Multipole expansions in the theory of surface waves

Abstract

Problems dealing with the generation of surface waves in water involve the consideration of singularities of different types in the liquid. In the case when bodies are present in the liquid, waves may be either generated by the movement of the body, or reflected from the body. The two cases are essentially equivalent, and the resulting motion can be described by a series of singularities placed within the body. The boundary conditions on the surface of the body give equations from which the exact form of the potential can be obtained. Ursell (10) has solved in this manner the problem, earlier discussed by Dean(1), of the generation of surface waves by a submerged circular cylinder. In this two-dimensional problem he used a series of complex potential functions arising from multipoles at the centre of the cylinder, but the velocity potential of the motion could have been described, without the introduction of the stream function, in terms of the velocity potentials of the multipoles.