New properties and dispersion relation in linear elasticity

In the case of an elastic strip we exhibit two properties of dispersion cur ves lambda(n), n greater than or equal to 1, that were not pointed out prev iously. We show cases where lambda'(n)(0) = lambda "(n)(0) = lambda'''(0) = 0 and we point out that these curves are not automatically monotoneous on R+. The non monotonicity was an open question (see [2], for example) and, f or the first time, we give a rigourous answer. Recall the characteristic pr operty of the dispersion curves: {lambda(n)(p); n greater than or equal to 1}is the set of eigenvalues of A(p) counted with their multiplicity. The op erators A(p) p is an element of R are the reduced operators deduced from th e elastic operator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relation D(p,lambda) = 0 i n a general framework, and not only for a homogeneous situation (in this la st case the relation is explicit). Recall that a dispersion relation is an implicit equation the solutions of which are eigenvalues of A(p). The main property of the function D that we build is the following one: the multipli city of an eigenvalue lambda of A, is equal to the multiplicity it has as a root of D(p,lambda) = 0. We give also some applications. AMS Subject Class ification. 73D25, 73D20, 35P.