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The cavity problem has been extensively studied due to its importance in the computation of radar cross section. A two-dimensional cavity can be used to model long seems or cracks in metallic surfaces which can significantly contribute to the overall radar profile of large objects. The cavity problem has been solved with a variety of methods including integral equations, finite elements, and Fourier methods. These approaches are primarily applied to a problem with a material filled cavity in a PEC ground place opening into an empty half space.
A neglected aspect of the problem is when the cavity is buried beneath a layered material. A two-dimensional cavity beneath a layered material can serve as model for seams and cracks in metallic surfaces which are covered by paint or materials applied during a manufacturing process. The material covering would mean that the cavities are invisible to a visual inspection. However, they may be revealed by understanding the scattering characteristics of the cavity when exposed to an electromagnetic field. The mathematical model can serve as a predictor of the scattering characteristics for use in nondestructive testing.
[FIGURE 1 OMITTED]
We want to compute the fields in the finite cavity region shown in Figure l.The computation can be done with integral equation methods. The disadvantage of this approach is that a layered Green's function must be used which adds difficulty to the problem and brings up convergence issues . A finite element approach can also be used but would require appropriate boundary conditions at the cavity opening . A key contribution of this manuscript is to provide the understanding of the Fourier approach in the overlayer setting on which an appropriate boundary condition is based.
In addition, the Fourier approach has some unique advantages. It is the sensible and natural approach for a rectangular geometry. Further, it is the only approach which can employ Morgan and Schwering's mode matching method and improved mode matching methods to get very fast approximations for large cavities and high frequencies. Very large problems with high wave numbers will become computationally cumbersome for any technique. Using the fast mode matching approach opens the door to dealing with very large problems which could not be dealt with by other conventional approaches [3,4].
2. Problem Setting
We consider a time harmonic plane wave with fixed frequency w incident onto a ideal metal or a perfect electrical conductor (PEC) half plane which is covered by a material layer(s) of uniform thickness. Embedded in the PEC half space is a cavity with an opening to the upper half space. As indicated in Figure 1, the region above the material layer is empty space and will be denoted by region 0, the region inside the material layer will be denoted by region 1, while the cavity will be called region 2. The opening of the cavity is the interval [0,L] in the x-direction. The depth of the cavity is denoted by d and top opening of the cavity is located at y = 0 while the bottom of the cavity is at y = -d. The cavity may be filled with a uniform material. The electric permittivities and magnetic permeabilities of the materials are denoted by [[member of].sub.k] and [[mu].sub.k] where k = 0,1,2 represents the region. Multiple material layers can be considered for both the inside of the cavity as well as the material covering the PEC half space. The paper will deal exclusively with a single material in both cases. The generalization to multiple layered materials follows easily once a single material is understood.
We will solve for the electric and magnetic fields denoted by E and H which satisfy Maxwell's equations  . In the two-dimensional setting, Maxwell's equations can be separated into two fundamental polarizations [ 5] . In the transverse magnetic (TM) polarization the electric field has only a z component which will be computed. We denote the z component of E by u. The other polarization is the transverse electric (TE) where the magnetic field H has only a z component which is denoted by v. In Sections 5 and 6, we will solve for the unknown functions u and v inside the cavity region. In both cases u and v will satisfy a Helmholtz equation with appropriate boundary conditions. Once the solutions are found in region 2 we will show how to compute far field values.
Note that in regions 0 and 1 the solutions can be decomposed into a superpositions of fields. We use a subscript to denote which region the field is contained. Thus [u.sub.k] and [v.sub.k] for k = 0,1,2 are the solutions in each of the respective regions. In region 0, the fields [u.sub.0] and [v.sub.0] are expressed as a sum of an incoming plane waves [u.sup.i.sub.0] and [v.sup.i.sub.0], outgoing plane waves [u.sup.r.sub.0] and [v.sup.r.sub.0], and a scattered fields [u.sup.s.sub.0] and [v.sup.s.sub.0]. The incoming waves in region 0 are known and specified. The remaining fields will be computed in following sections below. In region 1, the fields [u.sup.r.sub.1] and [v.sup.r.sub.1] also consist of incoming plane waves [u.sup.i.sub.1] and [v.sup.i.sub.1], outgoing plane waves [u.sup.r.sub.1] and [v.sup.r.sub.1] and scattered fields [u.sup.s.sub.1] and [v.sup.s.sub.0] all of which are unknown. When appropriate we will combine the incoming and outgoing plane waves into a single term = [u.sup.ir.sub.k] = [u.sup.i.sub.k] + [u.sup.r.sub.k] and [v.sup.ir.sub.k] = [v.sup.i.sub.k] + [v.sup.r.sub.k] for k = 0,1. Finally the fields [u.sub.2] and [v.sub.2] in the cavity region 2 will not be decomposed but solved as single fields.
When the known incident plane waves [u.sup.i.sub.0] and [v.sup.i.sub.0] come in from region 0 they interact with the material layer (region 1). The interaction produces reflected plane waves [u.sup.r.sub.0] and [v.sup.r.sub.0] as well as transmitted plane waves into region 1. The transmitted plane waves propagate into region 1 which produces plane waves propagating in the positive y-direction ([u.sup.r.sub.1] and [v.sup.r.sub.1]) and in the negative y-direction ([u.sup.r.sub.1] and [v.sup.r.sub.1]). While the plane waves [u.sup.i.sub.0] and [v.sup.i.sub.0] are given, the remainder of the plane waves must be computed. They are computed assuming that the cavity (region 2) is not present. Once the plane waves are completely known they are used as a source field which interacts with the cavity to produce scattered fields [u.sup.s.sub.k] and [v.sup.s.sub.k] with k = 0,1 in …