Multiple scattering by random configurations of circular
cylinders: Reflection, transmission, and effective interface
conditions
P. A. Martina)
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden,
Colorado 80401-1887
(Received 16 August 2010; revised 20 December 2010; accepted 4 January 2011)
In a previous paper, Linton and Martin [J. Acoust. Soc. Am. 117, 3413–3423 (2005)] obtained two
formulas for the effective wavenumber in a dilute random array of circular scatterers. They
emerged from a study of the problem of the reflection of a plane wave at oblique incidence to a
half-space containing the scatterers. Here, their study is extended to obtain formulas for the reflection and transmission coefficients and to investigate the average fields near the boundary of the
C 2011 Acoustical Society of America.
half-space. Comparisons with previous work are made. V
[DOI: 10.1121/1.3546098]
PACS number(s): 43.20.Fn, 43.20.Bi, 43.20.Hq [AMJD]
In recent years, there has been a revival of interest in
methods for estimating the effective wavenumber for propagation through random composites. In this paper, twodimensional acoustic problems are considered, with the
Helmholtz equation, (r2 þ k2)u ¼ 0, outside circular cylinders. Inside each cylinder, there is another Helmholtz
equation, ðr2 þ k02 Þu0 ¼ 0, with transmission conditions
connecting u and u0 across the circular boundaries. It is
assumed that every circle has radius a.
To analyze multiple scattering1 by random configurations of scatterers, the theory given by Linton and Martin2 is
developed further; their paper and formulas taken from it
will be identified by LM below. In LM, formulas for the
effective wavenumber, K, were derived. These formulas take
the form
2
2
K ¼ k þ n0 d1 þ
n20 d2 ;
(1)
where n0 is the number of cylinders per unit area; the area
fraction occupied by the scatterers, / : n0pa2, is assumed to
be small. Explicit formulas for d1 and d2 were given in LM.
They were obtained using the Lax quasicrystalline approximation (QCA), they compare favorably with experiments,3,4
they have been confirmed by an independent method that is
valid for weak scattering5 (see Sec. VIII), and they have
been used to estimate the dynamic effective mass density of
random composites.6 The LM formulas are accurate to second order in /. They require the solution of a scattering
problem for one scatterer; this scalar (transmission) problem
is discussed briefly in Sec. II.
The basic problem considered in LM consists of circular
scatterers distributed randomly in the (right) half-plane x > 0
with a plane wave incident obliquely from the (left)
a)
Author to whom correspondence should be addressed. Electronic mail:
pamartin@mines.edu
J. Acoust. Soc. Am. 129 (4), April 2011
half-plane x < 0. The goal of LM was to estimate the effective wavenumber, K, within the right half-plane. No information on the reflected and transmitted wavefields was given.
Here, we show that these wavefields can be determined by
extending the analysis in LM.
There is a specularly reflected plane wave. The (average) reflection coefficient is found to be
R ¼ n0 ði=k2 Þf ðp 2hin Þ þ Oðn20 Þ;
(2)
where hin is the angle of incidence and f(h) is the far-field
pattern for scattering by one cylinder. An explicit expression
for the Oðn20 Þ contribution is also obtained, see Eq. (42).
The average field transmitted to the right half-plane has
a more complicated form. For example, at normal incidence,
it is found to be
eiKx þ n0 pa2 A1 eiKx eikx þ Oðn20 Þ;
(3)
where an explicit formula for A1 is obtained, see Eq. (52).
To verify these results, comparisons with the independent
weak-scattering results in Ref. 5 are made, and agreement is
found.
In Sec. IX, the behavior of the fields across x ¼ 0 is
examined. It is found that the fields themselves are continuous but the slopes are discontinuous. An estimate for the
slope discontinuity is derived. This is then used to estimate
the effective density of the right half-plane.
II. LINTON–MARTIN (LM) FORMULATION
The LM approach (described in Sec. IV of LM2) starts
with an exact multiple-scattering method for an arbitrary
deterministic arrangement of N identical circular scatterers.
The total field outside the cylinders, u, is written as LM(47),
u ¼ uin þ
N X
1
X
Ajn Zn Hn ðkrj Þ einhj ;
(4)
j¼1 n¼1
0001-4966/2011/129(4)/1685/11/$30.00
C 2011 Acoustical Society of America
V
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Author's complimentary copy
I. INTRODUCTION
Pages: 1685–1695
for rj > a, j ¼ 1, 2, …, N, where (rj, hj) are the polar coordinates centered at rj ¼ (xj, yj), the center of the jth cylinder.
The incident field is taken as a plane wave at oblique incidence, LM(24), so that
uin ¼ eiðaxþbyÞ
with
a ¼ k cos hin ; b ¼ k sin hin :
(6)
with
Dn ¼ ðq0 =qÞHn0 ðkaÞJn ðk0 aÞ ðk0 =kÞHn ðkaÞJn0 ðk0 aÞ:
(7)
Here, k0 is the interior wavenumber, q0 is the interior density, and q is the exterior density. The interface conditions
are LM(50),
u ¼ u0
1 @u
1 @u0
¼
q @r q0 @r
and
1
X
Zn ¼ 4if ð0Þ;
d2 ¼ 4pib2
1
1
X
X
Zm Zn dmn ðkbÞ;
(14)
n¼1 m¼1
where b is the “hole radius” introduced in the conditional
averaging to prevent cylinders overlapping, the function
dn(x) is defined by LM(73),
dn ðxÞ ¼ Jn0 ðxÞHn0 ðxÞ þ ½1 ðn=xÞ2 Jn ðxÞHn ðxÞ;
(15)
and Jn is a Bessel function. Equation (13) is the well-known
Foldy–Lax estimate. The second-order contribution, Eq.
