Water-waves modes trapped in a canal by a near-surface rough body

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter epsilon > 0 while the distance of the body to the water surface is also of order epsilon. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely it is proved that, for any given d > 0 and integer N > 0, there exists a number epsilon(d, N) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case epsilon is an element of (0, epsilon (d, N)). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim