On the existence of waves guided by a cavity in an elastic film

Abstract

In the regime where linearised elasticity is a suitable approximation to the behaviour of an elastic body, the existence of a wave guided along the cavity in a film of a homogeneous and isotropic elastic material is proved, at least for a certain range of frequencies. Using the theory of linear self-adjoint operators, it is shown that the associated eigenvalue problem has a nontrivial solution in an appropriate Sobolev class, with the corresponding eigenvalue lying below the continuous spectrum. We study the existence of localised modes of this kind in two particular cases: under the assumption that the cavity is sufficiently narrow in the direction transverse to the film and for a rectangular cavity of arbitrary size.