### A Matrix Approach to the Energy-Norm Bisection in Wave Motion

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The influence of a seismic wave on a building is customarily described as a force, a function of the time, whose explicit expression is prescribed. We here suggest a one-dimensional model able to relate this force to the sudden onset of a fault in the rock layer on which the building is built.

A new model for propagation of long waves including the coastal area is introduced. This model considers only the motion of the surface of the sea under the condition of preservation of mass and the sea floor is inserted into the model as an obstacle to the motion. Thus we obtain a constrained hyperbolic free-boundary problem which is then solved numerically by a minimizing method called the discrete Morse semi-flow. The results of the computation in 1D show the adequacy of the proposed model.

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators An, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of An is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators ${A}_{n}$, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of ${A}_{n}$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

In this article, we derive a complete mathematical analysis of a coupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.