Estimating the dynamic effective mass density of random
composites
P. A. Martina兲
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden,
Colorado 80401-1887
A. Maurel
Laboratoire Ondes et Acoustique, UMR CNRS 7587, Ecole Supérieure de Physique et de Chimie
Industrielles, 10 rue Vauquelin, 75005 Paris, France
W. J. Parnell
School of Mathematics, Alan Turing Building, University of Manchester, Manchester M13 9PL,
United Kingdom
共Received 27 January 2010; revised 4 June 2010; accepted 7 June 2010兲
The effective mass density of an inhomogeneous medium is discussed. Random configurations of
circular cylindrical scatterers are considered, in various physical contexts: fluid cylinders in another
fluid, elastic cylinders in a fluid or in another solid, and movable rigid cylinders in a fluid. In each
case, time-harmonic waves are scattered, and an expression for the effective wavenumber due to
Linton and Martin 关J. Acoust. Soc. Am. 117, 3413–3423 共2005兲兴 is used to derive the effective
density in the low frequency limit, correct to second order in the area fraction occupied by the
scatterers. Expressions are recovered that agree with either the Ament formula or the effective static
mass density, depending upon the physical context.
© 2010 Acoustical Society of America. 关DOI: 10.1121/1.3458849兴
PACS number共s兲: 43.20.Fn, 43.20.Bi, 43.20.Hq 关JAT兴
I. INTRODUCTION
Methods for estimating the effective properties of random composites have been of interest for many decades. In
recent years, some of this interest has been directed at the
effective mass density, ␳eff. In some circumstances, a simple
共static兲 mixture rule is found. Thus, for a two-phase composite 共matrix of density ␳ and inclusions of density ␳0兲, the
st
static estimate is ␳eff
= 共1 − ␾兲␳ + ␾␳0, where ␾ is the fraction
of space 共area density or volume density兲 occupied by the
st
could be determined by weighing
inclusions. Evidently, ␳eff
samples.
However, one can also consider dynamic problems, followed by taking a low-frequency limit. This is appropriate if
one wants to estimate effective wavespeeds. In some circumstances, this “quasistatic” limit gives a different estimate,
namely,
␳eff ␳ + ␳0 − ␾共␳ − ␳0兲
.
=
␳
␳ + ␳ 0 + ␾ 共 ␳ − ␳ 0兲
共1兲
The three-dimensional version of this formula is due to
Ament.1 Wavespeed estimates based on Eq. 共1兲 have been
shown to agree with experiments.2 One purpose of this paper
is to provide independent analytical verification of Eq. 共1兲.
When is the static effective density inappropriate? If the
matrix is a fluid, inertia is important, and so one might not
expect to obtain the static effective density. On the other
a兲
Author to whom correspondence should be addressed. Electronic mail:
pamartin@mines.edu
J. Acoust. Soc. Am. 128 共2兲, August 2010
Pages: 571–577
hand, if the matrix is solid, there is a well-defined static
st
limit, and so one may expect to recover ␳eff
. Indeed, this is
what has been found in the literature, as reviewed in Sec. II.
In this paper, a variety of two-dimensional problems is
considered, and the low-frequency limit is examined. The
prototype problem consists of the Helmholtz equation, 共ⵜ2
+ k2兲u = 0, outside circular cylinders. Inside each cylinder,
there is another Helmholtz equation, 共ⵜ2 + k20兲u0 = 0, with
various transmission conditions connecting u and u0 across
the circular boundaries. For simplicity, it is assumed that
each circle has radius a.
To analyze multiple scattering by random configurations
of small scatterers, a formula for the effective wavenumber,
K, obtained recently by Linton and Martin3 is used. Their
formula 关see Eq. 共21兲 in Sec. IV兴 uses the Lax quasicrystalline approximation 共QCA兲, it compares favorably with
experiments,4 and it has been confirmed by an independent
method that is valid for weak scattering.5 The Linton-Martin
formula is accurate to second order in ␾. It requires the solution of a scattering problem for one scatterer; this scalar
共transmission兲 problem is discussed briefly in Sec. III.
The Linton-Martin formula is applied to four problems.
The first 共Sec. V兲 concerns fluid cylinders in a fluid matrix.
