On the evolution of a solitary wave in a gradually varying channel

Abstract

The adiabatic approximation for a solitary wave in a channel of gradually varying breadth b and uniform depth is tested by experiment and by numerical solution of the generalized Korteweg-de Vries (KdV) equation. The results for a linearly diverging channel show good agreement with the prediction α (dimensionless wave amplitude). The experiments and numerical solutions for the linearly converging channel show that the wave growth is well approximated by α ∞ b−½. The discrepancy between the diverging and converging channels is shown to be due to nonlinear effects associated with the choice of the spatial variable as the slow variable in the generalized KdV equation. The measured and computed profiles display the predicted ‘shelves’ of elevation and depression in the converging and diverging channels, respectively.