Non existence of further closed form 2-D sloshing modes in a symmetric triangular basin

The study of sloshing modes has a history dating back to Euler. Very few 2-D or 3-D closed form solutions have been constructed and these pertain to the simpler case of inviscid, small amplitude, fluid motion. In 2-D they exist for symmetric channels with vertical walls or straight walls inclined at 45∘ or (even modes only) 60∘ to the vertical. Fokas' extended transform method facilitates a solution of Laplace's equation in a convex polygon which, for favorable choices of polygon shape and boundary conditions, reduces the solution to a sum of residues in the complex transform-plane. For closed form sloshing modes, this requires all poles to be associated with a common eigenvalue, which is shown to be impossible in some standard cases and for other angles that are rational multiples of π.