Steady wave patterns on a non-uniform steady fluid flow

Abstract

A steady slightly non-uniform flow with a free surface is subject to a concentrated surface pressure which gives rise to a pattern of surface waves. (For gravity waves on deep water this is the well-known Kelvin ship-wave pattern.) The motion is assumed inviscid, and the waves are assumed small. A theory is developed for the wave pattern, based on the following assumptions: The stream velocity component normal to a wave crest is equal to the phase velocity based on the local wavelength; the separation between consecutive crests is equal to the local wave-length. These assumptions are expressed in mathematical form, and the existence of a set of characteristic curves (associated with the group velocity) is deduced from them. These characteristics are not identical with the crests. Let the additional assumption be made that the characteristics all pass through the point disturbance; the characteristics are then completely defined and may be constructed by a step-by-step process starting at the point disturbance. The same construction gives the direction of the wave crests at all points. The wave crests can then be deduced. Assumptions of the same type as (1) and (2) have long been familiar in various applications of ray tracing. For uniform flows the present theory gives the same pattern as the method of stationary phase.