## 1. Introduction

[2] 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 diaphragm. The 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 two semielliptical cylinders confocal to the cavity wall and with their flat surfaces flush to the ground plane.

[3] 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 **E** or **H** 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 **E** or **H** field is polarized parallel to the axis of the structure and the source is arbitrarily located in any of the four regions, but never exactly on any boundary. 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. The ability to obtain an exact analytical solution will provide insight into the interpretation of the electromagnetic field structure for problems where this geometry is applicable.

[4] 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. Preliminary results were given by *Erricolo and Uslenghi* [2005b]. The technique is an extension of that used by *Uslenghi* [2004a], for which numerical results were given by *Erricolo et al.* [2005a, 2005b]. The notation for the Mathieu function is that of *Stratton* [1941] and *Blanch* [1966] [*Staff of the Computation Laboratory*, 1967].

[5] This work is important because it provides the analytical solution to a new canonical problem and thus enriches the list of problems for which exact solutions are known. Furthermore, 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 codes. 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(+*iωt*) is omitted throughout.