Asymptotics of sloshing eigenvalues for a triangular prism

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues in a 3D sloshing problem on a triangular prism with specific angles, deriving expansions and conjectures based on quasimodes and lattice counting methods.

Contribution

It provides a two-term asymptotic expansion for the eigenvalue counting function and constructs quasimodes to approximate eigenvalues for specific prism angles.

Findings

Exact second-term value for the $rac{ p{rac{ ext{pi}}{4}}}$ case.

Conjectured asymptotic expansion for general angles.

Quasimodes closely approximate true eigenvalues, linked to lattice counting problems.

Abstract

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form $\frac{\pi}{2q}$, where $q$ is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are $\frac{\pi}{4}$, we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.