, Volume 1, Issue 2, pp 225-245

On the periodic solutions of a forced second-order equation

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Summary

We study the equations for two-dimensional irrotational motion induced in a rectangular tank of water which is forced to oscillate horizontally in a periodic fashion (“shallow water sloshing”). The problem reduces to a second-order non-autonomous ordinary differential equation. We rigorously show the existence of many solutions obtained previously by numerical computations, and in addition we show that there are many other bounded solutions. We use simple shooting to obtain irregular solutions of the kind obtained by the methods of dynamical systems. In addition, we obtain a kind of “kneading” theory whereby we can order the initial conditions according to the pattern of “spikes” exhibited by the solutions.

Communicated by Stephen Wiggins