Non–uniqueness in two–dimensional water wave problems: numerical evidence and geometrical restrictions

Abstract

In this paper numerical evidence for the existence of eigenvalues in a particular two‐dimensional water wave problem is presented, together with computations of the eigenvalues themselves. The geometry considered is that of a pair of identical inclined surface‐piercing barriers arranged symmetrically about a vertical line in infinitely deep water. Only oscillations symmetric about this line are considered so that the geometry is equivalent to an inclined surface‐piercing barrier next to a vertical wall. The problem is formulated as a hypersingular integral equation which is then solved numerically by expanding the unknown function, corresponding to the jump in potential across the barrier, as a series of Chebyshev polynomials. We also prove that for a more general class of bodies consisting of a pair of surface‐piercing bodies placed symmetrically about a vertical line midway between them, there are ranges of the spectral parameter for which no eigenvalues exist. The numerically computed eigenvalues for the inclined barriers are consistent with these results.