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 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 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.

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.