Surface Water Wave Scattering and the Hydrotope

We study the classical tree-level scattering amplitudes of deep-water surface gravity waves using the methods of high-energy physics. For scattering in one horizontal dimension and in the two-negative-wavenumber sector we obtain a closed formula for $n$-wave scattering. Up to a kinematic prefactor, the amplitude is the volume of a classic polytope -- a box sliced by a hyperplane, which we dub the hydrotope, whose purpose in life is simply to organize the sign patterns of the "chambers" characterizing all the different regions of the two-minus kinematic space. The general formula was discovered by Claude Opus 4.6 working under our guidance, beginning with our earlier discovery of a one-term expression valid in the "simplest" kinematic chamber. Our results resolve the puzzle raised by Y.V. Lvov's 1997 computation of the five-wave amplitudes, unifying and extending it to all multiplicities.