Compressible ocean waves generated by sudden seabed rise near a step-type topography.

This article focuses on modeling tsunami wave generation and propagation caused by vertical motion of the ocean floor over regions with variable bathymetry, incorporating the effects of ocean compressibility and static compression within linearized water wave theory. Using Fourier transformation and eigenfunction matching methods, the study derives analytic solutions for surface displacement and pressure profiles across regions of differing depths, extending previous constant-depth models to variable-depth scenarios. A novel energy balance relation is established, accounting for both pure-gravity and acoustic-gravity wave (AGW) modes, and the model is validated by convergence tests and limiting cases. Time-domain simulations reveal wave reflection at depth changes, standing wave formation in shallower regions, and significant pressure fluctuations linked to compressibility effects, with three-dimensional axisymmetric solutions showing more gradual tsunami wavefront slopes compared to two-dimensional cases. The work provides detailed mathematical formulations, numerical methods, and visualizations to enhance understanding of tsunami wave behavior near their generation zones.