Dispersion and the transverse aether drag

Abstract

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2$T/c$, (4$T$-3$V$)4 and (3$T$-2$V$)$c$ respectively, where $T$ and $V$ denote the kinetic and potential energies and $c$ is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian ($T$-$V$). The integral properties of the wave are related to the constants $Q$, $R$ and $S$ which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Pade approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.