On the two-dimensional periodic surface waves occurring in rectangular vessels: Theory versus experiment

This paper is a four part dialectic between theory and experimental observations and measurements of two dimensional periodic surface waves in a vessel of rectangular cross section. The first two parts consist of minor contributions to the existing theory and experimental measurements of elliptic (deep water) standing waves and the (shallow water) travelling hydraulic jump. A new theory is proposed in the third part which explains the transition from the elliptic standing wave to the hyperbolic traveling hydraulic jump. The result is a family of standing cnoidal waves. To lowest order the solution is composed of two non-interacting periodic progressive cnoidal waves travelling in opposite directions. At higher order the two wave trains interact but the interaction produces nothing of qualitative significance except when the cnoidal standing wave of highest amplitude is considered. This family of waves is validated by experimental observation in part four. In addition in part four observation of compound cnoidal standing waves leads to a conjecture about a sequence of continuous wave forms in parameter space leading from the linear sinusoidal wave to the travelling hydraulic jump.