Abstract
Within the context of the linearised theory of time-harmonic water waves in three dimensions, a number of identities are obtained that are satisfied throughout the fluid domain by the velocity potential in scattering problems. This is done for both incident plane waves and for incident cylindrical waves. The implications of these results for the solution of time-domain scattering problems by the method of expansion in generalised eigenfunctions are discussed. In particular, it is demonstrated explicitly, for both two and three dimensions, that two different formulations of the generalised eigenfunction method are equivalent. Further, a new representation is given for the time-domain solution as an integral over the angles of incidence for particular generalised eigenfunctions.
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