Communicated by J. C. NedelecWe consider the two-dimensional scalar problem of the diffraction of a plane wave by an infinite grating of conducting bodies immersed in a periodical dielectric medium. A Fredholm-type formulation is derived and studied. The existence of a solution is proved and some uniqueness results are established. A detailed description of the guided modes of the grating is carried out. Finally, various non-uniqueness examples for the diffraction problem are exhibited.

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.

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ENSTA; SMP; URA 853-CNRS-32 bd Victor; 75739 Paris Cedex 15; France Communicated by J.C. Nedelec SUMMARY A generalized mode matching method that applies to a wide class of scattering problems is developed in the time harmonic two-dimensional Helmholtz case. This method leads by variational means to an integro-di erential formulation whose unknown is the trace of the ÿeld on an unbounded onedimensional interface. The well-posedness is proved after a careful study of the rather original functional framework. Owing to a fundamental density result-based upon some properties of a singular integral operator similar to the Hilbert transform-the limiting absorption principle related to this original formulation is established. Finally, two other situations are emphasized.

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