On the non-existence of surface waves trapped by submerged obstructions having exterior cusp points

Trapped modes, i.e. localized unforced oscillations of fluid in presence of floating structures and bottom topography, have been a topic of considerable interest over many years and substantial effort has been put, for a variety of different geometries, into finding the solutions and conditions of their existence or non-existence. However, the class of geometries having cusp points has been avoided in existing proofs of non-existence of trapped modes, though problems of scattering of water waves by such structures have been considered and examples of trapped modes are known. In the work we consider the case when the depth profile or submerged obstacles have exterior cusp points with horizontal tangents. Under some geometrical restrictions a proof of non-existence of trapped modes is given with the help of the so-called Maz'ya's integral identity. The identity suggested in [1, 2, 3] states that a quadratic form in the potential of trapped mode is equal to zero, which yields non-existence of trapped modes if the form, depending on the geometry, is non-negative. For the obstruction under consideration a generalization of the identity is derived including coefficients of local asymptotics of the trapped mode potential near cusp points of the contour.