A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

Abstract

In this article, we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg–de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-down $${\bar{\eta }}$$ η ¯ and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works.

Rigorous estimates on mechanical balance laws in the Boussinesq–Peregrine equations

“…In fact, having an appropriate expression for the pressure is essential for the formulation of the momentum and energy balance (see Sections 3.2 and 3.3). Substituting ( 14) with ( 16) in (27) and dropping all terms of order 𝜀 2 , we find…”

“…Differential balance laws in continuum mechanics play a pivotal role, as they give a mathematical expression of the fundamental principles of mass, momentum, and energy conservation. For example, mechanical balance laws can be used to study the problem of wave shoaling, when waves enter a shallower area 22,27 as well as de-shoaling (see, e.g., Refs. 10,31).…”