Alternative Form of Boussinesq Equations for Nearshore Wave Propagation

TLDR

Boussinesq‐type equations model nonlinear surface wave transformation in shallow water, and different velocity variables yield distinct linear dispersion relations. The study derives a new form of the Boussinesq equations using a velocity at an arbitrary depth instead of depth‑averaged velocity. The new equations are solved with a finite difference scheme and validated against numerical and experimental wave propagation on a constant slope beach. The new equations improve linear dispersion over a broader depth range and accurately capture nonlinear shoaling effects such as harmonic amplification and wave asymmetry.

Abstract

Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow water including the amplification of the forced lower‐ and higher‐frequency wave harmonics and the associated increase in the horizontal and vertical asymmetry of the waves.