Irregular frequencies and the motion of floating bodies

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Abstract
A floating horizontal cylinder of infinite length is performing simple harmonic oscillations of small amplitude in the free surface of a uniform inviscid fluid under gravity. The cylinder intersects the mean free surface at right angles, and the fluid is bounded below by a fixed horizontal plane. Let the corresponding two-dimensional velocity potential be expressed as a distribution of simple wave sources over the boundary of the body. Then it is known that the source density satisfies a Fredholm integral equation of the second kind which has a unique solution except at those frequencies (the irregular frequencies) at which the Fredholm determinant vanishes. The present work is concerned with the irregular frequencies. Let the simple wave sources be replaced by a fundamental solution which consists of a simple wave source together with additional wave singularities inside the cylinder. It is shown how irregular frequencies can be eliminated by an appropriate choice of these singularities.

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