In the case of an elastic strip we exhibit two properties of
dispersion curves λn,n ≥ 1, that were not pointed out
previously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on
${\mathbb{R}}_{+}$. The non monotonicity was an open question (see [2],
for example) and, for the first time, we give a rigourous answer. Recall the
characteristic property of the dispersion curves: {λn(p);n ≥ 1} is
the set of eigenvalues of Ap, counted with their multiplicity. The
operators Ap, $p\in{\mathbb{R}}$, are the reduced operators deduced from the elastic
operator A using a partial Fourier transform. The second goal of this article
is the introduction of a dispersion relation
D(p,λ) = 0 in a general framework, and not only for a homogeneous situation
(in this last case the relation is explicit). Recall that a dispersion
relation is
an implicit equation the solutions of which are eigenvalues of Ap. The main
property of the function D that we build is the following one: the
multiplicity of an eigenvalue λ of Ap is equal to the multiplicity it
has as a root of D(p,λ) = 0. We give also some applications.