P. A. Martin
Department of Mathematical and
Computer Sciences,
Colorado School of Mines,
Golden, CO 80401-1887
e-mail: pamartin@mines.edu
1
Scattering by a Cavity in an
Exponentially Graded Half-Space
An inhomogeneous half-space containing a cavity is bonded to a homogeneous halfspace. Waves are incident on the interface and the problem is to calculate the scattered
waves. For a circular cavity in an exponentially graded half-space, it is shown how to
solve the problem by constructing an appropriate set of multipole functions. These functions are singular on the axis of the cavity, they satisfy the governing differential equation
in each half-space, and they satisfy the continuity conditions across the interface between
the two half-spaces. Seven recent publications are criticized: They do not take proper
account of the interface between the two half-spaces. 关DOI: 10.1115/1.3086585兴
Introduction
Consider two half-spaces, x ⬎ 0 and x ⬍ 0, welded together
along the interface at x = 0. The left half-space 共x ⬍ 0兲 is homogeneous. The right half-space is inhomogeneous. If a wave is incident from the left, it will be partly reflected and partly transmitted
into the right half-space. We assume that these fields can be calculated.
Suppose now that the right half-space contains a cavity or some
other defect 共see Fig. 1兲. How are the basic fields described above
modified by the presence of the cavity? In general, it is not easy to
answer this question, as the associated mathematical problem is
difficult, in general.
In some recent papers, Fang et al. claimed to solve a variety of
such problems. All concern “exponential grading,” meaning that
the material parameters are proportional to e−␤x for x ⬎ 0, where ␤
is a given constant. The papers concern antiplane shear waves
关1–4兴, thermal waves 关5,6兴, and shear waves in a piezoelectric
material 关7兴. All of these papers assume that the effect of the
interface on the cavity can be found by introducing simple image
terms, as if the interface were a mirror or a rigid wall. Unfortunately, this assumption is incorrect.
In this paper, we outline how the problems described above can
be solved. We do this in the context of antiplane shear waves with
exponential grading and a circular cavity. The main technical part
concerns the derivation of suitable multipole potentials; these reveal the complicated image system.
The study of problems involving scatterers near boundaries or
interfaces has a long history. For linear surface water waves interacting with a submerged circular cylinder, see the famous paper
by Ursell 关8兴. For plane-strain elastic waves in a homogeneous
half-space with a buried circular cavity, see Ref. 关9兴. There are
also many papers on the scattering of electromagnetic waves by
objects near plane boundaries; see, for example, Ref. 关10兴.
Some problems involving objects near plane boundaries can be
solved using images. However, determining the strength and location of the images may be difficult: Doing so will depend on the
governing differential equations and on the conditions to be satisfied on the plane boundary. For two interesting examples where
the location of the images is not obvious, we refer to Chap. 8 of
Ting’s book 关11兴 共construction of static Green’s functions in anisotropic elasticity兲 and a paper by Stevenson 关12兴 共construction
of Green’s function for the anisotropic Helmholtz equation in a
half-space兲.
The basic scattering problem is formulated in Sec. 2. The
reflection-transmission problem 共for which the cavity is absent兲 is
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received November 12, 2007; final
manuscript received March 10, 2008; published online March 10, 2009. Review
conducted by Professor Sridhar Krishnaswamy.
Journal of Applied Mechanics
solved in Sec. 3. The solution of this problem gives the “incident”
field that will be scattered by the cavity. To solve the scattering
problem, we construct an appropriate set of multipole functions
共Sec. 4.1兲. Each of these satisfies the governing differential equations and the interface conditions, and is singular at the center of
the circular cavity. Each multipole function is defined as a contour
integral of Sommerfeld type; for a careful discussion of similar
functions, see Refs. 关13,9兴. In Sec. 4.2, the multipole functions are
combined so as to satisfy the boundary condition on the cavity,
leading to an infinite linear system of algebraic equations. The
far-field behavior of the multipole functions is deduced in Sec.
4.3. Closing remarks are made in Sec. 5.
2
Formulation
We consider the antiplane deformations of two elastic halfspaces, bonded together. In terms of Cartesian coordinates 共x , y兲,
the half-space x ⬍ 0 is homogeneous, the half-space x ⬎ 0 is inhomogeneous 共“graded”兲, and the interface is at x = 0. The homogeneous region has shear modulus ␮0 and density ␳0 共both constants兲. The inhomogeneous region has shear modulus ␮共x兲 and
density ␳共x兲 given by
␮共x兲 = ␮0e2␤x
and
␳共x兲 = ␳0e2␤x
共1兲
where ␤ is a constant. Thus, the material parameters are continuous across the interface.
