A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling

Martin O Paulsen,Henrik Kalisch

doi:10.1007/s42286-021-00053-1

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 }} 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.

Highlights

In this article, the development of surface waves across a gently sloping bottom is in view
Given a steady wavetrain of wave height H, wavelength λ and period T, the variation of the wave as it propagates from depth h A to h B is obtained by imposing conservation of T, conservation of the energy flux qE, and balance of forces using the bottom forcing by the pressure force, and the radiation stress
We have shown how to derive expressions for the energy flux and radiation stress of a wavetrain in the context of the well-known Korteweg–de Vries (KdV) equation

Summary

Introduction

The development of surface waves across a gently sloping bottom is in view. Rather than following a wave in space and time as it propagates over the slope, we estimate wave properties as functions of the depth using conservation of period T and energy flux, and a prescribed change in momentum due to the effect of the radiation stress These conservation equations are due to the assumptions stated above, and they form the basis for the description of the shoaling process as first envisioned by Rayleigh [44], and subsequently used by a number of authors. Given a steady wavetrain of wave height H , wavelength λ and period T , the variation of the wave as it propagates from depth h A to h B is obtained by imposing conservation of T , conservation of the energy flux qE , and balance of forces using the bottom forcing by the pressure force, and the radiation stress To execute this plan, one needs to have in hand formulations for the energy flux qE and the radiation stress Sxx in the context of the KdV equation. In the Conclusion we put our work into context and mention possibilities for further work

Momentum and Energy Balance in the KdV Approximation

Radiation Stress in the KdV Approximation

The Nonlinear Shoaling Equations

Implementation of the Shoaling Equations

Conclusion

A Integrals of Cnoidal Functions