Agreement with the Ament formula, Eq. 共1兲, is found, correct
to second order in ␾. Then, in Sec. VI, antiplane motions
共SH waves兲 in a solid-solid composite are considered: the
static estimate for ␳eff is obtained. In Sec. VII, elastic cylinders in a fluid matrix are considered, whereas movable rigid
cylinders in a fluid are considered in Sec. VIII. In both of
these cases, agreement with the Ament formula, Eq. 共1兲, is
found, correct to second order in ␾.
0001-4966/2010/128共2兲/571/7/$25.00
© 2010 Acoustical Society of America
571
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The methods used herein can be extended to threedimensional problems, with random configurations of
spheres. The appropriate estimate for the effective wavenumber is the Lloyd-Berry formula.6,7 The basic scattering problems for one sphere can be solved by standard methods. For
one elastic sphere in a fluid, see Faran’s paper.8 For one
movable rigid sphere in a fluid, see the paper by Hickling
and Wang.9
Consider acoustic scattering in two dimensions. In the
exterior, there is a homogeneous compressible fluid with
density ␳. The scatterers are homogeneous with density ␳0.
Scattering by random configurations of scatterers is of interest.
The main focus of this paper is on formulas for the
effective density of the random medium, ␳eff. There are many
such formulas. The simplest is the static estimate,
共1 − ␾兲␳ + ␾␳0 = ␳ + ␾⌬,
where
⌬ = ␳0 − ␳ 共2兲
is the density difference and ␾ is the area fraction occupied
by the cylinders. In three dimensions 共many identical spherical scatterers兲, ␾ is the volume fraction occupied by the
spheres. Although 0 ⬍ ␾ ⬍ 1 , ␾ is usually regarded as being
small.
Ament1 obtained another estimate in 1953, incorporating
dynamic and viscous effects. In the absence of viscosity, his
A
estimate, ␳eff
, becomes 关see Eq. 共9兲 in Ref. 1 or Eq. 共1.5兲 in
Ref. 10兴
A
st
= ␳eff
−
␳eff
2⌬2␾共1 − ␾兲
.
2⌬共1 − ␾兲 + 3␳
共3兲
This formula is for three-dimensional problems. Substituting
st
and ⌬, Eq. 共3兲 reduces 共exactly兲 to
for ␳eff
A
␳eff
1 − ␾Q3
=
␳
1 + 2␾Q3
␳ − ␳0
Q3 =
;
␳ + 2␳0
with
共4兲
the subscript 3 denotes three dimensions. Also, if Eq. 共4兲 is
approximated for small ␾兩Q3兩,
A
⯝
␳eff
␳共1 − 3␾Q3 + 6␾
2
Q23兲.
共5兲
In 1961, Waterman and Truell 关Eq. 共3.35兲 in Ref. 11兴
obtained the estimate
WT
= ␳共1 − 3␾Q3兲,
␳eff
␳+
␳
KT
2␳eff
=
1−␾
B
␳eff
+
2␳
+
␾
B
␳eff
+
2␳0
.
=␾
␳0 − ␳
,
␳ + 2␳0
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
B
−␳
␳eff
␳0 − ␳
=␾ B
,
B
␳eff + 2␳0
3␳eff
共8兲
which may be compared with Eq. 共7兲. Equation 共8兲 is a quaB
dratic equation for ␳eff
. For small ␾, it gives
冉
B
= ␳ 1 − 3␾Q3 + 6␾2Q23
␳eff
冊
3␳0
,
␳ + 2␳0
which differs from Eq. 共5兲 in the ␾2 term.
More recently, Sheng and his colleagues2,15,16 have
given a new derivation of the two-dimensional version of Eq.
共7兲, which is
␳0 − ␳
␳eff − ␳
=␾
.
␳0 + ␳
␳eff + ␳
共9兲
They derive Eq. 共9兲 by starting from a consideration of
waves in a periodic square arrangement of cylinders. They
also mistakenly refer to Eqs. 共7兲 and 共9兲 as defining the “Berryman effective mass density:” Berryman’s formula is different. The same mis-attribution of Eq. 共9兲 has also been made
by Torrent and Sánchez-Dehesa;17 these authors have also
given a generalization of Eq. 共9兲 covering non-random distributions of cylinders.
Solving Eq. 共9兲 gives
␳eff 1 − ␾Q2
=
␳
1 + ␾Q2
with
Q2 =
␳ − ␳0
.
