On trapping of surface water waves by cylindrical bodies in a channel

A channel of infinite length and depth and of constant width contains ideal inviscid heavy fluid having a free surface. The fluid is bounded internally by cylindrical bodies with the cross-section of arbitrary shape; the cylinders span the channel and have their generators normal to the sidewalls. The linearised problem of time-harmonic motion of the fluid is considered and the range of frequencies of oscillations below the continuous spectrum is studied. A monotonicity principle, which compares eigenfrequencies for two nested fluid domains having the same free surface, is proved. The principle also yields existence of eigensolutions (so-called trapped modes) for the smaller fluid domain if they exist for the bigger one. This allows us to improve substantially known results on existence of trapped modes, and bounds for the eigenfrequencies. In particular, the existence of trapped modes for systems, including both totally and partly submerged bodies, is first proved.