Bidirectional Whitham Equations as Models of Waves on Shallow Water

Hammack & Segur [1] conducted a series of surface water-wave experiments in which the evolution of long waves of depression was measured and studied. This present work compares time series from these experiments with predictions from numerical simulations of the KdV, Serre, and several unidirectional and bidirectional Whitham-type equations. These comparisons show that accurate predictions come from models that contain an accurate reproduction of the Euler phase velocity, sufficient nonlinearity, and surface tension effects. The focus is to compare the quality of several bidirectional Whitham models. Most interestingly, the comparisons show that the bidirectional Whitham equations can provide predictions that are more accurate than the unidirectional models even though the experiments were essentially one dimensional.