bulk waves and hosting a
random configuration of scatterers
F. Luppéa , J.-M. Conoirb and A. N. Norrisc
a
LOMC-GOA, Université du Havre, place R. Schuman, 76610 Le Havre, France, Metropolitan
b
CNRS, Institut Jean Le Rond d’Alembert - UMR CNRS 7190, Université Pierre et Marie
Curie - 4 place Jussieu, 75005 Paris, France
c
Rutgers University, Mechanical and Aerospace Engineering, Piscataway, NJ 08854, USA
francine.luppe@univ-lehavre.fr
Wave propagation through an isotropic host medium containing a large number of randomly and uniformly
located scatterers is considered at low frequency and for low concentrations of spheres, and the dispersion
relation of the coherent waves is obtained. The same problem had been addressed by Lloyd and Berry for
spheres in an ideal fluid, and more recently by Linton and Martin for cylinders in an ideal fluid, and by Conoir
and Norris for cylinders in an elastic solid. Here, the dispersion relation is derived in the case of spheres, and
extended to that of cylinders, from the comparison of the 3d and 2d cases in an elastic solid. The host medium
considered may support the propagation of P different types of bulk waves, as for example a thermo-viscoelastic medium or a poro-elastic medium (P=3). As in the previous works mentioned above, the hole correction
of Fikioris and Waterman is taken into account, along with the quasi-crystalline approximation. The method
follows exactly that used by Conoir and Norris.
1
Introduction
This paper summarizes some of the results obtained in
Refs.[1,2] on the propagation of coherent waves in
homogeneous media that contain distributions of either
spherical (d=3) or cylindrical (d=2) inhomogeneities acting
as scatterers. It is focused on the low frequency and low
concentration approximations of the dispersion equation of
the coherent wave associated with the “fastest wave” in the
host medium (more precisely, the wave which has the
smallest modulus of all complex wave numbers). In a porovisco-elastic medium obeying Biot’s theory, for example,
that would be the fast longitudinal wave.
The method used to obtain the equations that govern the
coherent fields follows that of Fikioris and Waterman’s
paper [2]. A harmonic plane wave propagating in the host
medium is supposed to be normally incident upon some
(d )
semi-infinite region x > 0 hosting a uniform
concentration of scatterers. In case of spherical scatterers,
x(3) will be equal to z, while in case of infinitely long
cylinders, x(2) will be equal to x, and the axis of the
cylinders will be parallel to the z axis, so that the problem
will be of dimension d=2. This harmonic plane wave gives
rise to a multiple scattering process and to scattered waves
of different polarization types denoted by a natural number
p. When averaged over all possible locations of the
scatterers, the total field of a given type p represents the
coherent wave of type p. For a low enough concentration of
scatterers, the coherent waves are supposed to be plane
(d )
waves that propagate in the same direction x as the
original incident plane wave.
In the following, we suppose that the host medium
supports the propagation of P different types of waves (P=1
in an ideal fluid, P=2 in an elastic solid, P=3 in a thermovisco-elastic medium or a poroelastic solid); they are
numbered from p=1 to p=P, and their complex wave
numbers at the given angular frequency ω are kp, with
∀p ≠ 1, k1 < k p ,
(1)
so that the coherent we shall focus on in the last section
corresponds to p=1. We shall also suppose, with no loss of
generality but in order to simplify the d=3 dimensional
( p)
problem, that the original incident plane wave, ϕinc , is also
of type p =1, i.e. a longitudinal wave. An e-iωt time
dependence of all fields, while supposed, will be omitted
everywhere for the sake of brevity :
( )
ϕinc
= δ p1e
p
d
ik p x( )
, k p = k p ′ + ik p ′′ , k p ′ > 0, k p ′′ ≥ 0 . (2)
2
Coherent fields in the framework
of the Fikioris and Waterman theory
The waves are described by scalar displacement
potentials. In the d=3 dimensional problem, these are the
Debye potentials [4] of the longitudinal and shear waves :
the Debye potentials of transverse waves are not taken into
account for symmetry reasons linked to the longitudinal
nature of the incident wave of Eq.(2), as explained in
Ref.[2]. In the d=2 dimensional case, the scalar potentials
are the displacement potential of the longitudinal waves and
the z-component [5] of the vector potential of the shear
waves.
