Obliquely interacting solitary waves and wave wakes in free-surface flows

TL;DR

This paper analyzes obliquely interacting solitary waves in shallow water using the Benney--Luke equation, deriving modulation equations and analytical formulas for wave phenomena, with numerical validation and implications for wave-topography interactions.

Contribution

It introduces modulation equations for the BL equation, characterizes Mach phenomena, and provides analytical formulas for critical angles and wavefront shapes, linking theory with numerical results.

Findings

Derived modulation equations for the BL equation.

Characterized Mach expansion and reflection phenomena.

Numerical results agree with theoretical predictions.

Abstract

This paper investigates the weakly nonlinear isotropic bi-directional Benney--Luke (BL) equation, which is used to describe oceanic surface and internal waves in shallow water, with a particular focus on soliton dynamics. Using the Whitham modulation theory, we derive the modulation equations associated with the BL equation that describe the evolution of soliton amplitude and slope. By analyzing rarefaction waves and shock waves within these modulation equations, we derive the Riemann invariants and modified Rankine--Hugoniot conditions. These expressions help characterize the Mach expansion and Mach reflection phenomena of bent and reverse bent solitons. We also derive analytical formulas for the critical angle and the Mach stem amplitude, showing that as the soliton speed is in the vicinity of unity, the results from the BL equation align closely with those of the Kadomtsev--Petviashvili (KP) equation. Corresponding numerical results are obtained and show excellent agreement with theoretical predictions. Furthermore, as a far-field approximation for the forced BL equation -- which models wave and flow interactions with local topography -- the modulation equations yield a slowly varying similarity solution. This solution indicates that the precursor wavefronts created by topography moving at subcritical or critical speeds take the shape of a circular arc, in contrast to the parabolic wavefronts observed in the forced KP equation.