The interaction of solitary-wave solutions of a model equation for long waves in dispersive media is examined numerically. It is found that the waves do not emerge from the interaction unscathed. Instead, two new solitary waves, having slightly different amplitudes from the original waves, together with a small dispersive tail are generated as a result of the interaction.

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a one-dimensional representation of weakly nonlinear, dispersive waves in shallow water. A model for such flows was proposed by Korteweg & de Vries (1895) and this has provided the theoretical basis for a number of laboratory experiments. Some recent studies that have been made in the area are those of Zabusky & Galvin (1971), Hammack (1973) and Hammack & Segur (1974). In each case the theoretical model gave a good qualitative account of the experiments, but the quantitative comparisons were not very extensive. One of the purposes of this paper is to provide a more detailed quantitative assessment of a particular model than has been given to date. An important aspect of the formulation of the theoretical model is the specification of the initial conditions and the boundary conditions for the equation. Zabusky & Galvin (1971) considered an initial-value problem having spatial periodicity, whereas Hammack (1973) and Hammack & Segur (1974) considered an initial-value problem posed on the real line. In contrast, we shall consider an initial-value problem posed on the half line with boundary data specified at the origin. This problem was chosen to correspond with an experiment in which waves were generated at one end of a long channel, and obviates certain difficulties inherent in the other formulations.