Eigenmodes of water oscillations in the closed basin of variable depth

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 for the channel with functionally associated width and depth. 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.