Nonlinear sloshing and passage through resonance in a shallow water tank |
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Authors: | E A Cox J P Gleeson M P Mortell |
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Institution: | (1) Dept. of Mathematical Physics, University College, Dublin, Ireland;(2) Dept. of Applied Mathematics, University College, Cork, Ireland |
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Abstract: | 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.Received: December 16, 2003 |
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Keywords: | KdV variable coefficients nonlinear resonance evolution |
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