␳ + ␳0
共10兲
This can be seen as the two-dimensional version of Ament’s
formula, Eq. 共4兲. For small ␾兩Q2兩, Eq. 共10兲 gives
␳eff ⬃ ␳共1 − 2␾Q2 + 2␾2Q22兲.
共11兲
This formula defines what is called the small-␾ Ament estimate below.
III. SCATTERING BY ONE CYLINDER
共7兲
KT
A
= ␳eff
, exactly, as they noted. Kuster and
which gives ␳eff
Toksöz considered elastic spheres in a fluid, and they “em-
572
B
3␳eff
共6兲
using a Foldy-type method, including effects of multiple
scattering. Thus, they obtained the linear term in Eq. 共5兲 共but
did not cite Ament’s paper1兲. Later, Fikioris and Waterman
关Eq. 共4.15兲 in Ref. 12兴 obtained precisely Eq. 共4兲.
In 1974, Kuster and Toksöz 关Eq. 共25兲 in Ref. 13兴 used a
different argument in which multiple scattering was ignored.
KT
, is defined by
Their estimate, ␳eff
KT
−
␳eff
1
This equation can be rewritten as
II. REVIEW
st
=
␳eff
KT
phasized that 关␳eff
兴 is an effective inertial density” 共see p.
593 of Ref. 13兲.
Berryman 关Eq. 共32兲 of Ref. 14兴 also considered elastic
spheres in a compressible fluid. He used a self-consistent
B
,
method and he ignored multiple scattering. His estimate, ␳eff
is given by
For one circular scatterer, the exact solution can be constructed by separation of variables. Thus, consider one circle
of radius a, centered at the origin. Then 共ⵜ2 + k2兲u = 0 for r
⬎ a and 共ⵜ2 + k20兲u0 = 0 for r ⬍ a. The interface conditions are
u = u0
and
1 ⳵ u 1 ⳵ u0
=
␳ ⳵ r ␳0 ⳵ r
on r = a.
共12兲
These are appropriate for a fluid cylinder surrounded by a
different fluid; u is the pressure. Outside the cylinder, u
= uin + usc where uin = eikx and usc satisfies the Sommerfeld radiation condition. Then3
Martin et al.: Estimating effective density of random composites
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⬁
兺
uin共r, ␪兲 =
i Jn共kr兲e
n
in␪
共13兲
,
Zn ⬃
n=−⬁
⬁
兺
usc共r, ␪兲 =
AnZnHn共kr兲e
in␪
,
n=−⬁
⬁
u0共r, ␪兲 =
BnJn共k0r兲ein␪ ,
兺
n=−⬁
共14兲
where Jn is a Bessel function, Hn = H共1兲
n is a Hankel function
and
Zn =
共␳0/␳兲J⬘n共ka兲Jn共k0a兲 − 共k0/k兲J⬘n共k0a兲Jn共ka兲
共␳0/␳兲H⬘n共ka兲Jn共k0a兲 − 共k0/k兲J⬘n共k0a兲Hn共ka兲
.
共15兲
Also, Zn = Z−n. The interface conditions, Eq. 共12兲, yield equations for An and Bn; in particular, An = −in. Note that, from
Eq. 共14兲, AnZn could have been replaced by a single quantity.
However, Zn is retained for two reasons. First, it was used in
Ref. 3. Second, Zn characterizes scattering by a single cylinder, for any incident field, and so it arises naturally when
scattering by many identical cylinders is considered.
The far-field pattern, f共␪兲, is defined by
usc ⬃ 冑2/共␲kr兲f共␪兲exp共ikr − i␲/4兲
as
r → ⬁.
Hence, Eq. 共14兲 gives
⬁
f共␪兲 = −
Znein␪ .
兺
n=−⬁
共16兲
A. Behavior of Zn for small ka
The behavior of Zn 关defined by Eq. 共15兲兴 is studied for
small ka, in various situations, starting with n = 0:
共␳0/␳兲J1共ka兲J0共k0a兲 − 共k0/k兲J1共k0a兲J0共ka兲
Z0 =
.
共␳0/␳兲H1共ka兲J0共k0a兲 − 共k0/k兲J1共k0a兲H0共ka兲
1
J0共ka兲
⯝
;
H0共ka兲 H0共ka兲
共17兲
this corresponds to a soft scatterer 共Dirichlet condition兲.
