On the mass, momentum, energy and circulation of a solitary wave. II

By accurate calculation it is found that the speed $F$ of a solitary wave, as well as its mass, momentum and energy, attains a maximum value corresponding to a wave of less than the maximum amplitude. Hence for a given wave speed $F$ there can exist, when $F$ is near its maximum, two quite distinct solitary waves. The calculation is made possible, first, by the proof in an earlier paper ($I$) of some exact relations between the momentum and potential energy, which enable the coefficients in certain series to be checked and extended to a high order; secondly, by the introduction of a new parameter $\omega $ (related to the particle velocity at the wave crest) whose range is exactly known; and thirdly by the discovery that the series for the mass $M$ and potential energy $V$ in powers of $\omega $ can be accurately summed by Pade approximants. From these, the values of $F$ and of the wave height $\epsilon $ are determined accurately through the exact relations $3V=(F^{2}-1)M$ and $2\epsilon =(\omega +F^{2}-1$. The maximum wave height, as determined in this way, is $\epsilon _{\max}$ = 0.827, in good agreement with the values found by Yamada (1957) and Lenau (1966), using completely different methods. The speed of the limiting wave is $F$ = 1.286. The maximum wave speed, however, is $F_{\max}$ = 1.294, which corresponds to $\epsilon =0.790$. The relation between $\epsilon $ and $F$ is compared to the laboratory observations made by Daily & Stephan (1952), with reasonable agreement. An important application of our results is to the understanding of how waves break in shallow water. The discovery that the highest solitary wave is not the most energetic helps to explain the qualitative difference between plunging and spilling breakers, and to account for the marked intermittency which is characteristic of spilling breakers.