On uniqueness in the problem of gravity–capillary water waves above submerged bodies

In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.

The effect of surface tension on localized free-surface oscillations about surface-piercing bodies

It is well known that the inclusion of surface tension in the linear water-wave problem introduces an additional term in the free-surface boundary condition. Furthermore, if the fluid contains one or more partially immersed bodies, edge conditions describing the motion of the fluid at each contact line need to be applied. In this paper, an inverse procedure is used to construct examples of two-dimensional, surface-piercing trapping structures for non-zero values of surface tension. The problem is considered with two separate edge conditions that are appropriate for investigations involving trapped modes. The first edge condition fixes each contact line and the procedure used forces the bodies to be horizontal at the contact points; it is shown that results can be found for all values of surface tension. The second condition forces the free-surface slope to be zero at the contact points, and results are obtained for a restricted range of surface tension values.

The effect of surface tension on trapped modes in water-wave problems

In this paper the effect of surface tension is considered on two two-dimensional water-wave problems involving pairs of immersed bodies. Both models, having fluid of infinite depth, support localized oscillations, or trapped modes, when capillary effects are excluded. The first pair of bodies is surface-piercing whereas the second pair is fully submerged. In the former case it is shown that the qualitative nature of the streamline shape is unaffected by the addition of surface tension in the free surface condition, no matter how large this parameter becomes. The main objective of this paper, however, is to study the submerged body problem. For this case it is found, by contrast, that there exists a critical value of the surface tension above which it is no longer possible to produce a completely submerged pair of bodies which support trapped modes. This critical value varies as a function of the separation of the two bodies. It can be inferred from this that surface tension does not always play a qualitatively irrelevant role in the linear water-wave problem.