We are interested in the theoretical study of a spectral problem
arising in a physical situation, namely interactions of fluid-solid
type structure. More precisely, we study the existence of solutions
for a quadratic eigenvalue problem, which describes the vibrations of a
system made up of two elastic bodies, where a slip is allowed on their
interface and which surround a cavity full of an inviscid
and slightly compressible fluid. The problem shall be treated like a
generalized eigenvalue problem. Thus by using some functional analysis
results, we deduce the existence of solutions and prove a spectral
asymptotic behavior property, which allows us to compare the spectrum
of this coupled model and the spectrum associated to the problem without
transmission between the fluid-solid media.