Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study

TL;DR

This study compares the numerical evolution of solitary surface waves over uneven bottoms using single wave equations and Boussinesq equations, revealing significant differences in wave amplitude and velocity predictions.

Contribution

It provides a detailed numerical comparison between KdV-type equations and Boussinesq equations for modeling surface waves over uneven bottoms, highlighting their differences.

Findings

Boussinesq equations show greater influence of bottom variations on wave properties.

Differences between approaches are significant in all studied parameter cases.

The study enhances understanding of wave behavior over uneven terrains.

Abstract

The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters $\alpha,\beta,\delta$. For the flat bottom, these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-order KdV (KdV5), and the Gardner equation (GE). In all studied cases, the influence of the bottom variations on the amplitude and velocity of a surface wave calculated from the Boussinesq equations is substantially more significant than that obtained from single wave equations.