An expansion theorem for water-wave potentials (Q1080304)

Consider an infinitely long, horizontal cylinder of arbitrary cross section, floating on the free surface of an inviscid, incompressible fluid of infinite depth. The fluid motion is assumed two-dimensional, irrotational and of small amplitude, and it is described by a wave potential satisfying the Laplace equation, the usual linearized free- surface and body-boundary conditions, as well as proper conditions at infinity. A general multipole expansion for the wave potential is derived, converging throughout the fluid domain. Conditions are also stated under which the corresponding expansion for the fluid velocity converges up to and on the body boundary. In this case the multipole expansion may be used in the numerical solution or in the theoretical study of various water-wave problems. To obtain the expansion, a decomposition of the wave potential in a regular wave, a wave source, a wave dipole and a regular wave-free part is first invoked. Subsequently, using Texeira's series and the conformal mapping between the semicircular region \(| \zeta | \geq 1\), Im \(\zeta\leq 0\), and the fluid domain, it is shown that the regular part of the wave potential can be represented by a convergent series of wave-free multipoles, which are given explicitly in terms of the mapping function.