Cauchy–Poisson problem in a homogeneous liquid layer over a magnetoelastic half-space

The present work investigates the generation and propagation of wave motion produced by initial disturbances in a finite-depth ocean with an elastic bottom, influenced by a constant magnetic field acting in the normal direction of wave propagation. The objective is to derive an analytical solution to the Cauchy–Poisson problem for an ocean over an elastic bottom, modeled as an elastic solid medium, in presence of a uniform magnetic field. The fluid is assumed to be incompressible and is bounded above by a free surface and below by a homogeneous magnetoelastic half-space. By applying linear theory of water waves and linear elasticity theory for solids, the physical problem is formulated as an initial boundary value problem. The Laplace–Fourier integral transform method is employed to obtain analytical expressions for the free surface elevation and the vertical displacement of the seabed in the form of multiple infinite integrals. The method of steepest descent approximation is then applied to evaluate these integrals asymptotically. The results, illustrated through figures, highlight the effects of various key physical parameters on wave behavior. The dispersion relation governing the wave motion is also derived and analyzed. The findings reveal that the magnetic field significantly alters wave characteristics and mitigates wave impact. Additionally, variations in pressure and shear wave velocities are found to have a notable influence on wave propagation. Validation is carried out by comparing the results with existing literature for the special case of a rigid seabed.