In this talk, we introduce an unusual eigenvalue problem that arises in fluid-structure interaction problems: the Jones eigenmode problem, described first by D.S. Jones in the eighties. The Jones eigenvalue problem was stated in the context of fluid-structure interaction problems where a bounded elastic body is immersed in an unbounded inviscid compressible fluid. In this model, there may exist frequencies such that the elastic obstacle sustains time-harmonic displacements whose normal components as well as tractions are identically zero on the boundary, and the fluid-structure problem fails to possess unique solutions.

The starting point of our work is the mathematical eigenvalue problem. In this unusual problem the existence of eigenvalues intimately depends on the shape of the boundary; indeed, it has been proved that almost all domains with infinitely smooth boundary do not possess such modes. The situation for Lipschitz domains has not been deeply studied. In this paper, we describe these eigenmodes for a range of planar domains. Analytic expressions are obtained for simple domains, and we confirm the existence of these modes for a range of other shapes numerically using a finite element strategy.