A Rigorous Interpretation of Approximate Computations of Embedded Eigenfrequencies of Water Waves

Abstract

In this paper, we will investigate embedded eigenvalues in the framework of the linearized theory of water waves. We assume that an approximation of an embedded eigenvalue is provided. To the question, whether there is a trapped mode near the computed solution, we provide an affirmative answer.

We will prove that, under certain assumptions on the data for a water wave problem in an infinite channel ${\Omega }^0$, the result ${\lambda }^0$ of the numerical computation can be justified as follows: if the computational error $\epsilon$ is sufficiently small, there exists a water domain ${\Omega }^{\epsilon }$ which is a local regular perturbation of ${\Omega }^0$ and has an eigenvalue ${\lambda }^{\epsilon }\in [{\lambda }^0-c\epsilon ,{\lambda }^0+c\epsilon ]$ embedded in the continuous spectrum. This conclusion is made by means of an asymptotic analysis of the augmented scattering matrix, whose properties guarantee a sufficient condition for the existence of trapped mode.