Numerical method for surface waves in a cylindrically perturbed elastic half-space. Part 1: Construction and analysis

We are interested in the numerical approximation of guided surface waves propagating in a three-dimensional elastic half-space with a local geometrical perturbation which is infinite and invariant under translation in one space direction (the propagation direction) and localized in the transverse directions. Such a problem amounts to solving a family of self-adjoint eigenvalue problems set in an unbounded domain, where ω+2$/ (ω being the frequency) plays the role of the eigenvalue and where the wave number β appears as a parameter. Our objective is to design a numerical method for the computation of such waves in order to study numerically the existence of and the dispersion relation of such surface waves. The main difficulty lies in the unboundedness of the transverse-cross section which does not allow a direct calculation. The main point of our method is thus to reduce the problem to an equivalent one posed in a bounded domain, namely, the geometrical perturbation. In order to do so, we construct a transparent boundary condition posed on the artificial interface. This boundary condition can be written as a generalized impedance condition via the introduction of a suitable pseudodifferential operator. The way we treat this operator numerically is nonclassical. It consists of looking at its inverse which, contrary to the original operator, appears to be analytically computable with the help of the Fourier transform and complex variable techniques. We then construct an approximation of the impedance operator as the inverse of an approximation of its inverse. We give a detailed description of this method and present its main properties. In a second paper we shall discuss more computational issues and present numerical results.