It is not our purpose here to discuss whether any real materials
can be well represented by the functional forms given in Eq. 共1兲.
Certainly, the choices in Eq. 共1兲 do lead to some mathematical
simplifications and they have been used in the past; see, for example, Refs. 关14,15兴.
For time-harmonic motions, with suppressed time-dependence
e−i␻t, the governing equation is
⳵ ␶xz ⳵ ␶yz
+
+ u␻2␳共x兲 = 0
⳵x
⳵y
where u共x , y兲 is the antiplane component of displacement and the
stress components are given by
␶xz = ␮共x兲
⳵u
⳵x
and
␶yz = ␮共x兲
⳵u
⳵y
Thus, in the homogeneous region, where we write u0 instead of u,
we obtain the two-dimensional Helmholtz equation
共ⵜ2 + k20兲u0 = 0
with
k20 = ␳0␻2/␮0
共2兲
In the inhomogeneous region, we obtain
ⵜ 2u + 2 ␤
⳵u
+ k20u = 0
⳵x
This equation is satisfied by writing
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y
x = b + r cos ␪
共8兲
y = r sin ␪
and
Thus, the cavity’s boundary is given by r = a, and the boundary
condition is
b
x
θ
⳵u
=0
⳵r
O
on
共9兲
r=a
where u is the total field in the inhomogeneous half-space.
To solve such a scattering problem, we write
u0 = uin + ure + v0,
Fig. 1 The scattering problem. The half-plane on the left of x
= 0 is homogeneous. The other half-plane is inhomogeneous.
The circular cavity has radius a. A plane wave is incident from
the left.
u共x,y兲 = e−␤xw共x,y兲
where w satisfies a different two-dimensional Helmholtz equation
共ⵜ2 + k2兲w = 0
with
k2 = k20 − ␤2
u0共0,y兲 = u共0,y兲 = w共0,y兲
冏 冏 冏 冏 冏 冏
⳵ u0
⳵x
3
=
x=0
⳵u
⳵x
⳵w
⳵x
=
x=0
共4兲
− ␤w共0,y兲
共5兲
x=0
Incident Field
Suppose that a plane wave is incident on the interface from the
homogeneous side. This wave is given by
uin共x,y兲 = e
ik0共x cos ␣0+y sin ␣0兲
where ␣0 is the angle of incidence, 兩␣0兩 ⬍ ␲ / 2; ␣0 = 0 gives normal
incidence. There will be a reflected wave ure and a transmitted
wave utr. Evidently,
ure共x,y兲 = Reik0共−x cos ␣0+y sin ␣0兲,
x⬍0
共6兲
utr共x,y兲 = Te−␤xeik共x cos ␣+y sin ␣兲,
x⬎0
共7兲
where R, T, and ␣ are to be found. Writing u0 = uin + ure and u
= utr, Eq. 共4兲 gives
1+R=T
and
共1 − R兲ik0 cos ␣0 = T共ik cos ␣ − ␤兲
Solving for R gives
k0 cos ␣0 − k cos ␣ − i␤
− i␤
=
k0 cos ␣0 + k cos ␣ + i␤ k0 cos ␣0 + k cos ␣
and then T = 1 + R.
For a simple check, put ␤ = 0; we obtain k = k0, ␣ = ␣0, R = 0,
and T = 1, as expected.
4
␾n =
再
e−␤x兵H共n1兲共kr兲ein␪ + ⌽n其, x ⬎ 0
x⬍0
⌿n ,
冎
共1兲
where Hn is a Hankel function and n is an arbitrary integer. We
require that ⌽n solves Eq. 共3兲 and ⌿n solves Eq. 共2兲. In addition,
⌽n and ⌿n are to be chosen so that ␾n satisfies the interface
conditions, Eqs. 共4兲 and 共5兲.
The use of polar coordinates is convenient for handling the
circular cavity but it is inconvenient when trying to impose the
conditions at x = 0. Therefore, we convert from polar coordinates
to Cartesian coordinates using an integral representation; see the
Appendix for details. In particular, if we insert Eq. 共8兲 in Eq. 共A4兲,
we obtain the integral representation
H共n1兲共kr兲ein␪ =
共− 1兲n
␲i
for
冕
⬁+␲i
ek共b−x兲sinh ␶−iky cosh ␶e−n␶d␶
−⬁
x ⬍ b,兩y兩 ⬍ ⬁
with
0⬍a⬍b
We also introduce polar coordinates, 共r , ␪兲, so that
031009-2 / Vol. 76, MAY 2009
共10兲
Notice that this formula is valid on the interface x = 0. The contour
of integration in Eq. 共10兲 is also described in the Appendix.
