## 1. Introduction

[2] The exact analytical solution to a new canonical problem is provided. Few exact analytical solutions exist for electromagnetic scattering problems, but they are very important because they provide useful insight into the interpretation of the behavior of the electromagnetic field. This new solution is possible due to the choice of isorefractive materials and of a particular geometry, which allows for the application of the boundary conditions using mode matching, thus leading to closed form solutions expressed as series. Additionally, the exact solution of this complicated problem, which involves sharp curved metallic edges, a cavity, and a curved surface separating different penetrable media provides a challenging test for the validation of frequency-domain computational software.

[3] The exact solution to a two-dimensional electromagnetic boundary-value problem involving a channel of semielliptical cross section with metallic walls flush-mounted under a metallic ground plane and coupled to the half-space above via a slot is considered. The half-space above the ground plane is separated from the material inside the channel by a multilayer diaphragm. The multilayer diaphragm, the material filling the channel, and the half-space are isorefractive to each other. The coupling slot occupies the interfocal distance in the cross section of the structure. The diaphragm is composed of *N* − 2 semielliptical shells confocal to the cavity wall and with their flat surfaces flush to the ground plane.

[4] This geometry is considered because it allows to model the penetration of electromagnetic radiation into a cavity or the radiation that escapes from an aperture of a cavity that contains an electromagnetic source. As an example, in a problem of electromagnetic compatibility, this geometry may model a wire inside a channel. The additional complication of the diaphragm models the presence of a mechanical cover that protects the cavity, similar to the function of a radome for a radar antenna.

[5] This is a two-dimensional geometry where the excitation is invariant with respect to the axial variable. The excitation is either a plane wave or a line source. For a plane wave source, the field is polarized with either or parallel to the axis of the structure and the direction of incidence is arbitrary in the plane of the cross section. For a line source, the or field is polarized parallel to the axis of the structure and the source is arbitrarily located in any of the *N* regions, but never exactly on any boundary.

[6] The exact solution is expressed in the form of series expansions involving Mathieu functions. The expansion coefficients in the series are determined analytically by imposing the boundary conditions, thereby leading to a canonical solution of the boundary-value problem. The notation for the Mathieu function is essentially the one of *Stratton* [1941], with the difference that the argument of the angular Mathieu function is the angle, rather than the cosine of the angle, as explained in *Blanch* [1966] and *Valentino and Erricolo* [2007].

[7] This work is an extension of the results by *Uslenghi* [2004a], where no diaphragm is considered, and *Valentino and Erricolo* [2007], where the diaphragm consists of only two layers. The technique used in the solution is similar to the one applied in *Erricolo and Uslenghi* [2005b], for which numerical results were given by *Erricolo et al.* [2005b, 2005a]. Prior related works consider two-dimensional cavities flush-mounted under a conducting ground plane with, however, the following differences: first, there is no diaphragm; second, the materials inside the cavity are usually the same as outside; third, the solution is not expressed in analytic closed form. For example, these prior works include: *Hinders and Yaghjian* [1991], who consider a semicircular channel with a dual series eigenfunction approach; *Lockard and Butler* [2004, 2006], who examine an arbitrarily shaped cavity with an integral equation approach; and *Scharstein et al.* [2007, 2008], who analyze rectangular trough with a variational approach.

[8] Numerical results based on the evaluation of the series of Mathieu functions are provided for the fields inside the channel, inside the diaphragm and in the open space above the structure. Issues such as the evaluation of the fields and surface currents on the metallic boundaries are analyzed in detail. The geometry of the problem is presented in section 2, and the solutions for plane wave incidence and line source excitation are shown in sections 3 and 4, respectively. Some numerical results for the different media are reported in section 5. The time-dependence factor exp(+*iω**t*) is omitted throughout.