Are all KdV-type shallow water wave equations the same with uniform solutions?

Abstract

ABSTRACT Cnoidal wave and its extreme case, the solitary wave, can be described by the KdV equation, which was first derived by Korteweg and de Vires with the first-order accuracy. Subsequently, different authors proposed their derivations and claimed that their equations, sharing similar expressions but different corresponding coefficients, were the same as the original one. After introducing a unified dimensionless frame, this study re-derived the KdV equation with respect to seven existing methods and confirmed that KdV equation indeed refers to a type of first-order equations, rather than a specified one. Differences in equations come from the influence of the second-order quantities associated with the derivation process. Regarding the cnoidal wave and solitary wave, the KdV-type equations obtained using different methods present the same first-order solution for wave profile. Nevertheless, in their directly derived results, different wave celerities and water particle velocities are presented due to the influence of second-order quantities. Additionally, comparing with the second-order solutions, all directly derived wave celerity solutions predict well for the Ursell number between 20 and 100. As for the first-order solution of the water particle velocity, all methods present the same result except Dean’s expression which contains a different coefficient.