(14), is approximated further in LM by taking the limit kb
! 0; the result is LM(7), a formula that is reminiscent of the
Lloyd–Berry formula for analogous three-dimensional
problems.7,8
Concerning the dependence of Fn on n0, the analysis in
Sec. IV C of LM shows that
Fn ¼ F þ n0 qn þ Oðn20 Þ;
on r ¼ a;
(13)
n¼1
(5)
The coefficients Ajn are to be found, Hn Hnð1Þ is a Hankel
function and Zn, which characterizes scattering by one isolated cylinder, is given by LM(49),
Zn ¼ ðRe Dn Þ=Dn ¼ Zn
d1 ¼ 4i
(8)
qn ¼ Q~ þ pb2 F
1
X
(16)
Zm dmn ðkbÞ;
(17)
m¼1
where u0 is the field inside the cylinder; these are appropriate
for a fluid cylinder surrounded by a different fluid, so that u
is the pressure. The far-field pattern, f, is given by LM(53) as
1
X
Zn einh ¼ Z0 2
n¼1
1
X
Zn cos nh:
(9)
III. DEPENDENCE ON ‘
n¼1
In LM, an exact linear system is derived for Ajn . Then, ensemble averages are taken, followed by letting N ! 1 and
imposition of the Lax QCA. (All the cylinders are in the
right half-plane, x > 0.) The key quantity is hAjn ij , the average of Ajn conditional on there being a cylinder at rj. It is
expressed as LM(56),
hAjn ij
n ibyj
¼i e
Un ðxj Þ;
Un ðxÞ ¼ Fn einu eikx
xj > 0:
Cn ð‘Þ ¼
ð‘
1
2n0 X
Zn einhin Cn ð‘Þ;
a n¼1
Un ðtÞeiat dt þ
for x > ‘;
and
b ¼ K sin u ¼ k sin hin :
(19)
This relation was not used in LM. As the main results in
LM do not depend on ‘, we shall assume that a continuity
argument can be used to assert that Eq. (11) should hold for
all ‘ > 0, leading to a useful simplification. So, setting ‘ ¼ 0
in Eq. (18) gives
(11)
1 ¼
1
2in0 X
Fn Zn einðhin uÞ :
aðk aÞ n¼1
(20)
This equation will be used in Sec. IV to obtain information on Fn.
(12)
The coefficients Fn were then shown to solve an infinite
homogeneous system of linear algebraic equations, LM(71).
Analysis of this system led to formulas for d1 and d2 in Eq.
(1), LM(77) and LM(80), respectively,
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
iFn einu iðkaÞ‘
e
:
ka
(18)
(10)
where the length ‘ is not (and need not be) specified; later,
we shall take ‘ ¼ 0 (see Sec. III). The quantities K and u are
defined by LM(31),
1686
1 ¼ B ¼
0
Then, in order to avoid possible difficulties near the
“interface” at x ¼ 0, Linton and Martin assumed that Un
could be written as LM(58),
k ¼ K cos u
On p. 3419 of LM, it is shown that B ¼ 1,
IV. INFERENCES FROM K
Given the LM formula for K, Eq. (1) with Eqs. (13) and
(14), estimates for u and k can be obtained. To begin,
rewrite Eq. (1) as
P. A. Martin: Scattering by random arrays of cylinders
Author's complimentary copy
f ðhÞ ¼ where F and Q~ are n0-independent constants. The values of
these two constants were not determined; this will be done in
Sec. IV.
ðK=kÞ2 ¼ 1 þ /j1 þ /2 j2 ;
(21)
where f 0 (0) ¼ 0 has been used [see Eq. (9)] and the function
K is defined by
where / ¼ pa2n0, and
pðkaÞ2
¼
4i f ð0Þ
pðkaÞ2
and
j2 ¼
d2
(22)
ðpka2 Þ2
are dimensionless. Ignoring terms that are O(/3) as / ! 0,
Eq. (21) gives K=k ¼ 1 þ 12 /j1 þ 18 /2 ð4j2 j21 Þ.
Put C ¼ cos hin, S ¼ sin hin, and T ¼ tan hin. To solve Eq.
(12)2, (K/k)sin u ¼ S, for u, write
2
u ¼ hin þ /p1 þ / p2 :
(23)
Then
2
2
S ¼ ðK=kÞ S cosð/p1 þ / p2 Þ þ C sinð/p1 þ / p2 Þ
1 2 2
2
¼ ðK=kÞ S 1 / p1 þ Cð/p1 þ / p2 Þ
2
1
¼ S þ /ðSk1 þ 2Cp1 Þ
2
1 2
þ / ½Sð4j2 j21 Þ þ 4Cp1 j1 þ 8Cp2 4Sp21 :
8
Hence,
1
p1 ¼ Tj1
2
and
1
p2 ¼ T½ð3 þ T 2 Þj21 4j2 : (24)
8
Next, calculate k ¼ K cos u, using Eqs. (23) and (24),
1
1
k=k ¼ C þ /j1 C1 þ /2 ð4j2 C1 j21 C3 Þ:
2
8
This approximation is used in Eq. (20). Thus,
2in0
4i
j1
j2
:
¼
1þ/
aðk aÞ pðkaÞ2 j1
4C2 j1
Zm Zn dmn ðkbÞ eind ;
(29)
n¼1 m¼1
so that Kð0Þ ¼ j2 [see Eqs. (14) and (22)]. (Note that
limkb!0 KðdÞ can be expressed as an integral, similar to that
for Kð0Þ as in Sec. IV C of LM.) Substituting Eqs. (26) and
(28) in Eq. (20) gives
j1
j2
j2
1þ/ Qþ
1 ¼ F 1 þ /
4C2 j1
j1
h
i
j1
¼ F 1 þ / Q þ 2 þ Oð/2 Þ:
4C
Thus,
F ¼ 1
Q¼
and
j1
i f ð0Þ
¼
:
2
4C
pðkaÞ2 C2
Substituting back in Eq. (28) gives
1
X
Fn Zn e
inðhin uÞ
n¼1
i
j1
j2
2
¼ pðkaÞ j1 1 /
4C2 j1
4
þ Oð/2 Þ:
(30)
This approximation will be used in Sec. VII.
V. RESULTS FROM FOLDY THEORY
(26)
Classical Foldy theory assumes isotropic scattering. The
effective wavenumber is given by LM(1),
K 2 ¼ k2 4ign0 ;
(27)
2
~
where Q ¼ Q=ðpa
FÞ, a dimensionless constant. Hence
1
1 X
Fn Zn einðhin uÞ
F n¼1
1
1
X
X
Zn einðhin uÞ
Zn einðhin uÞ þ /Q
¼
(31)
where g is the (dimensionless) scattering coefficient. For the
scattering problem, we have
huðx; yÞi ¼
eiðaxþbyÞ þ ReiðaxþbyÞ ;
AeiðkxþbyÞ ;
x < 0;
x > 0;
(32)
where R is the (average) reflection coefficient and A is the
(average) transmission coefficient.