For ␳0 ⫽ 0, one uses J1共x兲 ⬃ x / 2 and H1共x兲 ⬃ 2 / 共␲ix兲 as
x → 0, giving
Z0 ⬃
共19兲
with no dependence on k0 at leading order. In particular,
when ␳0 = 0, comparison with Eq. 共17兲 shows that Z0 is dominant in this case.
When ␳0 ⫽ 0 and ␳0 ⫽ ␳, Z0 and Z1 are both O共共ka兲2兲 as
ka → 0: in general, small cylinders behave as a combination
of a source and a dipole. This is the generic situation. It
includes sound-hard scatterers 共Neumann condition兲 by setting ␳ = 0.
Exceptionally, when ␳ = ␳0 ⫽ 0, one must use
冉
冊
Jn共x兲 ⬃
共x/2兲2
xn
1
−
,
2nn!
n+1
J⬘n共x兲 ⬃
共n + 2兲x2
xn−1
1
−
,
2n共n − 1兲!
4n共n + 1兲
冉
as x → 0, giving
Zn ⬃
冊
冉 冊
k2
␲i 共ka/2兲2n+2
1 − 20 .
2 n!共n + 1兲!
k
Comparison with Eq. 共18兲 shows that Z0 is dominant when
the density is constant, ␳ = ␳0. 共These results are consistent
with a paper by Exner and Šeba,18 who consider only the two
special cases ␳ = ␳0 and ␳0 = 0.兲
As is well known, the sound-soft case, ␳0 = 0, is atypical:
the scattering is isotropic. This suggests that care is needed
with the commutativity of the two limits ␳0 → 0 and ka → 0.
For investigations in this direction, see Refs. 19 and 20.
IV. SCATTERING BY RANDOM ARRANGEMENTS OF
CIRCULAR CYLINDERS
For scattering by random arrangements of scatterers, one
can use Foldy-type theories.21 If n0 is the number of scatterers per unit area, the effective wavenumber, K, is given by
共20兲
K2 = k2 − 4in0 f共0兲.
For ␳0 = 0,
Z0 =
␲i共ka/2兲2n ␳ − ␳0
,
n!共n − 1兲! ␳ + ␳0
冉
␲i
共ka兲2 1 −
4
␳k20
␳ 0k 2
冊
共18兲
.
For n ⬎ 0, one can use the approximations
Jn共x兲 ⬃
xn
,
2nn!
Hn共x兲 ⬃
2n共n − 1兲!
,
␲ixn
J⬘n共x兲 ⬃
xn−1
,
2n共n − 1兲!
Hn⬘共x兲 ⬃ −
2nn!
,
␲ixn+1
as x → 0. Then
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
This formula is linear in n0 and it is derived using Foldy’s
“closure assumption.”
Linton and Martin3 obtained a second-order correction
to Eq. 共20兲:
K2 = k2 − 4in0 f共0兲 +
8n20
␲k2
冕
␲
0
cot共␪/2兲
d
关f共␪兲兴2d␪ .
d␪
共21兲
This formula is derived using the Lax QCA as the closure
assumption. It was derived in the limit b → 0, where b is the
“hole radius” in the hole correction used to ensure that
circles do not overlap during the averaging process. Therefore, as b ⱖ 2a, it is natural to approximate the far-field pattern f共␪兲 assuming that ka is small. Assuming that ␳0 ⫽ 0 and
␳0 ⫽ ␳, Z0 and Z1 = Z−1 are retained 共see Sec. III A兲, giving
f共␪兲 ⯝ − Z0 − 2Z1 cos ␪
共22兲
with Z0 given by Eq. 共18兲,
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573
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Z1 =
␲i
共ka兲2Q2
4
and
关f共␪兲兴2 = Z20 + 4Z0Z1
As
in Eq. 共21兲 is
冕
␲
0
cos
Q2 =
␳ − ␳0
.
␳ + ␳0
␪ + 4Z21
cos2 ␪, the integral term
d
cot共␪/2兲 关f共␪兲兴2d␪
d␪
冕
冕
cos2共␪/2兲共Z0 + 2Z1 cos ␪兲d␪
0
␲/2
− 16Z1
共Z0 − 2Z1 + 4Z1 cos2 ␰兲cos2 ␰d␰
0
= − 4␲Z1共Z0 − 2Z1兲 − 12␲Z21
= − 4␲Z1共Z0 + Z1兲.
Hence, Eq. 共21兲 becomes
K2 = k2 + 4in0共Z0 + 2Z1兲 − 32n20k−2Z1共Z0 + Z1兲.
