An inhomogeneous half-space containing a cavity is bonded to a homogeneous half-space. 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.

## Introduction

Consider two half-spaces, $x>0$ and $x<0$, welded together along the interface at $x=0$. The left half-space $(x<0)$ 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 (see Fig.1). 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 $e\u2212\beta x$ for $x>0$, where $\beta $ is a given constant. The papers concern antiplane shear waves (1,2,3,4), thermal waves (5,6), and shear waves in a piezoelectric material (7). 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 (8). For plane-strain elastic waves in a homogeneous half-space with a buried circular cavity, see Ref. 9. There are also many papers on the scattering of electromagnetic waves by objects near plane boundaries; see, for example, Ref. 10.

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 (11) (construction of static Green’s functions in anisotropic elasticity) and a paper by Stevenson (12) (construction of Green’s function for the anisotropic Helmholtz equation in a half-space).

The basic scattering problem is formulated in Sec. 2. The reflection-transmission problem (for which the cavity is absent) is 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 (Sec. 4A). 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. 13 9. In Sec. 4B, 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. 4C. Closing remarks are made in Sec. 5.

## Formulation

It is not our purpose here to discuss whether any real materials can be well represented by the functional forms given in Eq. 1. Certainly, the choices in Eq. 1 do lead to some mathematical simplifications and they have been used in the past; see, for example, Refs. 14 15.

## Incident Field

For a simple check, put $\beta =0$; we obtain $k=k0$, $\alpha =\alpha 0$, $R=0$, and $T=1$, as expected.

## 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.

### Multipole Functions

Note that when $\beta =0$, $k=k0$, $\Delta (\tau )=\u2212k\u2009sinh\u2009\tau $, $A=0$, $B=1$, and $\Psi n=Hn(1)(kr)ein\theta $, as expected.

### Imposing the Boundary Condition

### Far-Field Behavior of Ψn

When $\beta =0$, we obtain $Bn(\u2212k0\u2009sin\u2009\Theta ;0)=ineik0b\u2009cos\u2009\Theta e\u2212in\Theta $. Then, Eq. 20 agrees with the known far-field expansion of $Hn(1)(kr)ein\theta $, when one takes into account that $\theta \u223c\pi \u2212\Theta $ and $r\u223cR+b\u2009cos\u2009\Theta $ as $R\u2192\u221e$.

### Near-Field Behavior of Φn

## 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. 17. The system matrix is very complicated: One has to calculate $(d/dr)Vmn(r)$ at $r=a$, where $Vmn$ is defined by Eq. 16 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. 1. 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.