We analyze the approximation of a vibro-acoustic eigenvalue problem for an elastic body which is submerged in a compressible inviscid fluid in R 3. As model the time-harmonic elastodynamic and the Helmholtz equation are used and are coupled in a strong sense via the standard transmission conditions on the interface between the solid and the fluid. Our approach is based on a coupling of the field equations for the solid with boundary integral equations for the fluid. The coupled formulation of the eigenvalue problem leads to a nonlinear eigenvalue problem with respect to the eigenvalue parameter since the frequency occurs nonlinearly in the used boundary integral operators for the Helmholtz equation. The nonlinear eigenvalue problem and its Galerkin discretization are analyzed within the framework of eigenvalue problems for Fredholm operator-valued functions where convergence is shown and error estimates are given. For the numerical solution of the discretized nonlinear matrix eigenvalue problem the contour integral method is a reliable method which is demonstrated by some numerical examples.

As a vibro-acoustic interface model problem we consider a three-dimensional elastic body, e.g., a submarine, which is completely immersed in a full space acoustic region, e.g., water [5]. Other applications that we have in mind are the sound radiation of passenger car bodies, where the acoustic region is bounded, or of partially immersed bodies such as ships, where the acoustic region is a half space [2].In this paper, we consider both a direct simulation of the interface problem by using a symmetric coupled finite and boundary element approach, and an eigenvalue analysis to determine the eigenmodes of the coupled system. The time-harmonic vibrating structure in˝s is modeled by the Navier equations in the frequency domain, while the acoustic fluid in the unbounded exterior domain˝f is described by the Helmholtz equation,In (1), and are the Lamé parameters, % s and % f are the densities of the structure and of the acoustic fluid, respectively, ! is the frequency, and Ä D !=c 2 R is the wave number. Note that˝s