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

Abstract

By accurate calculation it is found that the speedFof a solitary wave, as well as its mass, momentum and energy, attains a maximum value corresponding to a wave oflessthan the maximum amplitude. Hence for a given wave speedFthere can exist, whenFis 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 parameterω(related to the particle velocity at the wave crest) whose range is exactly known; and thirdly by the discovery that the series for the massMand potential energyVin powers ofωcan be accurately summed by Padé approximants. From these, the values ofFand of the wave heightєare determined accurately through the exact relations 3V= (F2- 1)Mand 2є= (ω+F2- 1). The maximum wave height, as determined in this way, isєmax= 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 isF= 1.286. The maximum wave speed, however, isFmax= 1.294, which corresponds toє=0.790. The relation betweenєandFis 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.