Cauchy–Poisson problem of wave propagation in an ocean with an elastic bottom

The classical two-dimensional Cauchy–Poisson problem for an ocean modelled as an incompressible fluid with an elastic bottom is considered here. In accordance with the linear theory, the problem is formulated as an initial-value problem for the velocity potential in the fluid region, dilation potential, and rotational potential in the elastic medium below the fluid region. The Laplace transform in time and the Hankel transform in space are used in the mathematical analysis to obtain the form of the free surface depression and ocean bed vertical displacement component in terms of multiple infinite integrals. These integrals are evaluated asymptotically by the method of steepest descent. Variation of the ratio of the ocean bed amplitude to the free surface amplitude for different forms of the prescribed initial axially symmetric surface depression or the impulse for different values of elasticity parameters is investigated. The results obtained in the study are compared to the analytical solution of the problem in the case with a rigid bottom.