Sloshing in triangular basins

• Closed-form sloshing spectra and mode shapes are derived for triangular basins. • Area normalisation gives scale-free wavenumbers/frequencies for fair comparisons. • Shallow- and deep-water limits of the fundamental frequency follow from dispersion. We present closed-form sloshing spectra for three uniform-depth triangular basins: the equilateral, the hemi-equilateral, and the right-isosceles triangle. Within linear potential theory, the free-mode problem reduces to a Neumann Helmholtz eigenproblem on the horizontal planform. For each basin we give an explicit real trigonometric eigenbasis and a closed-form expression for the wavenumbers, yielding fully explicit mode shapes and sloshing frequencies through the dispersion relation. To compare geometries on an equal footing, we introduce an area-based normalisation and tabulate the first ten non-zero area-normalised wavenumbers for each basin; the low-lying spectral ordering and interleaving are further illustrated over the first thirty modes. The depth dependence of the fundamental frequency is interpreted via its shallow- and deep-water limits.