The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass. © 2007 Society for Industrial and Applied Mathematics.