An analytical model of periodic waves in shallow water

An explicit, analytical model is presented of finite amplitude waves in shallow water. The waves in question have two independent spatial periods, in two independent horizontal directions. Both short-crested and long-crested waves are available from the model. Every wave pattern is an exact solution of the Kadomtsev-Petviashvill equation, and is based on a Riemann theta function of genus 2. These bi-periodic waves are direct generalizations of the well-known (simply periodic) cnoidal waves. Just as cnoidal waves are often used as one-dimensional models of typical nonlinear, periodic waves in shallow water, these bi-periodic waves may be considered to represent typical nonlinear, periodic waves in shallow water without the assumption of one-dimensionality.