Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.