The partition of energy in standing gravity waves of finite amplitude

Abstract

Our previous solution through fourth order for periodic two-dimensional standing gravity waves in an incompressible liquid of arbitrary uniform depth is applied to determine the kinetic and potential energies of the wave motion as functions of time, fluid depth, and wave amplitude. For a depth-wavelength ratio greater than 0.17 the temporal mean potential energy is always greater than the mean kinetic energy. For smaller depth-wavelength ratios the mean potential is less than the mean kinetic energy for small amplitudes, whereas for larger amplitudes the mean potential may be the greater energy. The distortion of the time variation of the energies from that predicted by infinitesimal-amplitude theory is investigated and is contrasted with that for axisymmetric gravity waves. For amplitudes up to the maximum, numerical verification is given for the hypothesis that (TM - VM)/E (i.e., the ratio of the mean Lagrangian to the total energy) is approximately ½(ω - ωo)/ωo. Examination of other gravity wave problems, as well as problems outside the field of fluid mechanics, suggests that the hypothesis may be applicable to a wide class of nonlinear oscillations of conservative systems.