Higher order corrections for shallow-water solitary waves: elementary derivation and experiments

TL;DR

This paper introduces an elementary method to derive higher-order equations for shallow-water solitary waves, solving them to predict wave shapes and velocities, and validating the second-order theory with experimental data.

Contribution

It provides a simplified derivation of higher-order shallow-water wave equations and demonstrates improved experimental agreement with second-order predictions.

Findings

Second-order theory matches experimental velocities better.

First-order equation is equivalent to KdV equation.

Numerical solutions of second-order equation align with observed wave shapes.

Abstract

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg-de Vries (KdV) equation, while the second-order equation is solved numerically. The propagation velocity found for the solitary waves of the second-order equation coincides with a known expression, but it is obtained in a simpler way. By measuring the propagation velocity of solitary waves in the laboratory, we demonstrate that the second-order theory gives a considerably improved fit to experimental results.