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 ε> 0 while the distance of the body to the water surface is also of order .ε 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 ε(d,N) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case ε∈ (0, ε(d,N)). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes