On the null-field equations for water-wave radiation problems

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Abstract
Consider a rigid body which is performing simple harmonic oscillations of small amplitude in the free surface of deep water under gravity. Under certain geometrical conditions on ∂D, the wetted surface of the body, it is known that the linear boundary-value problem [Pscr ] for a corresponding velocity potential ϕ is uniquely solvable at all frequencies. The usual method for solving [Pscr ] is to derive a Fredholm integral equation of the second kind over ∂D. There are two familiar ways of doing this: (i) represent ϕ as a distribution of simple wave sources over ∂D, leading to an integral equation for the unknown source strength; (ii) apply Green's theorem to ϕ and a simple wave source; when the field point lies on ∂D, this gives an integral equation for the boundary values of ϕ. It is well known that both of these integral equations have unique solutions, except at the same infinite discrete set of frequencies (the irregular frequencies).In this paper, we shall describe an alternative method for solving [Pscr ]: when the field point, in (ii), lies inside the body, we obtain an integral relation. If the simple wave source has a suitable bilinear expansion, this integral relation may be reduced to an infinite set of equations for the boundary values of ϕ. These equations, called the ‘null-field equations for water waves’, appear to be new; equations of this type were first obtained by Waterman for electromagnetic and acoustic scattering problems. The required bilinear expansion has been given by Ursell (1981) for two dimensions, and is given here for three dimensions. Using these, we show that the null-field equations always have a unique solution – irregular frequencies do not occur. This result is proved here for water waves in two and three dimensions. Similar results may be obtained for water of constant finite depth.

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