The form of Eq. 共10兲 suggests using a similar integral representation for ⌽n, and so we write
⌽n共x,y兲 =
共− 1兲n
␲i
冕
⬁+␲i
A共␶兲ekx sinh ␶e−iky cosh ␶+kb sinh ␶−n␶d␶,
x⬎0
−⬁
共11兲
where A共␶兲 is to be found; ⌽n solves Eq. 共3兲 automatically for any
reasonable choice of A.
We shall also need a similar integral representation for ⌿n共x , y兲
in x ⬍ 0, where the wavenumber is k0. However, in order to match
solutions across the interface at x = 0, we shall require the same
dependence on y as in Eq. 共11兲. Thus, we consider
⌿n共x,y兲 =
共− 1兲n
␲i
冕
⬁+␲i
B共␶兲ex⌬共␶兲e−iky cosh ␶+kb sinh ␶−n␶d␶,
x⬍0
−⬁
共12兲
where B共␶兲 is to be found,
Scattering by a Buried Cavity
Next, we investigate how the wavefields of Sec. 3 are modified
if there is a cavity in the inhomogeneous half-space, x ⬎ 0. See
Fig. 1.
We suppose that the cavity’s cross section is circular, with
boundary
共x − b兲2 + y 2 = a2
4.1 Multipole Functions. To represent the scattered field, we
introduce functions ␾n of the form
k0 sin ␣0 = k sin ␣
Then, Eq. 共5兲 gives
R=
r⬎a
where v0 solves Eq. 共2兲, v = we−␤x, and w solves Eq. 共3兲. Also, v0
must satisfy the Sommerfeld radiation condition and v must decay
with x.
共3兲
For simplicity, we assume that k20 ⬎ ␤2 and write k = + 冑k20 − ␤2.
The interface conditions require that the displacements and normal stresses be continuous, so that
x ⬎ 0,
u = utr + v,
x⬍0
⌬共␶兲 = 共k2 cosh2 ␶ − k20兲1/2 = 共k2 sinh2 ␶ − ␤2兲1/2
and the square root is taken so that Re ⌬ ⱖ 0 on the contour.
Notice that Eq. 共2兲 is satisfied automatically for any reasonable
choice of B.
We are now ready to enforce the interface conditions. Continuity of ␾n across x = 0 gives 1 + A = B whereas continuity of ⳵␾n / ⳵x
gives
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− k sinh ␶ − ␤ + 共k sinh ␶ − ␤兲A共␶兲 = ⌬共␶兲B共␶兲
n
Vm
共r兲 = 共− 1兲nIm−n共␤r兲H共n1兲共kr兲 +
兺f I
n
s m−s共␤r兲Js共kr兲
共16兲
s
Hence
A共␶兲 =
Thus, Eq. 共15兲 and orthogonality of 兵eim␪其 give
k sinh ␶ + ⌬ + ␤
k sinh ␶ − ⌬ − ␤
共13兲
兺 c V ⬘共a兲 = − U⬘ 共a兲,
n
n m
and
m
all
共17兲
m
n
B共␶兲 =
2k sinh ␶
k sinh ␶ − ⌬ − ␤
共14兲
which is a linear system of algebraic equations for the coefficients
c n.
These formulas complete the construction of the multipole functions ␾n.
Note that when ␤ = 0, k = k0, ⌬共␶兲 = −k sinh ␶, A = 0, B = 1, and
共1兲
⌿n = Hn 共kr兲ein␪, as expected.
