A fast method for the solution of a hypersingular integral equation arising in a waveguide scattering problem

In this paper we study the time-harmonic electromagnetic scattering problem associated with a T-junction between two rectangular waveguides. This junction is composed by an infinite (primary) waveguide and by a semi-infinite (secondary) waveguide with the same height and coupled through a common aperture. A standard calculation reduces Maxwell equations to a simpler scalar two-dimensional non-homogeneous Helmholtz equation, to which the domain decomposition technique is applied. In each waveguide the scattered electric field satisfies a non-homogeneous Helmholtz equation with homogeneous boundary conditions. An integral representation for this field is then obtained, from which the corresponding expression for the scattered magnetic field follows. By enforcing the continuity of the tangential component of the total electric and magnetic fields generated in the two separate waveguides at their interface, we obtain a hypersingular integral equation defined on an interval. From the solution of this equation the scattering matrix of the junction is easily computed. The integral equation has two kernels given in terms of series expansions. Our analysis determines the relevant singular components and shows that besides the standard second-order hypersingularity, one of the kernels also has a fixed-point second-order hypersingularity at each endpoint of the interval of integration. The equation is finally solved by means of a Galerkin method, whose implementation is performed quite efficiently, so that the overall numerical method is very fast and accurate. The complete numerical solution is presented in the form of parametric plots and convergence results are discussed.