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.

The two-dimensional time-harmonic acoustic scattering by a semi-infinite waveguide composed of two parallel rigid plates is considered. An original mode matching method is developed to avoid the use of the Wiener–Hopf technique, generalizing usual mode matching methods to the case of unbounded media. A brief description of the method is given by means of Fourier decomposition leading to a well-posed variational problem with unknown being the trace of the solution on a well-chosen interface. From the numerical point of view, a local approximation is first considered by using Lagrange P1 finite elements on a segment of the interface. This leads to the computation of oscillatory integrals involving Fourier transform and complex square root functions. As a matter of accuracy, a special function is added to the finite element space, in order to take into account the asymptotic behavior of the solution. Finally, this method is extended to deal with local perturbations of the media by coupling the previous method to a classical integral one.

International audienceL'objet de cette Note est de proposer une méthode pour l'étude de la diffraction par un réseau de plaques horizontales semi-infinies localement perturbé par un obstacle. La méthode proposée couple une équation variationnelle posée dans un domaine borné entourant l'obstacle et une équation pseudo-différentielle écrite sur la droite verticale située à l'extrémité des plaques. Après avoir donné une formulation variationnelle du problème, on montre que celui-ci relève de l'alternative de Fredholm, en dehors des fréquences de résonance du réseau

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I was lucky to meet Yoichiro Nambu at the beginning of my scientific activity. The experience of working with him influenced my subsequent research and in the following I will try to convey what he transmitted to me. It was also a friendship that continued for decades in spite of the rare occasions to meet after our collaboration.