The total potential associated with a wave of type p is
due to the incident wave, if of the same type, and to all the
scattered waves of type p, so that, considering a given
number N of scatterers,
N
(1)
ϕ( p) r = δ p1ϕinc
+ ∑ ϕS( p) r ; rk .
()
k =1
( p)
Here, as in Ref.[1], ϕS
( )
(3)
(r; r ) represents the wave (of
k
type p) that is scattered by a scatterer centered at rk and
( p)
observed at r . Letting ϕE
(r; r ) denote the field (of type
k
p) that is exciting a scatterer centered at rk and observed at
r , the following equation defines the linear scattering
(qp)
rk of the scatterers
operators T
( )
P
ϕS( p) r ; rk = ∑ T (qp) rk ϕ(Eq) r ; rk .
( )
q=1
( ) ( )
(4)
The exciting field acting on the k-th scatterer is the sum
of the incident field and the scattered waves from all the
other scatterers:
P
N
(1)
ϕ(Ep) r ; rk = δ p1ϕinc
r + ∑∑ T (qp) rj ϕEq r ; rj ,(5)
( )
()
q=1 j =1
j≠k
( ) ( )
and the average exciting field on the 1st scatterer (supposed
fixed), within the quasi-crystalline approximation, is given
by
p
(1)
ϕ(E ) r ; r1 = δ p1ϕinc
r +
( )
∫
()
d rj n rj , r1
P
with
,
(6)
( ) ∑ T (r ) ϕ (r; r )
(qp)
( q)
j
E
N ( p) (ξk ) = ξk bj ′ (ξk b) h(1) ( k pb) − k pbj (ξk b) h(1)′ ( k pb)
j
q =1
with the integration performed over the whole region that
hosts the centers of the scatterers and n rj , r1 , the
conditional number density of scatterers at rj if one is
known to be at r1 , given by the “hole correction” [3] :
( )
n0 for r − rj > b
n r , rj =
with b>2a.
0 otherwise
(7)
The effective potentials are expressed as infinite series
that respect the symmetries of both the incident wave and
the scatterers (see section II.B in Ref.[2]),
d = 3,
+∞
ϕE( p) r ; rj = ∑ An( p) rj jn ( k p ρ j ) Pn cos θ ρ j
d = 2,
ϕEp r ; rj =
( )
+∞
∑
n=−∞
(
( )
n =0
( )
( )
( ))
d = 2, An(
p)
P
An( )e
pk
n
j
.
(9)
iξ k X j
pk
P
+∞ +∞
4n0 πb
ξk2 − k p2
(qp)
ν
q =1 ν =0 =0
pk )
=
2πn0
2
ξk − K p2
q
+∞
+∞
∑ i J (k r ) e
n
n
, (13)
inθ
q
n=−∞
qp )
,
+∞
ϕ(qp) = ∑ i n (2n + 1)Tn(qp)hn(1) ( k p r ) Pn (cos θ )
n=0
d = 2,
+∞
∑ i T(
n
n
qp)
, (14)
H n(1) ( k p r ) einθ
by a single scatterer centered at x=y=z=0.
