Compressible water-wave evolution equations for coupled gravity–acoustic modelling of long ocean waves

Semi-analytical studies have demonstrated that water compressibility, seabed elasticity, and gravitational potential modify tsunami phase speed and can explain systematic arrival-time deviations observed in farfield measurements [1]. However, operational and research tsunami models remain based on incompressible formulations, preventing explicit simulation of acoustic modes and limiting investigation of gravity–acoustic coupling in large-scale free-surface flows.

We present the derivation and numerical implementation of a compressible set of water-wave evolution equations compatible with the widely used finite-volume tsunami modelling frameworks. Starting from the compressible Euler equations, the formulation retains weak compressibility and acoustic propagation while preserving the long-wave structure required for basin scale simulations. Particular attention is given to the pressure closure, dispersion relation, and numerical consistency with existing solvers.

The equations are being implemented within an open-source solver and validated against analytical limits and controlled numerical benchmarks. Preliminary results demonstrate stable coexistence of surface-gravity and acoustic modes, recovery of expected dispersion behaviour, and improved consistency of wavefront propagation speed relative to incompressible formulations. Synthetic impulsive source experiments of landslides illustrate the generation and radiation of coupled hydroacoustic–surface wave fields and their sensitivity to compressibility effects.

The proposed framework provides a physically consistent pathway for extending dispersion based corrections into fully time-dependent numerical models, which enables systematic investigation of gravity–acoustic coupling, compressibility effects, and wave–acoustic energy partitioning in long-wave ocean dynamics. The formulation also establishes a foundation for coupling numerical wave physics with hydroacoustic observations in future integrated modelling studies.

Reference

[1] A. Abdolali, U. Kadri, & J. Kirby, 2019. Effect of Water Compressibility, Sea-floor Elasticity, and Field Gravitational Potential on Tsunami Phase Speed. Scientific Reports, 9 (1), 1-8.