  We are concerned with a coupled-physics spectral problem arising in the
coupled propagation of acoustic and elastic waves, which is referred to as the
acoustic-elastic transmission eigenvalue problem. There are two major
contributions in this work which are new to the literature. First, under a mild
condition on the medium parameters, we prove the existence of an
acoustic-elastic transmission eigenvalue. Second, we establish a geometric
rigidity result of the transmission eigenfunctions by showing that they tend to
localize on the boundary of the underlying domain. Moreover, we also consider
interesting implications of the obtained results to the effective construction
of metamaterials by using bubbly elastic structures and to the inverse problem
associated with the fluid-structure interaction.