3
Low frequency and low
concentration approximation for all
coherent wave dispersion equations
The low frequency approximation corresponds to small
values of k pb , whatever that of p. In the low concentration
approximation, it is assumed that ξ p and kp are close
enough for the expansion of ξ p2 − k p2 in terms of powers of
∀p ∈ {1, 2,3} ,
∑∑∑ (−1) (2ν + 1)T
d = 2, An(
scattering of a type q incident wave ϕi( ) ,
n0 to be accurate enough at order 2:
∀p ∈ {1, 2,...P} , ∀k ∈ {1, 2,...P}
=
the modal coefficient associated with the
n=−∞
Determination of the system of equations the An( )
amplitudes obey involves the decomposition of the incident
plane wave, Eq.(2), into either spherical or cylindrical
functions, the writing of the wave scattered by one scatterer
as a sum of waves incident on the other scatterers via an
addition theorem [6,7], and leads (see Refs.[1,2] for
details), for identical scatterers, to
pk )
qp )
ϕ(qp) =
k =1
d = 3, An(
and Tν(
into a type p wave ϕ(
in both cases,
k =1
=0
d = 3,
P
iξ X
rj = i n ( 2n + 1) ∑ An( pk )e k j
(r ) = i ∑
Pn ( cos θ ) Pν ( cos θ ) = ∑ G ( 0,ν | 0, n | ) P ( cos θ ) , (12)
,(8)
(d )
( )
and the Gaunt coefficients G (0, ν | 0, n | ) defined from
d = 2, ϕi =
Descartes laws of refraction, i.e. they propagate in the x
direction :
d = 3, An
1
(11)
( q)
and we assume the coherent waves obey the Snell-
( p)
1
n =0
( )
with ρ j = r − rj and θ ρ j = Arg ρ j
N mp (ξk ) = ξk bJ m′ (ξk b) H m( ) ( k pb) − K pbJ m (ξk b) H m( )′ ( k pb)
d = 3, ϕi(q) = ∑ i n (2n + 1) jn ( kqr ) Pn (cos θ )
inθ(ρ j )
A rj J n ( K p ρ j ) e
p
n
d = 2,
+∞
( )
( )
d = 3,
P
(qk )
( p)
Aν N (ξk )G (0, ν | 0, n | ),
+∞
∑ ∑ T(
q =1 m=−∞
qp)
m
Am( ) N mp−n (ξk )
qk
(10)
⎛
⎟⎞
P
⎜⎜ (d )
.(15)
d) ⎟
(
ξ − k = d1 p n0 + ⎜⎜d 2 p + ∑ d 2 pq ⎟⎟⎟ n02 + O (n03 )
⎜⎜
⎟⎟
q=1
⎝⎜
⎠
q≠ p
2
p
2
p
(d )
Equations.(11) are infinite linear and homogeneous
pk
systems of equations for the unknown amplitudes An( ) ;
setting their determinants to zero provides the dispersion
equations of the p-type coherent wave (once again, see
Ref.[2] for details), which, under both assumptions of low
concentration and low frequency, leads to
(3)
d1 p = −4iπk
3
p
+∞
∑(2n +1)T
( pp)
n
,
Eq.(14) at a large distance r from a single scatterer centered
at x=y=z=0 :
d = 3,
ϕ( pq) (r , θ )
(16)
n=0
+∞ +∞ +∞
ϕ
n=0 m=0 ν =0
( pq )
(r , θ )
( pp) ( pp)
(2m +1)(2n +1)Tn Tm νG (0, m | 0, n | ν )
d 2(3pq) = −16π 2
k 9p
+∞ +∞ +∞
(17)
∑∑∑
kq (k p2 − kq2 ) n=0 m=0 ν =0
ikq r
f pq(3) (θ )
r
kq
.
−iπ / 4
e
e
( 2)
2 f pq (θ )
π kq1/ 2
ikq r
r
(r →+∞)
(20)
(2)
with, in case of a circular cylinder, f pq an even function of
θ if p and q are associated both to either longitudinal or
shear waves, and an odd function if only one is associated
with a longitudinal (or shear) wave.
ν
⎛ ⎞
⎜⎜ ⎟ G (0, m | 0, n | ν )
⎜⎝ kq ⎟⎠
(qp) ( pq) ⎜ k p ⎟
⎟
(2m +1)(2n +1)Tn Tm
e
d = 2,
d 2( p) = −8π 2 k p6 ∑∑∑
3
(r →+∞)
−i
(d )
(d )
(2)
While d1 p , d 2 p , and d 2 pq have all been written [810,1] in terms of the far field form functions defined by
(3)
Eq.(20), we have not managed to do the same for d 2 pq ,
for the spherical case, and to
d1(2p) = −4ik p2
+∞
∑ T(
pp)
n
,
(18)
unless p=1 and obeys Eq.(1) ; this is the reason why we
focus now on the coherent wave that is associated with the
fastest longitudinal wave p=1. We found indeed [2] :
n=−∞
d 2( p) = −8k p4
2
+∞
+∞
∑ ∑
m − nTn( )Tm(
pp
π
f1(q ) (θ ) f q(1 ) (θ )
8π 2
sin θd θ ,(21)
d 21q =
k p ∫0 (k p2 + kq2 − 2k p kq cos θ )3/ 2
pp)
n=−∞ m=−∞
⎛ k ⎞⎟
⎜⎜ p ⎟
d 2 pq = −16 2
⎟
2 ∑ ∑ ⎜
k p − kq n=−∞ m=−∞ ⎜⎝ kq ⎠⎟
k p6
(2)
+∞
+∞
,(19)
m−n
(qp) ( pq)
Tn Tm
(2)
2
while the equivalent term in the d=2 case was [1] :
π
( )
( )
( )
( )
8 f1q (θ ) f q1 (−θ ) + f q1 (θ ) f1q (−θ )
d 21q = ∫
dθ .
