An Analytical Model of Periodic Waves in Shallow Water

Abstract

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‐Petviashvili equation, and is based on a Riemann theta function of genus 2. These biperiodic 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 biperiodic waves may be considered to represent “typical” nonlinear, periodic waves in shallow water without the assumption of one‐dimensionality.

Three-Dimensional Long Water-Wave Phenomena

“…For N = l one recovers the two-dimensional cnoidal waves, while N = 2 yields doubly periodic waves that may be interpreted as two obliquely interacting cnoidal waves (Segur & Finkel 1985). These In related experimental work, Hammack et al (1989) studied doubly periodic wave disturbances in the laboratory and made a comparison with the corresponding genus-2 solutions of the KP equation.…”

Finite-gap solutions of the Davey-Stewartson equations

“…We shall not discuss the relation of the parameters of these solutions to real physical values. This interesting problem for the simplest solutions can be solved in a similar way to that used for the KP equation [6].…”

Soliton interaction as a possible model for extreme waves in shallow water

“…The resonance in this context is related to two interacting solitons (Miles, 1977a, b) resulting in a single soliton that was resonant with both interacting waves. In this framework, many authors have demonstrated that the amplitude of the water elevation at the intersection point of two solitons may exceed that of the sum of incoming solitons (see, for example, Segur and Finkel, 1985;Haragus-Courcelle and Pego, 2000;Tsuji and Oikawa, 2001;Chow, 2002) but neither the limits of the elevation nor the spatial occupancy of the high elevation have been analysed in detail. This model allows to consider the solitons with arbitrary amplitudes whereas in the case of the Mach reflection the amplitudes of the counterparts generally are related to each other.…”