Wave resistance

1. The object of this paper is to give more direct proofs of certain expressions for wave resistance which have been used in previous calculations; further, in view of other possible applications, the expressions are generalised so that one can obtain the wave resistance for any set of doublets in any positions or directions in a uniform stream, or for any continuous distribution of doublets or equivalent sources and sinks. The only limitation is the usual one that the additional velocities at the surface are small compared with the velocity of the stream. One might take a simple source as the unit, but to avoid certain minor difficulties it would be necessary to assume an equal sink at some other point. The possible applications are to bodies either wholly, or with certain limitations partially, submerged. The image system in such a case consists of a distribution of sources and sinks of zero aggregate strength, and so may be replaced by an equivalent distribution of doublets. Hence it is simpler to use the doublet as the unit from the beginning. The wave resistance of a submerged sphere was obtained previously both by direct calculation of pressures on the sphere and by an analogy with the effect of a certain surface distribution of pressure. The latter method was then generalised to give the wave resistance of any distribution of horizontal doublets in a vertical plane parallel to the direction of the stream. In a recent paper Lamb has supplied a method for calculating wave resistance which avoids the comparison with an equivalent surface pressure; it consists in calculating the rate of dissipation of energy by a certain integral taken over the free surface when, as is usual in these problems, a small frictional force has been introduced into the equations of motion of the fluid. Lamb, however, deals only with a single doublet, to which a submerged body is equivalent to a first approxima­tion, and so does not obtain the interference effects which arise from an extended distribution of doublets; further, he carries out the necessary calculation by analysing first the surface distribution of velocity potential, or in effect analysing the wave pattern. In the following paper it is shown that this intermediate analysis may be avoided by a direct application of the Fourier double integral theorem in two dimensions. This step simplifies the extension of the calcula­tion to any distribution of doublets in any positions and directions; various cases, which it is hoped to use later, are given in some detail for deep water, and one case of a single doublet in a stream of finite depth.