Foldy theory (see Sec. III A in LM) predicts that R ¼ RF
and A ¼ AF , where
n¼1
n¼1
2
þ/
pðkaÞ2
(25)
From Eqs. (16) and (17),
1
X
Fn
Zm dmn ðkbÞ;
¼ 1 þ /Q þ /ðb=aÞ2
F
m¼1
1
1
X
4iðb=aÞ2 X
pðkaÞ
Kðhin uÞ
4i
1
¼ f ðhin uÞ /Qf ð0Þ i/pðkaÞ2 Kð0Þ
4
1
0
¼ ff ð0Þ /p1 f ð0Þg /Qf ð0Þ i/pðkaÞ2 j2
4
1
1
1
2
2
¼ ipðkaÞ j1 i/QpðkaÞ j1 i/pðkaÞ2 j2
4
4
4
pj1
j2
2
(28)
;
ðkaÞ 1 þ / Q þ
¼
4i
j1
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
RF ¼
ak
kþa
and
AF ¼
2a
¼ 1 þ RF :
kþa
(33)
From these formulas and Eq. (32), it follows that both hui
and (@/@x)hui are continuous across x ¼ 0.
Working to first order in /, u ¼ hin þ /p1, p1 ¼ 2igT/
[p(ka)2], k ¼ a /kp1/S, and Eq. (33) gives
RF ¼
i/g
pðkaÞ2 C2
¼
in0 g
a2
and
AF ¼ 1 þ R F :
P. A. Martin: Scattering by random arrays of cylinders
(34)
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Author's complimentary copy
j1 ¼
d1
KðdÞ ¼
If we put g ¼ f(0), which is the correct non-isotropic
extension of Eq. (31), it turns out that Eq. (34) does not give
the correct estimates for R or A, as we show below. [See the
text above Eq. (43) for R and the text below Eq. (52) for A:]
VI. THE AVERAGE REFLECTED FIELD
Calculating the ensemble average of Eq. (4) for x < 0 gives
ðð
1
N X
n0 X
Zn hAjn ij Hn ðkrj Þ
N j¼1 n¼1
einhj dxj dyj :
(35)
Here, x xj ¼ rj cos hj and y yj ¼ rj sin hj. Then, use
the indistinguishability of the scatterers, let N ! 1, and use
Eq. (10):
huðx; yÞi ¼ uin þ n0
1
X
in Zn
n¼1
ð1
ð1
Un ðx1 Þ
with f ðp hin uÞ ¼ f ðhre Þ þ 12 /Tj1 f 0 ðhre Þ þ Oð/2 Þ and
h ¼ p 2hin.
As F ¼ 1, Eq. (39) gives
R ¼ /R1 þ /2 R2 ;
where
pðkaÞ2 C2
The inner integral can be evaluated. In LM, it is shown
j1 0
i
Tf ðhre Þ þ pðkaÞ2 Kðhre Þ
2 2 2
4
pðkaÞ C
i
j1
2 f ðhre Þ :
2C
i
R2 ¼
¼
ð1
eibY Hn ðkRÞeinH dY
1
n inhin iaX
e
ð2=aÞðiÞ e
ð2=aÞin einhin eiaX ;
; X > 0;
X < 0;
ð1
1
2n0 iby X
ð1Þn Zn
e
a
n¼1
Un ðx1 Þeinhin eiaðx1 xÞ dx1 :
(38)
0
Hence, comparison with Eq. (32) gives
ð1
1
2n0 X
n inhin
R¼
Zn
ð1Þ e
Un ðx1 Þeiax1 dx1
a n¼1
0
1
2in0 X
Fn Zn einðphin uÞ :
¼
aðk þ aÞ n¼1
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
Bose10 has obtained similar formulas [see Eq. (24) in
Ref. 10] for a slab, 0 < x < H. They involve the quantity
e2ikHC, so that the limit H ! 1 cannot be taken. It appears
that this is due to the use of a “Born-type approximation” at
an early stage in the analysis.
The estimate R ¼ /R1 agrees with the Foldy estimate,
RF [see Eq. (34)], but only for isotropic scattering [where
f(h) does not depend on h]: particularly, using g ¼ f(0) gives
an incorrect result.
At normal incidence, we have hin ¼ 0, hre ¼ p, C ¼ 1,
and T ¼ 0, whence
R1 ¼
(39)
Le Bas et al.9 have obtained a formula for R for a slab,
0 < x < d; letting d ! 1 in their Eq. (41) gives agreement
with Eq. (39). The same paper9 also contains Eq. (20); see
their Eq. (31).
Equation (39) is an exact formula for the reflection coefficient. It can be used to approximate R for small /. From
Eq. (25),
2in0
i/
/j1
¼
1
:
aðk þ aÞ pðkaÞ2 C2
4C2
(42)
(37)
where a and b are defined by Eq. (5), X ¼ R cos H and
Y ¼ R sin H. Comparison with the inner integral in Eq. (36)
shows that we should take Y ¼ y1 y, R ¼ r1 , H ¼ h1 + p,
and X ¼ x1 x. As x < 0 and x1 > 0, X > 0 and so
huðx; yÞi ¼ uin þ
(41)
C ¼ cos hin, and
that
Ln ðXÞ ¼
;
(36)
1
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i f ðhre Þ
R1 ¼
0
eiby1 Hn ðkr1 Þ einh1 dy1 dx1 :
1
1
X
X
Fn inðphin uÞ
Zn e
ð1 þ /QÞZn einðphin uÞ
¼
F
n¼1
n¼1
1
i/pðkaÞ2 Kðp hin uÞ
4
¼ f ðp hin uÞ /Qf ðhre Þ
1
i/pðkaÞ2 Kðhre Þ;
4
(40)
i f ðpÞ
pðkaÞ
2
and
R2 ¼ KðpÞ 2f ð0Þf ðpÞ
;
2
4
½pðkaÞ2 (43)
where we have used Eq. (22) for j1. The formula for R1
agrees with an estimate from Angel et al.11 (see Appendix A
for comparisons with the work of Aristégui, Angel, and their
colleagues).
VII. THE AVERAGE TRANSMITTED FIELD
The field in the region to the right of x ¼ 0 is given by
Eq. (4) outside the cylinders. Inside the jth cylinder, the field,
uj, is given by LM(48),
uj ¼
1
X
Bjn Jn ðk0 rj Þ einhj ;
rj < a:
(44)
n¼1
P. A. Martin: Scattering by random arrays of cylinders
Author's complimentary copy
huðx; yÞi ¼ uin þ
Also,
The fact that there are different expansions in different
regions makes the calculation of hu(x, y)i for x > 0 less
straightforward than when x < 0.
In Appendix B, it is shown that
Combining Eqs. (26) and (30) shows that
P ¼ 1 þ Oð/2 Þ;
there is no linear term in /. Thus,
huðx; yÞi ¼ ð1 /Þuin þ huðx; yÞiext þ huðx; yÞiint ; (45)
ð1 /Þuin þ PeiðaxþbyÞ ¼ /eiðaxþbyÞ þ Oð/2 Þ
as / ! 0:
for x > a, where
1
X
Zn
n¼1
inh1
e
hA1n i1 Hn ðkr1 Þ
dx1 dy1 ;
1 ðð
X
huðx; yÞiint ¼ n0
x1 >0; r1 >a
n¼1
r1 <a
(46)
hB1n i1 Jn ðk0 r1 Þeinh1 dx1 dy1 :
(47)
(As x > a, the disc r1 < a lies in x1 > 0.)
For Qext , use K2 k2 ¼ k2 a2, Eqs. (26) and (40), and
a ¼ kC to obtain
2pn0
2
j2
þ Oð/2 Þ:
1
/
¼
k2 K 2
j1
ðkaÞ2 j1
From LM(72),
1
2
N n ðKa;kaÞ ¼ ð2i=pÞ 1 i/pðkaÞ j1 dn ðkaÞ þ Oð/2 Þ;
4
where dn is defined by Eq. (15). Then, using Eqs. (27) and (29),
1
X
A. Calculation of hu iext
n¼1
From Eqs. (10) and (11),
hA1n i1 ¼ in Fn einu eiðkx1 þby1 Þ :
(48)
Substitution in Eq. (46) gives
huðx; yÞiext ¼ n0
1
X
n¼1
in Fn Zn einu
n¼1
ðð
eiðkx1 þby1 Þ Hn ðkr1 Þ einh1 dx1 dy1 :
x1 >0; r1 >a
The double integral is similar to Mn in Sec. IV B of LM.
Its value is found to be
2iðiÞn iðaxþbyÞ inhin 2pðiÞn iðkxþbyÞ inu
e
e
e N n ðKa;kaÞ;
þ 2
e
aðk aÞ
k K2
where
N n ðKa; kaÞ ¼ kaHn0 ðkaÞJn ðKaÞ KaHn ðkaÞJn0 ðKaÞ:
Hence,
huðx; yÞiext ¼ PeiðaxþbyÞ þ Qext eiðkxþbyÞ ;
where
1
2in0 X
P¼
Fn Zn einðhin uÞ ;
aðk aÞ n¼1
1
2pn0 X
Fn Zn N n ðKa; kaÞ:
2
2
k K n¼1
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
1
X
Qext ¼ 1 þ /Q þ /
Zn dn ðkaÞ þ Oð/2 Þ:
n¼1
B. Calculation of huiint
Next, consider huiint, defined by Eq. (47) in terms of the
coefficients B1n in Eq. (44). In Sec. IV A of LM, a linear system for Ajn was obtained by applying the pair of transmission
conditions, Eq. (8), on each cylinder followed by elimination
of Bjn . Those calculations also yield a simple relation
between Ajn and Bjn , namely
Bjn ¼ cn Ajn ;
Hence
Qext ¼
1
2 X
Zn ð1 þ /QÞ
ip n¼1
h
i /
i
1 /pðkaÞ2 j1 dn ðkaÞ ðkaÞ2 j2
4
2
1
j
2
¼ ðkaÞ2 j1 1 þ /Q þ /
j1
2
1
X
Zn dn ðkaÞ :
þ/
Fn Zn N n ¼
with
cn ¼
2ðq0 =qÞ
;
pikaDn
for Dn, see Eq. (7). Using this relation and Eq. (48) in Eq.
(47) gives
huðx; yÞiint ¼ n0
1
X
in Fn cn einu
n¼1
ðð
eiðkx1 þby1 Þ Jn ðk0 r1 Þ einh1 dx1 dy1
r1 <a
¼ n0 eiðkxþbyÞ
1
X
Fn cn In ¼ Qint eiðkxþbyÞ ;
n¼1
P. A. Martin: Scattering by random arrays of cylinders
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Author's complimentary copy
huðx; yÞiext ¼ n0
ðð
say, where
n inu
ð a ð 2p
eiKRcosðHuÞ Jn ðk0 RÞeinH RdHdR
In ¼ ðiÞ e
0 0
ða
¼ 2p Jn ðKRÞJn ðk0 RÞRdR
0
¼ 2pðK 2 k02 Þ1 Mn ðKa;k0 aÞ
paper and formulas taken from it will be identified by MM
below. Particularly, MM confirms the LM formula for K2
and it contains an estimate for the transmitted field when
hin ¼ 0, MM(5.25),
hui ¼ AMM eiKx
Mn ðKa; k0 aÞ ¼ k0 aJn0 ðk0 aÞJn ðKaÞ KaJn ðk0 aÞJn0 ðKaÞ:
(49)
where
H¼
Hence,
1
X
cn
n¼1
Mn ðka; k0 aÞ
2
ðk0 aÞ ðkaÞ
(53)
1
1
AMM ¼ 1 þ m0 / þ m20 fP0 pðkaÞ2 Hg/ þ Oð/2 Þ;
4
4
(54)
and
Qint ¼ 2/
with
2
1
i X
J n dn ;
4 n¼1
(55)
J n ¼ Jn2 Jn1 Jnþ1 ¼ Jn02 f½n=ðkaÞ2 1gJn2 ;
þ Oð/2 Þ:
4P0 ¼ ðkaÞ2 þ 2piðkaÞ2
1
X
J n ðJn Hn dn Þ;
(56)
(57)
n¼1
C. Synthesis
huðx; yÞi ¼ A eiðkxþbyÞ / eiðaxþbyÞ ;
(50)
A ¼ 1 þ /A1 þ Oð/2 Þ;
(51)
with
A1 ¼
i f ð0Þ
1
X
Zn dn ðkaÞ
pðkaÞ2 C2 n¼1
1
X
Mn ðka; k0 aÞ
cn
þ2
:
ðk0 aÞ2 ðkaÞ2
n¼1
þ
(52)
and all functions have argument ka. (MM also contains a formula for the O(/2) correction to AMM .) Here, Eq. (54) will
be compared with the estimate found in Sec. VII, Eq. (50)
with Eqs. (51) and (52).
