Criteria for trapped modes in a cranked channel with fixed and freely floating bodies |
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Authors: | S A Nazarov K M Ruotsalainen |
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Institution: | 1. Mathematics and Mechanics Faculty, St. Petersburg State University, 198504, Universitetsky pr. 28, Stary Peterhof, Saint Petersburg, Russia 2. Mathematics Division, University of Oulu, P.O. Box 4500, 90014, Oulu, Finland
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Abstract: | 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. |
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