Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

TL;DR

This paper studies a dissipative Boussinesq system modeling wave phenomena like undular bores, classifies shock waves based on dispersion and dissipation, and compares numerical solutions with laboratory data to validate the model.

Contribution

It introduces a classification of diffusive-dispersive shock waves in a dissipative Boussinesq system and analyzes the accuracy of the Peregrine system in modeling undular bores.

Findings

Traveling wave solutions include oscillatory and regularized shock waves.

Numerical solutions match laboratory data across various phase speeds.

Error between dissipative and non-dissipative Peregrine systems is proportional to dissipation and time.

Abstract

We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accurately described in a wide range of phase speeds. Undular bores are often described using the original Peregrine system which, even if not possessing traveling waves tends to provide accurate approximations for appropriate time scales. To explain this phenomenon, we show that the error between the solutions of the dissipative versus the non-dissipative Peregrine systems are proportional to the dissipation times the observational time.