共24兲
This is the formula that is used below to estimate the effective mass density.
It is worth recalling the estimate for K obtained by Waterman and Truell,11 KWT, given by
2
= k2 − 4in0 f共0兲 + 共2n0/k兲2兵关f共␲兲兴2 − 关f共0兲兴2其.
KWT
共25兲
6
It is known that the term in n20 is incorrect but, nevertheless,
Eq. 共25兲 has been used to estimate effective properties. Using
Eq. 共22兲 gives
2
= k2 − 4in0共Z0 + 2Z1兲 − 16n20k−2Z0Z1 .
KWT
共26兲
Evidently, Eqs. 共24兲 and 共26兲 agree when 2共Z0 + Z1兲 = Z0, that
is, when f共0兲 = 0 according to the approximation Eq. 共22兲.
V. EFFECTIVE MASS DENSITY IN FLUID-FLUID
SYSTEMS
Define the area fraction occupied by the cylinders, ␾:
␾ = n 0␲ a 2 .
Then, as already noted, a 共static兲 effective density could be
defined by Eq. 共2兲. However, other definitions are possible,
especially in a dynamic context, and it is these that are of
interest here. Expressions for ␳eff will be extracted from the
small-ka approximations to the effective wavenumber, K,
and these will be compared with Eq. 共11兲. Notice that, in this
section, sound waves in a compressible fluid are considered.
共SH-waves in an elastic composite will be considered in Sec.
VI兲 Thus, k = ␻ / c and c2 = M / ␳, where ␻ is the frequency, c
is the wavespeed and M is the bulk modulus. Then, using
obvious notation,
K2 = ␻2␳eff/M eff,
k2 = ␻2␳/M,
k20 = ␻2␳0/M 0 ,
and, from Eq. 共18兲, Z0 = 共␲i / 4兲共ka兲2共1 − M / M 0兲. Evidently,
Z0 involves the bulk moduli only whereas Z1 关defined by Eq.
共23兲兴 involves the densities only.
Start with the Foldy estimate,
574
⫻共1 + 8in0k−2Z1兲 + O共n20兲.
共27兲
As 共K / k兲 = 共␳eff / ␳兲共M / M eff兲, this suggests that
2
1
1−␾ ␾
1
= 共1 + 4in0k−2Z0兲 =
+
M eff M
M
M0
共28兲
␳eff = ␳共1 + 8in0k−2Z1兲 = ␳共1 − 2␾Q2兲.
共29兲
and
␲
= − 8Z1
共K/k兲2 = 1 + 4in0k−2共Z0 + 2Z1兲 + O共n20兲 = 共1 + 4in0k−2Z0兲
共23兲
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
Thus, Eq. 共28兲 gives M eff as the harmonic average of M and
M 0, whereas Eq. 共29兲 agrees with the small-␾ Ament estimate, Eq. 共11兲, to first order in ␾.
It is noteworthy that Waterman and Truell11 used the
same argument 共but in three dimensions兲 and obtained the
estimate Eq. 共6兲. They regarded their estimate as “not so
readily interpretable” as Eq. 共28兲, and went on to consider
the limit ␳0 → ␳.
Next, consider the Linton-Martin small-ka estimate for
K, Eq. 共24兲. It gives
共K/k兲2 = 共1 + 4in0k−2Z0兲共1 + 8in0k−2Z1 − 32n20k−4Z21兲,
共30兲
correct to order n20. Note that the first factor on the right-hand
side is exactly the same as in Eq. 共27兲, so that the estimate
for M eff, Eq. 共28兲, is unchanged. The second factor gives a
refined estimate for ␳eff; it agrees precisely with the small-␾
Ament estimate, Eq. 共11兲.
Aristégui and Angel 关Eq. 共21兲 in Ref. 22兴 used the
Waterman-Truell estimate, Eq. 共25兲, to obtain an estimate,
AA
␳eff
, for the effective density,
AA
/␳ = 1 − 2in0k−2关f共0兲 − f共␲兲兴.
␳eff
共31兲
AA
It is noted that Eq. 共31兲 is linear in ␾ = n0␲a2; it gives ␳eff
⬃ 1 − 2␾Q2 as ka → 0. Aristégui and Angel 关Eq. 共18兲 in Ref.
23兴 have obtained a similar estimate for SH waves in a random composite: this problem is discussed in Sec. VI.
