The wave resistance of a spheroid

Abstract

1. A method which has been used to calculate the wave resistance of a submerged solid is to replace the solid by a distribution of sources and sinks, or of doublets, the distribution being the image system for the solid in a uniform stream. The cases which have been solved hitherto have been limited to those in which the image system is either a single doublet or a distribution of doublets lying in a vertical plane parallel to the direction of motion. It is shown here how to obtain the solution for an ellipsoid moving horizontally at given depth below the surface of the water, and with its axes in any assigned directions. The present paper deals specially with prolate and oblate spheroids moving end-on and broadside-on, the general case of an ellipsoid with unequal axes being left for a subsequent paper. In 2 it is shown that the image system for an ellipsoid in a uniform stream is a certain surface distribution of parallel doublets over the elliptic focal conic, the direction of the doublets being in general inclined to the direction of motion; if the motion is parallel axis, the doublets are in the same direction. For a spheroid the image system reduces to either a line distribution or to a surface distribution over a certain circle; explicit expressions are given in 3 for prolate and oblate spheroids when moving either in the direction of the axis of symmetry or at right angels to that axis.