A Boussinesq system for two-way propagation of interfacial waves

TL;DR

This paper derives a Boussinesq system modeling two-way interfacial wave propagation in layered fluids, accounting for quadratic and cubic nonlinearities, and studies solitary wave interactions numerically.

Contribution

It introduces a new Boussinesq model for interfacial waves that includes cubic nonlinearities near critical depth ratios, extending previous quadratic models.

Findings

The model captures both quadratic and cubic nonlinear effects.

Numerical simulations show solitary wave propagation and collision behaviors.

Cubic nonlinearities become significant near specific depth ratios.

Abstract

The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion. A Boussinesq system for two-way propagation of interfacial waves in a rigid lid configuration is derived. In most cases, the nonlinearity is quadratic. However, when the square of the depth ratio is close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. The propagation as well as the collision of solitary waves and/or fronts is studied numerically.