Water‐waves modes trapped in a canal by a near‐surface rough body
Giuseppe Cardone, Tiziana Durante, С. А. Назаров · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2010
Abstract 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.