From MM(2.24), we have an estimate for f(0) that can
be used in the first term in Eq. (52) (with C ¼ 1),
T1 i f ð0Þ
pðkaÞ2
1
1
¼ m0 m20 pðkaÞ2 H;
4
4
these contributions can be seen in Eq. (54).
From MM(2.18), we have an estimate for Zn that can be
used in the second term in Eq. (52),
T2 1
X
Zn dn
The first term in Eq. (52) constitutes the Foldy estimate [see
AF , given by Eq. (34)] if we take g ¼ f(0), but it is seen here
that the correct estimate of A at O(/) contains two additional terms.
Notice that the dependence of A1 on the angle of incidence appears only in the first term, via C ¼ cos hin.
The formula for the transmitted field, Eq. (50) with Eqs.
(51) and (52), is surprisingly complicated, especially as it is
only first order in /. (Indeed, the analysis above does not
give any information on the O(/2) contribution, unlike in
Sec. VI where we obtained the second-order contribution to
R.) Fortunately, we can check our calculations with an independent analysis that is valid for weak scattering: we do this
next.
where ln ¼ p(ka)2dn and Sn ¼ 2kaJn1Jn+1.
The third term in Eq. (52), denoted by T3, is more complicated. To begin, MM(2.12) and MM(2.14) give
VIII. WEAK SCATTERING
where Un ¼ 2ka(JnHn dn) + 2[i=(pka)][n2 (ka)2]. Then,
as (k0a)2 (ka)2 ¼ m0(ka)2 and Mn ðka; kaÞ ¼ 0, it is necessary to expand Mn ðka; k0 aÞ to third order in m0. Thus,
from Eqs. (6), (7), and (49),
The term “weak scattering” means here that
q ¼ q0
and
jm0 j 1;
where
m0 ¼ 1 ðk0 =kÞ2 :
Martin and Maurel5 (MM) have given results for weak
scattering, correct to second order in both / and m0; their
1690
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
n¼1
¼ m0 pðkaÞ2 H 1
m20 X
pkafiSn kaln J n gdn ;
16 n¼1
(58)
cn ¼ 2=ðpikaDn Þ
1
1
¼ 1 þ im0 ln þ m20 ðl2n pikaUn Þ;
4
16
(59)
Mn ðka;k0 aÞ ¼ kaReDn
1
1
1 2
2
¼ m0 ðkaÞ J n m0 Sn =ðkaÞþ m0 Xn ; (60)
2
4
16
P. A. Martin: Scattering by random arrays of cylinders
Author's complimentary copy
Substituting the results for huiext and huiint back in Eq.
(45) then gives the transmitted field as
2
4
Xn ¼ ð8 þ n2 z2 ÞJn02 þ zJn Jn0
3
3
2
2
2
2
fðn z Þðn z2 þ 8Þ þ 8z2 gz2 Jn2 ;
3
(61)
and we have written z : ka. Then, using Eqs. (59) and (60),
the third term in Eq. (52) becomes
1 X
i
m2
1 m0 ln 0 ðl2n pikaUn Þ
4
16
n¼1
2
m0
m0
Jn Sn þ
Xn
4ka
16
1
1 X
m0 X
1
Jn iln J n þ Sn
¼
4 n¼1
ka
n¼1
1 2 X
m0
Sn
2
iln J n ln þ pikaJ n Un þ Xn :
þ
16 n¼1
ka
T3 ¼
P
P
As n J n ¼ 1 and n Sn ¼ 0 [see Sec. 2.2 of MM or
Eq. (C1)],
T3 ¼ 1 m0 pðkaÞ2 H
1 m2 X
Sn
m2
iln J n l2n þ 0 S 1 ;
þ 0
ka
16 n¼1
16
(62)
P
where S 1 ¼ n ðpikaJ n Un þ Xn Þ and we have used Eq.
(55). Substituting for Un and comparison with Eq. (57) gives
S 1 ¼ 4P0 + S 2, where
S 2 ¼ ðkaÞ2 þ
1
X
f2½ðkaÞ2 n2 J n þ Xn g
n¼1
1
X
1
¼ ðkaÞ2 þ
Xn ;
2
n¼1
(63)
after use of Eq. (C2). Hence, adding Eqs. (58) and (62),
1
1
T2 þ T3 ¼ 1 þ m20 P0 þ m20 S 2 :
4
16
It is shown in Appendix C that S 2 ¼ 0. Thus, for weak
scattering and normal incidence,
/A1 ¼ /ðT1 þ T2 þ T3 Þ ¼ ðAMM 1Þ þ /;
which gives
hui ¼ AMM eiKx þ /ðeiKx eikx Þ:
(64)
This shows agreement with the MM estimate, correct to
first order in / and second order in m0. Note that the method
used in MM is based on an iterative solution of the governing Lippmann–Schwinger equation. It leads to an expression
of the form
ikx
hui ¼ e ðpolynomial in xÞ;
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
which is then set equal to AMM eiKx ; expanding about x ¼ 0
leads to expressions for both K and AMM . If this process is
applied to Eq. (64), it is easily seen that the last term is
O(/2) and so should be ignored if the goal is to determine
the amplitude correct to first order in /.
IX. EFFECTIVE INTERFACE CONDITIONS
In this section, the fields near the “interface” at x ¼ 0 are
investigated, working to first order in /.
The average total field in x < 0, evaluated at x ¼ 0, u, is
u ¼ ð1 þ /R1 Þeiby
and the corresponding x-derivative, u0 , is
u0 ¼ ikCð1 /R1 Þeiby :
From Eq. (50), the transmitted field at x ¼ d, say, u+, is
uþ ¼ fð1 þ /A1 ÞeiðkaÞd /geiad eiby
¼ f1 þ iðk aÞd þ /A1 /geiad eiby
1
¼ 1 þ / ikdj1 =C þ A1 1 eiad eiby
2
(65)
and the corresponding x-derivative, u0þ , is
n
o
u0þ ¼ ikC ðk=aÞð1þ/A1 ÞeiðkaÞd / eiad eiby
1
2 1
¼ ikC 1þ/ j1 =C þ ikdj1 =CþA1 1 eiad eiby :
2
2
(66)
To estimate these quantities, suppose further that ka 1
and k0a 1. Then, the terms containing kdj1 in Eqs. (65)
and (66) are smaller than the other terms (see below): ignoring them and letting d ! 0 gives
uþ u ¼ /ðA1 R1 1Þeiby ;
(67)
1
u0þ u0 ¼ /ikC A1 þ R1 þ j1 =C2 1 eiby :
2
(68)
These give the discontinuities in hui and its normal derivative across x ¼ 0.
