Polynomes de Tchebytchev et modes de transmission totale dans les multicouches périodiques

Abstract
The acoustic wave propagation in periodically layered media has been successfully described by Floquet theory during these last years. However, this method requires the determination of eigenvalues of the period transfer matrix, which may cost time for numerical developments. Moreover, a number of numerical (and experimental) observations, such as the existence and frequency distribution of total transmission modes, cannot be explained through the Floquet formalism. Here we propose an alternative approach, based on matricial algebraic properties, which enables to express the transfer matrix of the whole structure in terms of the period matrix and its trace as an argument of Tchebytchev polynomials of the second kind. Besides numerical facilities introduced by this new algorithm, the analytic form of the transfer matrix so obtained gives the key for the understanding of the distribution of eigenvibrations and total transmission modes of the N-periodic structure. As a main result, two families of modes are shown to be present: the first one is associated with the basic period, the other depends on the whole structure.