One of the advantages of using formulas such as those
above, involving the far-field pattern f, is that they often
transpire to be useful well beyond the regime of small ka
under which they were derived. However they are, of course,
limited to the regime of small ␾. An alternative approach in
classical multiple scattering methods is to use the assumption
of small ka earlier on in the analysis and retain the leading
order terms in ka: classically this was done for twodimensional elasticity problems by Bose and Mal.24,25 This
approach gives an effective wavenumber which is physically
viable 共meaning that predictions of effective properties reside inside strict variational bounds兲 for all ␾. Using such an
approach leads to M eff as in Eq. 共28兲 and ␳eff as in Eq. 共10兲.
VI. SH WAVES IN A SOLID-SOLID COMPOSITE
Here, anti-plane 共SH兲 motions in a solid composite are
considered. The problem of scattering by one cylinder is almost the same as for the fluid-fluid problem discussed in Sec.
III. Thus, 共ⵜ2 + k2兲u = 0 for r ⬎ a and 共ⵜ2 + k20兲u0 = 0 for r ⬍ a,
where u and u0 are the out-of-plane displacement components. The shear wavenumbers, k and k0, are given by
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k 2 = ␻ 2␳ / ␮
and
k20 = ␻2␳0/␮0 ,
where ␮ and ␮0 are the shear moduli. The interface conditions differ from Eq. 共12兲; they are
u = u0
and
␮
⳵u
⳵ u0
= ␮0
⳵r
⳵r
on
r = a.
共32兲
Outside the cylinder, u = uin + usc 共as before兲 where uin = eikx
and usc satisfies the Sommerfeld radiation condition.
Comparing Eq. 共32兲 with Eq. 共12兲 shows that all the
results of Sec. III can be reused: replace ␳0 / ␳ by ␮ / ␮0
therein. Thus, for small ka, Eqs. 共18兲 and 共23兲 give
Z0 ⬃ 共␲i/4兲共ka兲2共1 − ␳0/␳兲
and
Z1 ⬃ 共␲i/4兲共ka兲2Q
with Q = 共␮0 − ␮兲 / 共␮0 + ␮兲, in agreement with Bose and Mal
关Eq. 共16兲 in Ref. 24兴. Then, proceeding as in Sec. V, one
defines an effective shear modulus ␮eff by K2 = ␻2␳eff / ␮eff.
Working correct to order n20, the Linton-Martin estimate, Eq.
共30兲, gives
␳eff = ␳共1 + 4in0k−2Z0兲 = 共1 − ␾兲␳ + ␾␳0
共33兲
and
should be expected, for the in-plane two-dimensional elasticity problem 共the so-called P/SV problem兲 the static effective
density ␳eff, as in Eq. 共33兲, is recovered.25,31
VII. FLUID-SOLID PROBLEM
Consider an elastic cylinder in a compressible inviscid
fluid. The transmission conditions at r = a are
1 ⳵u
= ␻2r̂ · u0
␳ ⳵r
1 ⳵u
= ␻2ur0,
␳ ⳵r
on
r = a.
⬁
共36兲
n=0
where ⑀0 = 1, ⑀n = 2 for n ⱖ 1, Zn = bn / Dn,
冨
冨
n
n
共␳/␳0兲xT2Jn共x兲 a12
a13
bn =
which agrees with the formula of Bose and Mal 关Eq. 共39兲 in
Ref. 24兴, namely,
共35兲
correct to second order in ␾Q. The expression Eq. 共35兲 was
derived in Ref. 24 using the assumption of ka Ⰶ 1 from the
outset, as described at the end of Sec. V. Also, it is noted that
Eq. 共35兲 is the well-known static result of Hashin and Rosen
关Eq. 共71兲 in Ref. 26兴. Yang and Mal27 have noted that it
cannot be obtained from the Waterman-Truell estimate, KWT,
defined by Eq. 共25兲.
