Boussinesq's equations for (2+1)-dimensional gravity-surface waves in an ideal fluid model

TL;DR

This paper derives and analyzes (2+1)-dimensional Boussinesq equations for gravity surface waves in an ideal fluid, including effects of surface tension, and connects them to known (1+1)-dimensional wave equations.

Contribution

It systematically derives (2+1)D Boussinesq equations for gravity waves with various parameters and relates them to classical wave equations in reduced dimensions.

Findings

Boussinesq equations cannot be simplified to a single wave equation for surface elevation.

They can be reduced to a nonlinear PDE for an auxiliary potential function.

Limiting to (1+1)D recovers KdV and related equations.

Abstract

We study the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation parameter $\delta$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations can not be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function $f(x,y,t)$ which determines the velocity potential but is not directly observed quantity. The solution $f$ of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.