Short surface waves due to an oscillating immersed body

Abstract

A long circular cylinder of radius a , with its axis horizontal, is half-immersed in a fluid under gravity and is making periodic vertical oscillations of small constant amplitude and of period 2 π /σ about this position. It is required to find the resulting fluid motion when the parameter N = σ 2 a / g is large; the method of an earlier paper (Ursell 1949) is then unworkable. The present solution is made to depend on an integral equation (3∙15) which can be chosen to have a kernel tending to zero with N -1 , and which is solved by iteration. Successive terms in the iteration are of decreasing order, and the convergence of the method for sufficiently large N is proved. Expressions are given for the virtual-mass coefficient (5∙1) and for the wave amplitude at infinity (5∙7). The present work appears to be the first practical and rigorous solution of a short-wave problem when a solution in closed form is not available. It is suggested that a similar technique may be applicable to the diffraction problems of acoustics and optics, which have hitherto been treated by the approximate Kirchhoff-Huygens principle.