The static effective density in Eq. 共33兲 is recovered by
employing asymptotic homogenization for elasticity problems, e.g., the SH problem solved by Parnell and
Abrahams.28 In this case, because the arrangement of scatterers was periodic, the effective shear modulus was anisotropic in general. Reciprocity between the SH and acoustics
problems, as described above, therefore gives a static effective bulk modulus and an anisotropic effective density in the
case of acoustics. This latter point is particularly pertinent to
note, considering recent comments regarding possible applications in metamaterials.29 The static effective density is also
recovered by application of a recently proposed scheme of
homogenization applied to the SH problem: the so-called
integral equation method proposed by Parnell and
Abrahams.30 This method also recovers Eq. 共35兲 at leading
order, with successive lattice corrections for materials where
scatterers are positioned on a periodic lattice. Finally, as
␶r0␪ = 0
and
u = 兺 ⑀nin兵Jn共kr兲 − ZnHn共kr兲其cos n␪ ,
␮eff/␮ ⬃ 1 + 2␾Q + 2␾2Q2
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
0
=−u
␶rr
The problem of plane-wave scattering by an elastic cylinder in a fluid can be solved by standard methods.8 For a
convenient resolution, see p. 46 in Ref. 32, where the total
pressure field in the fluid is given by
共34兲
␮eff 1 + ␾Q
,
=
␮
1 − ␾Q
− ur̂ = T0u0 ,
where ␳ is the fluid density, r̂ is a unit vector in the radial
direction, u0 is the elastic displacement in the cylinder and
T0 is the traction operator. With u0 = 共ur0 , u0␪兲, these conditions
become
␮/␮eff = 1 + 8in0k−2Z1 − 32n20k−4Z21 = 1 − 2␾Q + 2␾2Q2 .
Thus, one obtains the static estimate for ␳eff, Eq. 共2兲, whereas
Eq. 共34兲 gives
and
冨
n
n
a22
a23
,
xJn⬘共x兲
n
n
a32
a33
0
n
n
共␳/␳0兲xT2Hn共x兲 a12
a13
xHn⬘共x兲
Dn =
n
a32
an33
0
冨
n
an23 ,
a22
n
= − 2xLJn⬘共xL兲 + 共2n2 − xT2兲Jn共xL兲,
a12
n
a22
= − xLJn⬘共xL兲,
n
a32
= 2n关xLJn⬘共xL兲 − Jn共xL兲兴,
n
a13
= − 2n关xTJn⬘共xT兲 − Jn共xT兲兴,
n
= nJn共xT兲,
a23
n
a33
= 2xTJn⬘共xT兲 − 共2n2 − xT2兲Jn共xT兲,
x = ka, xT = kTa, xL = kLa, kT is the shear wavenumber, kL is
the compressional wave-number and ␳0 is the solid density.
Note that Zn = Z−n, so that Eq. 共16兲 remains valid.
For n ⱖ 1, all anij are O共xn兲 as x → 0. Hence, a rough
calculation gives
Zn ⬃
J⬘n共x兲
Hn⬘共x兲
⬃−
␲i共x/2兲2n
.
n!共n − 1兲!
This is a correct estimate for n ⱖ 2 but there is some cancellation for n = 1. Thus,
冏
1
a122 a23
1
a32
1
a33
冏
1
⬃ xLxT共xL2 − xT2兲
8
and
Martin et al.: Estimating effective density of random composites
575
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冏
1
a12
a113
1
1
a32
a33
冏
1
⬃ xLxT3共xL2 − xT2兲,
8
Fx = −
冏 冏 冏
冏 冏 冏
1
1
1
␳xT2x a122 a23
x a12 a13
−
1
1
1
2␳0 a32
2 a32
a33
a133
Z1 ⬃
1
1
1
1
a23
2␳xT2 a22
2 a12 a13
+
1
1
1
␳0␲ix a32
␲ix a132 a33
a33
⬃
␲i 2
x Q2 ,
4
exactly as in Eq. 共23兲.
0
= 0, and so Z0 = b̃0 / D̃0, where
For n = 0, a013 = a023 = a32
b̃0 =
冏
0
共␳/␳0兲xT2J0 a12
− xJ1共x兲
0
a22
冏
D̃0 =
,
冏
0
共␳/␳0兲xT2H0 a12
− xH1共x兲
0
a22
冏
,
0
0
= xLJ1共xL兲. Then, a12
⬃ xL2
a012 = 2xLJ1共xL兲 − xT2J0共xL兲 and a22
2
0
2
− xT, a22 ⬃ xL / 2,
b̃0 ⬃
冏
冏
共␳/␳0兲xT2 xL2 − xT2
D̃0 ⬃
xL2/2
− x2/2
冏
,
共␳/␳0兲xT2H0共x兲 xL2 − xT2
xL2/2
2i/␲
冉
冏
⬃
2 2
共x − x2 兲,
␲i L T
冊
x
2
共xL2
−
xT2兲
=−
M
.