π 0
k12 + kq2 − 2k1kq cos θ
2
(2)
for the cylindrical case.
(3)
2
(3)
2
2
2
(22)
In both cases, the first order terms d1 p and d1 p , that
involve no mode conversion, do not depend on the
complexity of the host medium. The second order terms
that exhibit as well no mode conversions had been given by
Lloyd and Berry for spheres in an ideal fluid [8,9], and by
Linton and Martin for cylinders [10,1]. The coupling term
(2)
for cylinders, d 2 pq ,was obtained by Conoir and Norris [1].
In Ref.[2], we used the matrix formalism developed by the
latter to obtain its spherical counterpart.
While Eqs.(15-19) are the most suitable for numerical
computations, the dispersion equations are most often
presented in terms of the far-field scattering functions of the
scatterers. This is the object of next section.
4
Low frequency and low
concentration approximation of the
dispersion equation of the fastest
coherent wave in terms of far-field
scattering functions
Equations.(20) define the far-field scattering functions
(d )
f pq (θ ) from the expression of scattered fields as in
Reducing the d=2 case to circular cylinders whose far
field functions have the parity properties afore mentioned,
and
defining
vectors
k1 .kq = k1kq cos θ , and
k1 = k1 X , kq such that
k1q = k1 − kq , the dispersion
equation of the coherent p=1 wave may be recast from
Eqs.(15-22) into
2
n02
ξ12
(d ) n0
(d ) n0
δ
δ
=
1
+
+
+
11
21
k12
k1d
k12 d k1d
P
( )
δ21
q
q =2
k1d
∑∫
d
Ω( )
d
d Ω (d ) ,
(23)
(d )
with Ω the solid angle in the d-dimensional space.
The first order terms and the uncoupling 2nd order terms
are, as given in Refs.[8-10] (in Ref.[1], all integrals
(2)
corresponding to δ2 p should extend from 0 to π, instead of
2π) :
δ11(3) = −4iπ f11(3) (0)
δ11( ) = −4if11( ) (0)
2
2
,
(24)
π
(3)
δ21 = 4π
2
References
2
d
1
3
f11( ) (θ ) d θ
⎛ θ ⎞ dθ
sin ⎜⎜ ⎟⎟⎟
⎜⎝ 2 ⎠
(
∫
0
)
2
2⎤
⎡
+4π 2 ⎢ f11(3) (0) − f11(3) (π ) ⎥
,
⎢⎣
⎥⎦
π
⎛θ ⎞ d
8
2
2
(2)
f11( ) (θ ) d θ
δ21
= ∫ cot ⎜⎜ ⎟⎟⎟
⎜
⎝ 2⎠ dθ
π 0
(
) (
)
(
(25)
)
and the second order coupling terms may be written, from
Eqs.(21-23), in a way that exhibits the similarities of the
spherical and circular cylindrical cases :
(3)
δ21q = 4π
(2)
21q
δ
f1(q3) (θ ) f q(13) (−θ )
k1d
k1d
( 2)
( 2)
2 f1q (θ ) 2 f q1 (−θ )
=4
π k11/d 2
π k11/d 2
.
(26)
Looking back at Eqs.(24,16), one can only wonder if it
is possible to write Eq.(25) in a similar way, such as
()
δ21
= 4π ∫ x (θ ) f11( ) (θ ) f11( )′ (θ ) dΩ ( )
3
3
3
3
3
Ω( )
( )
δ21
= 4 ∫ x (θ )
2
2
Ω( )
2 ( 2)
2 (2)′
2
f11 (θ )
f11 (θ ) dΩ ( )
π
π
.(27)
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