Zahra Khorsand: Linear and nonlinear theories for wave shoaling and setdown

Abstract

Waves transport mass, momentum and energy. In shoaling processes, wave energy is generally conserved, while wave momentum may vary as the waves propagate towards the beach. The linear theory of wave shoaling utilizes energy conservation to obtain changes in waveheight as a function of the local undisturbed depth. The momentum balance is usually expressed in terms of the radiation stress tensor, and it can  be shown using the radiation stress that momentum transport in a shoaling wavetrain varies with decreasing depth, leading to local changes in the mean water level. The change in the mean water level is usually called setdown.

We will review the classical linear theory of shoaling, and then explain some attempts to extend this theory to the nonlinear case. For the nonlinear case, momentum and energy balances in the context of the KdV equation are used in conjunction with periodic cnoidal wave solutions.