**MSC2010** 33D45, 34B24, 35C07, 35C10, 35L05

### Eigenmodes of water oscillations in the closed basin of variable depth

#### A. V. Bagaev^{1}, E. N. Pelinovsky^{2}

Annotation | Eigenvalue problem for variable coefficient wave equation describing small oscillations of an incompressible ideal single-layer or two-layer fluid in a closed basin with uneven bottom is discussed. Eigenmodes of oscillations are founded in the channel, the width and depth of which are functionally associated and vanish on a frontier of channel. It is shown that such eigenmodes are expressed through Chebyshev polynomials of the second kind. Some properties of the eigenmodes are found. In particular, eigenmodes are described for the following configurations of channel: 1) constant width, 2) constant depth, 3) coherent channel of variable width and depth. In the first case the parametric form and in two other cases the explicit form of eigenmodes are found. In conclusion, the physical interpretation and the feasibility of the obtained solutions are discussed. |
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Keywords | varible-coefficient wave equation, Klein–Gordon equation, Sturm–Liouville problem, Chebyshev polynomials of the second kind, water oscillations in the closed basin |

^{1}**Andrey V. Bagaev**, Associate Professor, Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev (24 Minin Str., Nizhny Novgorod, 603950, Russia), PhD (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-5155-4175, a.v.bagaev@gmail.com^{2}**Efim N. Pelinovsky**, Professor, Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev (24 Minin Str., Nizhny Novgorod, 603950, Russia), Professor, Department of Information Systems, Nizhny Novgorod Branch of High School of Economics (25/12 Bolshaya Pecherskaya Str., Nizhny Novgorod, 603155, Russia), Chief Scientist, Institute of Applied Physics (46 Ul'yanov Str., Nizhny Novgorod, 603950, Russia), PhD (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-5092-0302, pelinovsky@gmail.com

**Citation**: A. V. Bagaev, E. N. Pelinovsky, "[Eigenmodes of water oscillations in the closed basin of variable depth]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 126–138 (In Russian)

**DOI** 10.15507/2079-6900.19.201701.126-138