Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.     

Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing

The forced Korteweg-de Vries equation with Burgers’ damping (fKdVB) on a periodic domain, which arises as a model for water waves in a shallow tank with forcing near resonance, is considered. A method for construction of asymptotic solutions is presented, valid in cases where dispersion and damping are small. Through variation of a detuning parameter, families of resonant solutions are obtained providing detailed insight into the resonant response character of the system and allowing for direct comparison with the experimental results of Chester and Bones (1968).     

Resonant hydraulic sloshing in a tank of variable depth

The forced resonant oscillations of a fluid in a tank of variable depth are considered within the hydraulic approximation. It is shown that for certain bottom topographies a continuous periodic output dominated by the first normal mode is possible. This contrasts with the case of a tank of constant depth, where hydraulic jumps are a feature of the motion. The amplitude and frequency of the output are connected by a cubic equation. The fluid response can act like that of a hard or soft spring, depending on the bottom topography. There is also a critical bottom topography that yields a higher order response amplitude.