  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.