For small ka and k0a, Z0 and Z+1 are dominant, in general, and they are O((ka)2) (see Sec. III A in Ref. 6). Thus,
f ðhÞ ’ Z0 2Z1 cos h.
From Eq. (22), j1 ¼ 4if(0)=[p(ka)2] ¼ O(1) as ka ! 0,
which justifies discarding the kdj1 terms above.
From Eq. (41), f(hre) ¼ f(p 2hin) is needed to calculate
R1. As cos hre ¼ 1 2C2,
R1 ¼
¼
i
pðkaÞ2 C2
i f ð0Þ
pðkaÞ
2
C2
½Z0 2Z1 ð1 2C2 Þ
þ
4iZ1
pðkaÞ2
:
P. A. Martin: Scattering by random arrays of cylinders
1691
Author's complimentary copy
where the first two terms can be found below MM(2.15),
Mn ðka; k0 aÞ ðkaÞn ðk0 aÞn
ðkaÞ2 ðk0 aÞ2
;
22nþ1 n! ðn þ 1Þ!
n 0;
with Mn ¼ Mn . Also, c0 1 and cn 2(k/k0)n for
n > 0, with cn ¼ cn. Hence, the dominant contribution to
the third term in Eq. (52) comes from n ¼ 0; as
M0 ðka; k0 aÞ 12 ½ðkaÞ2 ðk0 aÞ2 ,
A1 ¼
i f ð0Þ
pðkaÞ
2
C2
þ
4iZ1
pðkaÞ2
þ 1:
Use of these approximations for R1 and A1 gives
A1 ¼ 1 þ R 1 ;
(69)
so that Eq. (67) gives uþ u ¼ O(/2). In other words, there
is no discontinuity in hui across x ¼ 0, for any angle of
incidence.
Similarly, from Eq. (68),
u0þ u0 ¼ /kC
8Z1
pðkaÞ
2
eiby 2/ikC
q0 q iby
e ; (70)
q0 þ q
using Eq. (23) from Ref. 6. Thus, at this level of approximation, there is a jump in the normal (x) derivative of hui across
x ¼ 0 (unless q0 ¼ q). Moreover, for normal incidence, it is
seen that Eq. (70) agrees with the estimates of Aristégui and
Angel12 [see Eq. (A3)], in the low-frequency, small-/ limit.
As a reviewer noted, the discontinuity in slope at x ¼ 0
could be used to predict the effective density, qeff, of the
effective medium occupying x > 0:
0
q1 u0 ¼ q1
eff uþ :
(71)
0
Using the estimates for u6
given above,
qeff 1 þ /ðR1 þ 12 j1 =C2 Þ
j1 1 þ / 2R1 þ 2 :
¼
1 /R1
2C
q
penetrates [the second term on the right-hand side of Eq.
(50)]. This result was checked by comparing with an independent calculation, valid for weak scattering (and normal
incidence). Effective interface conditions at the boundary of
the half-space were also obtained. It is anticipated that extensions to three-dimensional problems can be made.
APPENDIX A: ARISTÉGUI AND ANGEL
Aristégui, Angel, and their colleagues have written several papers11–14 in which waves are normally incident on a
finite slab, h < y2 < h, containing circular scatterers. Comparisons with their work, in the limit of a semi-infinite slab,
will be given here.
To begin, write the averaged field as
8
< u0 eiky2 þ u0 R0 eiky2 ; y2 < h;
Uðy2 Þ ¼ Cþ eiKy2 þ C eiKy2 ; h < y2 < h;
:
y2 > h:
u0 T 0 eiky2 ;
Put x ¼ y2 + h, U(y2) ¼ u(x) and u0 ¼ eikh:
8
x < 0;
< eikx þ R0 e2ikh eikx ;
uðxÞ ¼ Cþ eiKh eiKx þ C eiKh eiKx ; 0 < x < 2h;
: 0 ikx
Te ;
x > 2h:
Aristégui and Angel13 have given expressions for R0 , C6
and T 0 . Using these gives
R R0 e2ikh ¼ e2iðkKÞh 1 e4iKh ðk2 K 2 Þ=D;
A Cþ eiKh ¼ 2ke2iðkKÞh ðK þ kÞ=D;
A C eiKh ¼ 2ke2iðkKÞh e4iKh ðK kÞ=D;
and T0 ¼ 4kK=D, where
D ¼ e2iðkKÞh fðk þ KÞ2 ðk KÞ2 e4iKh g:
Letting h ! 1 (using Im K > 0) gives
uðxÞ ¼
Substituting for R1 and j1 gives
qeff =q 1 þ 8i/Z1 =½pðkaÞ2 1 2/ðq q0 Þ=ðq þ q0 Þ;
in agreement with Ament’s formula for the effective density;
see Eq. (11) in Ref. 6. This agreement provides a further
check on the calculations.
X. CONCLUSIONS
A plane wave is incident on a half-space containing a
dilute random arrangement of identical scatterers. An
expression for the average reflection coefficient has been
derived: it involves the far-field pattern for a single scatterer.
The average field within the half-space has also been calculated: it is found that a small amount of the incident wave
1692
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
eikx þ Reikx ; x < 0;
x > 0;
AeiKx ;
(A1)
where
R¼
1H
;
1þH
A¼
2
¼1þR
1þH
and
H¼
K
:
k
These agree with the Foldy estimates, Eq. (34), when hin ¼ 0
and K2 ¼ k2 4ign0.
In later papers,11,14 formulas for non-isotropic scattering
were obtained, using K 2 ¼ k2 4in0 f ð0Þ þ Oðn20 Þ. In
particular,11
1
H ¼ ðK=kÞ 1 ð2in0 =k2 Þ½f ð0Þ f ðpÞ
¼ 1 ð2in0 =k2 Þf ðpÞ þ Oðn20 Þ;
giving
P. A. Martin: Scattering by random arrays of cylinders
Author's complimentary copy
For A1 , use Eq. (52), containing three terms. For the sec2
ond term,
P use dn(ka) 2i|n|=[p(ka)2 ] [see above LM(82)] to
obtain n Zn dn ðkaÞ 4iZ1 =½pðkaÞ . For the third term, use
R¼
in0
f ðpÞ
k2
and
A¼1þR¼1þ
in0
f ðpÞ:
k2
(A2)
This expression for R agrees with Eq. (43) but the estimate for A is incorrect.
