Criteria for trapped modes in a cranked channel with fixed and freely floating bodies

Trapped modes in the linearized water wave problem are localized free oscillations in an unbounded fluid with a free surface. For sometime, it has been known that certain structures, fixed or freely floating, can support such modes. In this paper, we consider the problem on a channel, which consists of a finite part and two cylindrical outlets into infinity. The finite (bounded) part may contain some submerged and/or surface-piercing bodies. Since the ordinary scattering matrix can by no means contribute any information on trapped modes, we introduce the fictitious scattering operator and present a criterion for the existence of trapped modes. The criterion states that the number of trapped modes is the difference of the multiplicities of the eigenvalue 1 of the fictitious scattering operator and the eigenvalue − i of the scattering matrix.