4.3 Far-Field Behavior of ⌿n. We should expect cylindrical
waves in the homogeneous half-space. These arise from the farfield behavior of ⌿n共x , y兲, for x ⬍ 0. Thus, put
4.2 Imposing the Boundary Condition. In the homogeneous
half-space, we write
Then, making the substitution k cosh ␶ = k0 cosh s in Eq. 共12兲
gives ⌬ = −k0 sinh s and
兺c ␾
u0 = uin + ure +
n
n
x = − R cos ⌰,
⌿n =
n
where ⌺n denotes summation over all integers n. Similarly, in the
graded half-space, we write
u共r, ␪兲 = utr +
兺c ␾
n
1
␲i
冕
n
⳵ utr
⳵r
=−
r=a
共15兲
r=a
To proceed, we write both sides of this equation as Fourier series
in ␪. For the right-hand side, we have
utr = e
−␤x
T be
ikr cos共␪−␣兲
=e
−␤x
Tb
兺
m
i Jm共kr兲e
im共␪−␣兲
B n共 ␨ ; ␤ 兲 =
where Tb = T exp共ikb cos ␣兲 and Jn is a Bessel function. Also, we
have the expansion
is
s
s
where In is a modified Bessel function. Hence,
utr共r, ␪兲 = e−␤b
兺 共− 1兲
m
冉
2共␨2 − k20兲1/2 exp兵− b共␨2 − k2兲1/2其 ␨ + 共␨2 − k2兲1/2
共− k兲
共␨2 − k2兲1/2 + ␤ + 共␨2 − k20兲1/2
冊
n
⌿n ⬃
⬃
1
Bn共− k0 sin ⌰; ␤兲eik0R
␲i
冑
冕
exp兵 2 ik0R共s − s0兲2其ds
1
2 i共k R−␲/4兲
Bn共− k0 sin ⌰; ␤兲
e 0
␲ k 0R
as
R→⬁
共19兲
共20兲
where the contour of integration in Eq. 共19兲 passes through the
saddle point.
When ␤ = 0, we obtain Bn共−k0 sin ⌰ ; 0兲 = ineik0b cos ⌰e−in⌰.
共1兲
Then, Eq. 共20兲 agrees with the known far-field expansion of Hn
⫻共kr兲ein␪, when one takes into account that ␪ ⬃ ␲ − ⌰ and r ⬃ R
+ b cos ⌰ as R → ⬁.
兺 共− 1兲 I 共␤r兲e ␪
s
兩⌰兩 ⬍ ␲/2
the square roots being defined to have non-negative real parts.
The formula for ⌿n, Eq. 共18兲, is convenient for estimating ⌿n
when k0R Ⰷ 1, as we can use the saddle-point method 共关16兴, Chap.
1
8兲. There is one relevant saddle point at s = s0 where s0 = i共 2 ␲
+ ⌰兲. As cosh s0 = −sin ⌰ and sinh共s0 + i⌰兲 = i, the standard argument gives
m
e −␤x = e −␤b
Bn共k0 cosh s; ␤兲ek0R sinh共s−i⌰兲ds,
−⬁
where
冏 冏 冏 冏
⳵ ␾n
⳵r
⬁+␲i
n
Then, by construction, the governing partial differential equations
and the interface conditions along x = 0 are all satisfied. It remains
to determine the coefficients cn using the boundary condition on
r = a, Eq. 共9兲; this gives
n
兩⌰兩 ⬍ ␲/2
共18兲
n
兺c
y = R sin ⌰,
Um共r兲eim␪
m
where
Um共r兲 = Tb
兺 共− i兲 I
s
m−s共␤r兲Js共kr兲e
−is␣
s
In a similar way, we obtain
⌽n共r, ␪兲 =
兺 共− 1兲
m n
f mJm共kr兲eim␪,
0⬍r⬍b
m
with
n
fm
共− 1兲n
=
␲i
冕
H共n1兲共kr⬘兲ein共␲−␪⬘兲 =
共− 1兲n
␲i
冕
⬁+␲i
ekx sinh ␶e−iky cosh ␶+kb sinh ␶−n␶d␶
−⬁
共21兲
⬁+␲i
A共␶兲e
2kb sinh ␶ −共m+n兲␶
e
−⬁
Hence
␾n共r, ␪兲 = e−␤b
4.4 Near-Field Behavior of ⌽n. As the expression for ⌽n,
Eq. 共11兲, is similar to Eq. 共10兲, it is reasonable to ask if ⌽n corresponds to a simple image term. To see that it does not, let us
define polar coordinates centered at the mirror-image point,
共x , y兲 = 共−b , 0兲: x = −b + r⬘ cos ␪⬘, y = r⬘ sin ␪⬘. Then, calculations
similar to those described in the Appendix show that
兺 共− 1兲
m
where
Journal of Applied Mechanics
m n
Vm共r兲eim␪
d␶
for 兩␪⬘兩 ⬍ ␲ / 2. The integral on the right-hand side of Eq. 共21兲
should be compared with the integral defining ⌽n, Eq. 共11兲. For
them to be equal, the function A共␶兲, defined by Eq. 共13兲, would
have to be constant: It is not, and it is not well approximated by a
nonzero constant. Thus, it is not justified to replace ⌽n with a
simple image term: We notice that Fang et al. 关2兴 used image
terms similar to those on the left-hand side of Eq. 共21兲, with ␲
− ␪⬘ replaced with ␪⬘.
