  The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns
unusual modes in bounded elastic bodies: time-harmonic displacements whose
tractions and normal components are both identically zero on the boundary. This
problem is usually associated with a lack of unique solvability for certain
models of fluid-structure interaction. The boundary conditions in this problem
appear, at first glance, to rule out {\it any} non-trivial modes unless the
domain possesses significant geometric symmetries. Indeed, Jones modes were
shown to not be possible in most $C^\infty$ domains (see article by T. Harg\'e
1990). However, we should in this paper that while the existence of Jones modes
sensitively depends on the domain geometry, such modes {\it do} exist in a
broad class of domains. This paper presents the first detailed theoretical and
computational investigation of this eigenvalue problem in Lipschitz domains. We
also analytically demonstrate Jones modes on some simple geometries.
