A dispersive effective equation for transverse propagation of planar shallow water waves over periodic bathymetry

TL;DR

The work addresses long-wavelength shallow-water waves over a bottom that is periodic in the transverse direction. By applying a multiscale perturbation expansion and y-averaging, it derives a constant-coefficient two-dimensional Boussinesq-type system with a dispersive term determined by the bathymetry, governing the mean dynamics of waves traveling along $x$. The model predicts solitary waves whose mean profile is close to $\mathrm{sech}^2$, with a bathymetry-induced correction capturing two-dimensional structure; the full 2D shape is expressed via a simple correction term, yielding a one-parameter family of solutions. Numerical experiments using pseudospectral and finite-volume methods show excellent agreement with the full variable-bottom shallow-water system across smooth and piecewise-constant bathymetries. The results provide a practical homogenized framework to predict 2D wave shapes over periodic bottoms and motivate extensions to dispersive-water-wave models.

Abstract

We study the behavior of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves traveling along the constant direction are known to evolve into solitary waves, due to an effective dispersion. We apply multiple-scale perturbation theory to derive an effective constant-coefficient system of equations, showing that the transversely-averaged wave approximately satisfies a Boussinesq-type equation, while the lateral variation in the wave is related to certain integral functions of the bathymetry. Thus the homogenized equations not only accurately describe these waves but also predict their full two-dimensional shape in some detail. Numerical experiments confirm the good agreement between the effective equations and the variable-bathymetry shallow water equations.

Figures (11)

• Figure 1: Geometry of the problem studied herein. The bathymetry $b(y)$ is shown in brown and repeats periodically with period $\delta$ in the $y$ direction. The regime studied herein is that for which $\lambda \gg \delta$.
• Figure 2: Comparison of homogenized and direct solutions, for sinusoidal bathymetry \ref{['sinbath']}. The surface elevation $\eta - \eta^0$ is shown.
• Figure 3: Comparison of homogenized and direct solutions, for piecewise-constant bathymetry \ref{['pwc-setup']}. The surface elevation $\eta - \eta^0$ is shown.
• Figure 4: Three-dimensional rendering of the FV solution shown in the last panel of Figure \ref{['fig:comparison']}. Results shown are to scale, except that the $x$-axis has been compressed by a factor of 8 to make it easier to see the waves.
• Figure 5: Closeup of the surface waves shown in Figure \ref{['fig:iso2']}. The vertical variation has been exaggerated by 10x and the $x$-axis has been compressed by a factor of 8 to make it easier to see the waves.