Evidently, Eqs. (A1) and (A2) show that u(x) is continuous across x ¼ 0, whereas
u0 ð0þÞ u0 ð0Þ ¼ iKA ikð1 RÞ
¼ iðK kÞ þ iðK þ kÞR
’ ikfðK=kÞ 1 þ 2Rg
’ ð2n0 =kÞff ð0Þ f ðpÞg:
(A3)
These results are consistent with those found in Sec. IX.
APPENDIX B: SOME ENSEMBLE AVERAGING
In this appendix, notation from Sec. II of LM2 is used.
Start with N scatterers located at r1, r2, …, rN; denote this
configuration by KN. The ensemble average of any quantity
F(r | KN) is defined by LM(8),
ð
hFðrÞi ¼
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1N ;
The first term in this expression involves integration for
which both r1 > a and r2 > a. In that case, an expansion of
the following form is available [cf. Eq. (4)],
FðrjKN Þ ¼ F0 ðrÞ þ
’
pðr2 ; …; rN jr1 Þ FðrjKN Þ dV2N ;
where p(r1, r2, …, rN) ¼ p(r1) p(r2, …, rN j r1) defines
p(r2, …, rN j r1) and p(r) ¼ n0=N.
For clarity, suppose first that N ¼ 2. Equation (B1)
reduces to
hFðrÞi ¼
ðð
¼
ðð
r2 >a r1 >a
F0 ðrÞðn0 =2Þ2 ½ð2=n0 Þ
þ
(B1)
ðN1Þ
(B4)
with N ¼ 2, where F0 does not depend on KN and F1, … ,FN
are small, O(n0). Hence
ð
ð
pðr1 ; r2 Þ FðrjK2 Þ dV12
r2 >a r1 >a
ð
ð
¼ F0 ðrÞ
pðr1 ; r2 Þ dV12
r2 >a r1 >a
ð
ð
þ
pðr1 ; r2 Þ F1 dV21
r1 >a r2 >a
ð
ð
þ
pðr1 ; r2 Þ F2 dV12
þ
ð
pa2 2
pðr1 ; r2 Þ F1 dV21
r >a
ð
pðr1 ; r2 Þ F2 dV12
r2 >a
ð
ð1
’ ð1 n0 pa2 ÞF0 ðrÞ
ð
ð
n0
n0
þ
hF1 i1 dV1 þ
hF2 i2 dV2
2 r1 >a
2 r2 >a
ð
hF1 i1 dV1 ;
¼ ð1 n0 pa2 ÞF0 ðrÞ þ n0
where the subscript (N) indicates that the integration is over
N copies of the region BN containing N scatterers, and
dV1N ¼ dV1 dVN. (BN has area N=n0.) Similarly, the average of F(r j KN) over all configurations for which the first
scatterer is fixed at r1 is given by LM(9),
hFðrÞi1 ¼
Fj ðrjKN Þ;
j¼1
ðNÞ
ð
N
X
r1 >a
using the indistinguishability of the scatterers in the last step.
Here, two approximations were made, in which Oðn20 Þ contriÐbutions were discarded. First, the inner integrals
rj >a dVj ðj ¼ 1; 2Þ of small quantities (F1 or F2) over the
(large) region B2 with a (small) hole were replaced by integrals over B2 (no hole). Second, the term (n0=2)2(pa2)2F0 was
ignored.
Similarly, the second term in Eq. (77) is approximately
pðr1 ; r2 Þ FðrjK2 Þ dV12
ð
pðr1 ; r2 Þ FðrjK2 Þ dV12
r2 <a
ð
pðr1 ; r2 Þ FðrjK2 Þ dV1 dV2 ¼
n0
2
ð
r2 <a
hFi2 dV2 :
r <a
þ
ð 1ð
pðr1 ; r2 Þ FðrjK2 Þ dV12 :
(B2)
Substituting back yields
2
hFðrÞi ¼ð1 n0 pa ÞF0 ðrÞ þ n0
ð
þ n0
hFðrÞi1 dV1 :
r1 >a
The first term in Eq. (B2) is
ð
r1 >a
hF1 i1 dV1
(B5)
ð
pðr1 Þ pðr2 jr1 ÞFðrjK2 Þ dV21 ¼
r1 <a
n0
2
ð
r1 <a
hFðrÞi1 dV1 :
The second term in Eq. (B2) is split as
ð
ð
r2 >a r1 >a
þ
ð
pðr1 ; r2 Þ FðrjK2 Þ dV12
ð
pðr1 ; r2 Þ FðrjK2 Þ dV12 :
(B3)
This is the result for N ¼ 2. It holds for any N 2, as
will be shown next.
From the definition, Eq. (B1),
ð
ð
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1 …N
hFðrÞi ¼
ðN1Þ r1 <a
ð
ð
þ
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1 …N :
ðN1Þ r1 >a
(B6)
r2 <a r1 >a
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
P. A. Martin: Scattering by random arrays of cylinders
1693
Author's complimentary copy
r1 <a
ð
The first term is ðn0 =NÞ
split as
ð
ð
Ð
r1 <a hFi1
Also, squaring,
dV1 . The second term is
2
2
16ðn2 z2 Þ2 Jn2 ¼ z4 ðJn2
Þ
þ 4Jn2 þ Jnþ2
ð
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1N
ðN2Þ
ð r2 >að r1 >að
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1N :
þ
ðN2Þ
r2 <a r1 >a
2
2
þ Jnþ1
Þ þ cross terms;
þ 4z2 ðJn1
where “cross terms” denotes terms of the form Jm Jn with
m = n. Hence, using Eq. (C1),
(B7)
16
The second term in Eq. (B7) is approximately
X
ðn2 z2 Þ2 Jn2 ¼ 6z4 þ 8z2 :
2
2
,
2Jn1 Jnþ1 þ Jnþ1
As 4½Jn0 2 ¼ Jn1
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV13 …N2
2
r2 <a ðN1Þ
¼
n0
N
ð
r2 <a
hFi2 dV2 :
rN >a
þ
ð
j¼2
rj <a
hFij dVj :
(B8)
2
2
þ cross terms,
Jnþ1
As 16Jn000 Jn0 ¼ 8½Jn0 2 Jn1
2
16
X
r1 >a
Collecting up the results, Eq. (B5) is obtained again, but
now for any N.
