On the Null-field Equations for Water-wave Scattering Problems

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Abstract
Consider a rigid body which is partially immersed in the free surface of deep water, the body is fixed and a given time-harmonic wave is incident upon it. The corresponding linear boundary-value problem is solved using the null-field method and two systems of null-field equations are obtained; each system is always uniquely solvable—irregular frequencies do not occur. These results can be proved in two or three dimensions, for water of infinite or finite depth, and for any incident wave. Next, it is assumed that the water is deep and that the incident wave is a regular surface wave. With these assumptions, one system of null-field equations takes on a very simple form. This system is solved using the “method of projection” (Gregory & Gladwell, 1982), leading to a sequence of computable approximations to the solution of the null-field equations; this sequence is guaranteed to converge. It seems worthwhile to establish this last result, because (at present) there is a notable absence of such convergence proofs in the general theory of the null-field method. As an example of this approach, the two-dimensional problem of the interaction between a regular surface wave and a fixed, half-immersed, elliptic cylinder is solved.

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