TABLE OF CONTENTS In this modelling example we study a fascinating phenomenon, vividly described by J.Scott Russell in his Report on Waves (1844):
I believe I shall best introduce this phaenomenon by describing the circumstances of my own acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel.
This solitary water wave can completely be analysed mathematically. We use it to illustrate several techniques dealt with in this book: global analysis of the phase plane ( V.6), Hamiltonian systems ( XI.3, 4), and one-step integration methods ( III.1).
(i) Modelling
In 1895 Korteweg and De Vries modelled the phenomenon described...