X
3 2X
½ð8 þ n2 z2 Þz2 Jn02 þ 2z3 Jn Jn0
z
Xn ¼
2
fðn2 z2 Þðn2 z2 þ 8Þ þ 8z2 gJn2 X
X
¼ 8z2
ðn2 z2 ÞJn02
Jn02 þ z2
X
X
ðn2 z2 Þ2 Jn2
Jn Jn0 þ 2z3
X
X
8
ðn2 z2 ÞJn2 8z2
Jn2
In this appendix, all functions have argument z and all
sums are from n ¼ 1 to n ¼ þ1. The basic sums are15
Jn2 ¼ 1
and
X
Jn Jnþm ¼ 0;
m 6¼ 0:
X
2X 2 2 2 2
ðn2 z2 ÞJn02
ðn z Þ Jn 2
2
z
1
1
3
¼ 2 ð6z4 þ8z2 Þ ð86z2 Þ ¼ z2 : (C2)
8z
8
2
2ðz2 n2 ÞJ n ðzÞ ¼
Then, using Eq. (61),
APPENDIX C: SOME SUMS OF PRODUCTS OF
BESSEL FUNCTIONS
X
X
ðn2 z2 ÞJn02 ¼ 8 6z2 :
These sums are sufficient to evaluate the sums needed in
Sec. VIII. First,
N
N ð
n0 X
þ
hFj ij dVj
pa
N
N j¼1 rj >a
n0
ð
2
hF1 ij dV1 :
’ ð1 n0 pa ÞF0 ðrÞ þ n0
0
(C1)
P
Differentiating the first of these gives Jn Jn0 ¼ 0.
The differential equation for Jn(z) gives
z2
1
ð8 6z2 Þ ð6z4 þ 8z2 Þ
16
16
þ 4z2 8z2
3
¼ z4 :
4
¼ 4z2 þ
4ðn2 z2 ÞJn ¼ 4z2 Jn00 þ 4zJn0
0
0
Þ þ 4zJn0
Jnþ1
¼ 2z2 ðJn1
¼ z2 ðJn2 2Jn þ Jnþ2 Þ þ 2zðJn1 Jnþ1 Þ;
using 2Jm0 ¼ Jm1 Jmþ1 . Hence
X
4
ðn2 z2 ÞJn2 ¼ 2z2 :
1694
Jn0 Jn00 ¼ 0:
n
X
X
z2 Jn000 þ 3zJn00 þ Jn0 þ 2zJn Jn0
ðn2 z2 ÞJn02 ¼
X
1
Jn000 Jn0
¼ þ z2
2
n
Using the expansion (B4), the first term in Eq. (B8)
becomes, approximately,
n N N
X
whence
pðr1 ; r2 ; …; rN Þ FðrjKN Þ dV1N
N
and
ðn2 z2 ÞJn0 ¼ z2 Jn000 þ 3zJn00 þ Jn0 þ 2zJn
r1 >a
ð
N
X
n0
Jn02 ¼ 1
Differentiating the differential equation for Jn(z) gives
The pattern is now clear. The splitting process is
repeated on the first term in Eq. (B7). This shows that the
second term in Eq. (B6) is approximately
ð
X
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
Thus,
1
P
Xn ¼ z2 =2 and so Eq. (63) gives S 2 ¼ 0.
P. A. Martin, Multiple Scattering (Cambridge University Press, Cambridge, 2006), p. 312.
P. A. Martin: Scattering by random arrays of cylinders
Author's complimentary copy
ð
ð
C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective
wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
3
A. Derode, V. Mamou, and A. Tourin, “Influence of correlations between
scatterers on the attenuation of the coherent wave in a random medium,”
Phys. Rev. E. 74(3), 036606 (2006).
4
J.-Y. Kim, “Models for wave propagation in two-dimensional random
composites: A comparative study,” J. Acoust. Soc. Am. 127(4), 2201–
2209 (2010).
5
P. A. Martin and A. Maurel, “Multiple scattering by random configurations of circular cylinders: Weak scattering without closure assumptions,”
Wave Motion 45(7–8), 865–880 (2008).
6
P. A. Martin, A. Maurel, and W. J. Parnell, “Estimating the dynamic effective mass density of random composites,” J. Acoust. Soc. Am. 128(2),
571–577 (2010).
7
P. Lloyd and M. V. Berry, “Wave propagation through an assembly of
spheres: IV. Relations between different multiple scattering theories,”
Proc. Phys. Soc. 91(3), 678–688 (1967).
J. Acoust. Soc. Am., Vol. 129, No. 4, April 2011
8
C. M. Linton and P. A. Martin, “Multiple scattering by multiple spheres:
A new proof of the Lloyd–Berry formula for the effective wavenumber,”
SIAM J. Appl. Math. 66(5), 1649–1668 (2006).
9
P.-Y. Le Bas, F. Luppé, and J.-M. Conoir, “Reflection and transmission by
randomly spaced elastic cylinders in a fluid slab-like region,” J. Acoust.
Soc. Am. 117(3), 1088–1097 (2005).
10
S. K. Bose, “Ultrasonic plane SH wave reflection from a uni-directional
fibrous composite slab,” J. Sound Vib. 193(5), 1069–1078 (1996).
11
Y. C. Angel, C. Aristégui, and M. Caleap, “Sound propagation in a solid
through a screen of cylindrical scatterers,” arXiv:0903.5191v1 (2005).
12
C. Aristégui and Y. C. Angel, “Coherent sound propagation across the effective interfaces of an immersed screen,” Wave Motion 47(4), 199–204 (2010).
13
C. Aristégui and Y. C. Angel, “New results for isotropic point scatterers:
Foldy revisited,” Wave Motion 36(4), 383–399 (2002).
14
Y. C. Angel and C. Aristégui, “Analysis of sound propagation in a fluid
through a screen of scatterers,” J. Acoust. Soc. Am. 118(1), 72–82 (2005).
15
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
(Dover, New York, 1965), p. 363.
P. A. Martin: Scattering by random arrays of cylinders
1695
Author's complimentary copy
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