Stationary wave patterns in deep water

Abstract

The classical problem of stationary wave patterns in deep water, generated by a “ship” or an obstacle in a stream, is revisited. The wave patterns are calculated using the results of the method of stationary phase. This allows for an elegant geometrical construction in which the reciprocal polar of the wave normal diagram reproduces the wave pattern. Analytical expressions are given for the Kelvin wedge deltoid (the gravity wave) and for the parabolic-like curves for the capillary wave. The conditions under which the Kelvin wedge shape disappears is calculated and linked to the “vanishing” of the point of inflection in the gravity wave normal curve. A further, unexpected, feature is the appearance of a “Froude” or “Mach” like line at the critical wave normal angle where the two modes coalesce. This line joins onto the “short wavelength” portion of the gravity wave branch in a cusped loop-like structure. These interesting new features are explained and highlighted in a series of figures which depict the evolution of stationary wave patterns as a function of the ratio of the stream speed to a critical speed (determined by a combination of surface tension and gravity) as it decreases from “high” to “low” values.