␭0 + ␮0
In summary, the dominant terms are Z0 and Z1 = Z−1,
where
Z0 =
冉
M
␲i 2
x 1−
4
␭0 + ␮0
冊
and
− Z1H1共ka兲其.
The cylinder oscillates in the x-direction with velocity U.
Thus, Newton’s law gives Fx = −i␻U共␲a2␳0兲, and so
U = 2共␻a␳0兲−1兵J1共ka兲 − Z1H1共ka兲其.
As the fluid velocity is 共i␻␳兲−1grad u,
1 ⳵u
= U cos ␪
i␻␳ ⳵ r
on r = a.
Applying this boundary condition, using Eq. 共36兲, determines Zn. For n ⫽ 1, Zn = J⬘n共ka兲 / H⬘n共ka兲, which is the same as
would be obtained for a sound-hard cylinder; in particular
Z0 ⬃ 共␲i / 4兲共ka兲2 as ka → 0. For n = 1,
Z1 =
共␳0/␳兲kaJ⬘1共ka兲 − J1共ka兲
共␳0/␳兲kaH⬘1共ka兲 − H1共ka兲
.
IX. CONCLUSION
To simplify, x2 = 共ka兲2 = 共␻a兲2␳ / M, xT2 = 共kTa兲2 = 共␻a兲2␳0 / ␮0
and xL2 = 共kLa兲2 = 共␻a兲2␳0 / 共␭0 + 2␮0兲. Hence
共␳/␳0兲xT2xL2
u共a, ␪兲cos ␪ ad␪ = − 2␲ia兵J1共ka兲
For small ka, one finds that Z1 behaves precisely as in Eq.
共23兲, implying that the effective properties for movable rigid
cylinders in a fluid are exactly the same as for the fluid-fluid
problem discussed in Sec. V.
共␳/␳0兲x2 x2
␲i 2
x 1 + 2 2 T 2L .
4
x 共xL − xT兲
Z0 ⬃
2␲
0
which are both smaller than first expected. Hence, a more
careful calculation gives
冏
冏
冕
Z1 =
␲i 2
x Q2 ,
4
with Q2 given by Eq. 共23兲. Using these estimates in Eq. 共30兲
gives
1−␾
1
␾
=
+
M eff
M
␭0 + ␮0
共37兲
for M eff and the small-␾ Ament estimate, Eq. 共11兲, for ␳eff.
Notice that, in Eq. 共37兲, the quantity ␭0 + ␮0 is the planestrain bulk modulus for the solid.
According to Ament,1 the dynamic 共low-frequency
limit兲 effective mass density of a random composite, ␳eff, is
given by Eq. 共1兲. This formula gives ␳eff → ␳ as ␾ → 0 and
␳eff → ␳0 as ␾ → 1, both of which are physically correct limits. In this paper, the small-␾ behavior has been explored in
more detail, retaining terms correct to second order in ␾. It
was found that, to this order, Ament’s formula agrees precisely with the predictions of an independent multiplescattering theory,3 for several 共two-dimensional兲 physical
systems. These systems have in common that the exterior
medium 共␾ = 0兲 is a compressible fluid. For SH waves in a
solid, however, the familiar static estimate, Eq. 共2兲, was recovered.
The detailed agreement between the two theories is
gratifying, given that the precise conditions for validity of
the Ament formula or the Linton–Martin multiple-scattering
formalism are unknown, at present. However, the same degree of agreement would not have been found if Ament’s
formula had been replaced with Berryman’s formula, Eq. 共8兲,
for example, or if the Linton–Martin theory had been replaced with the Waterman–Truell theory,11 for example. Consequently, analogous three-dimensional investigations are
warranted.
VIII. MOVABLE RIGID CYLINDERS IN A FLUID
1
Consider a rigid cylinder in a compressible inviscid
fluid. The cylinder is free to move, so that its equation of
motion is needed. The corresponding scattering problem has
been solved by Zhuk.33
As before, let u be the total pressure field in the fluid,
and suppose that it is expanded as Eq. 共36兲. The resultant
force per unit length on the cylinder due to the incident wave
uin = eikx is in the x-direction, and it is given by
576
J. Acoust. Soc. Am., Vol. 128, No. 2, August 2010
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