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5
Discussion
We have outlined how to solve the scattering problem for a
cavity buried in a graded half-space; the result is the infinite linear
algebraic system, Eq. 共17兲. The system matrix is very complin
n
cated: One has to calculate 共d / dr兲Vm
共r兲 at r = a, where Vm
is defined by Eq. 共16兲 as an infinite series of special functions with
coefficients given as contour integrals. In principle, the system
matrix could be computed but it is unclear whether this is a worthwhile exercise, given the limitations of the underlying model, with
both shear modulus and density varying exponentially; see Eq.
共1兲. However, it may be possible to extract asymptotic results
from the exact system of equations for small cavities or for cavities that are far from the interface: This remains for future work.
Appendix: Integral Representations
As explained in Sec. 4.1, we need to convert from polar coordinates to Cartesian coordinates in order to apply the interface
conditions at x = 0. This is done using certain integral formulas.
Thus, from Ref. 关17兴 共p. 178, Eq. 共2兲兲, we have the integral representation
1
␲i
H共n1兲共kr兲 =
冕
⬁+␲i
ekr sinh w−nwdw
共A1兲
−⬁
The integration is along any contour in the complex w-plane,
starting at w = −⬁ and ending at w = ␲i + ⬁. When w = ␰ + i␩, where
␰ and ␩ are real, 兩ekr sinh w兩 = ekr sinh ␰ cos ␩. Thus, we can generalize
Eq. 共A1兲 to
H共n1兲共kr兲 =
1
␲i
冕
⬁+i␩2
ekr sinh w−nwdw
共A2兲
−⬁+i␩1
where the constants ␩1 and ␩2 must satisfy
1
1
− 2 ␲ ⬍ ␩1 ⬍ 2 ␲
and
1
2␲
3
⬍ ␩2 ⬍ 2 ␲
In other words, we have some flexibility in our choice of contour,
flexibility that we shall exploit shortly.
Put w = ␶ + i共␪ − ␲兲. Then Eq. 共A2兲 becomes
H共n1兲共kr兲ein␪ =
共− 1兲n
␲i
冕
⬁+i␤2
e−kr共sinh ␶ cos ␪+i cosh ␶ sin ␪兲e−n␶d␶
−⬁+i␤1
共A3兲
where the constants ␤1 and ␤2 must satisfy
1
1
− 2 ␲ ⬍ ␤1 + ␪ − ␲ ⬍ 2 ␲
and
1
2␲
In particular, the choices ␤1 = 0 and ␤2 = ␲ show that
031009-4 / Vol. 76, MAY 2009
3
⬍ ␤2 + ␪ − ␲ ⬍ 2 ␲
H共n1兲共kr兲ein␪ =
共− 1兲n
␲i
for
冕
⬁+␲i
e−kr共sinh ␶ cos ␪+i cosh ␶ sin ␪兲e−n␶d␶
−⬁
1
2␲
3
⬍ ␪ ⬍ 2␲
共A4兲
This is the integral representation that we use in Sec. 4.1.
References
关1兴 Fang, X.-Q., Hu, C., and Du, S. Y., 2006, “Strain Energy Density of a Circular
Cavity Buried in Semi-Infinite Functionally Graded Materials Subjected to
Shear Waves,” Theor. Appl. Fract. Mech., 46, pp. 166–174.
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Cavity Buried in a Semi-Infinite Functionally Graded Material Subjected to
Shear Waves,” ASME J. Appl. Mech., 74, pp. 916–922.
关3兴 Fang, X.-Q., Hu, C., and Huang, W.-H., 2007, “Strain Energy Density of a
Circular Cavity Buried in a Semi-Infinite Slab of Functionally Graded Materials Subjected to Anti-Plane Shear Waves,” Int. J. Solids Struct., 44, pp.
6987–6998.
关4兴 Hu, C., Fang, X.-Q., and Huang, W.-H., 2008, “Multiple Scattering of Shear
Waves and Dynamic Stress From a Circular Cavity Buried in a Semi-Infinite
Slab of Functionally Graded Materials,” Eng. Fract. Mech., 75, pp. 1171–
1183.
关5兴 Hu, C., Fang, X.-Q., and Du, S.-Y., 2007, “Multiple Scattering of Thermal
Waves From a Subsurface Spheroid in Exponentially Graded Materials Based
on Non-Fourier’s Model,” Infrared Phys. Technol., 50, pp. 70–77.
关6兴 Fang, X. Q., and Hu, C., 2007, “Thermal Waves Scattering by a Subsurface
Sphere in a Semi-Infinite Exponentially Graded Material Using Non-Fourier’s
Model,” Thermochim. Acta, 453, pp. 128–135.
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