On the transport of energy in water waves

Theory is developed and utilized for the calculation of the separate transport of kinetic, gravity potential, and surface-tension energies within sinusoidal surface waves in water of arbitrary depth. In Sect. 2 it is shown that each of these three types of energy constituting the wave travel at different speeds, and that the group velocity, cg, is the energy-weighted average of these speeds, depth- and time-averaged in the case of the kinetic energy. It is shown that the time-averaged kinetic energy travels at every depth horizontally either with (deep water), or faster than the wave itself, and that the propagation of a sinusoidal wave is made possible by the vertical transport of kinetic energy to the free surface, where it provides the oscillating balance in surface energy just necessary to allow the propagation of the wave. The propagation speed along the surface of the gravity potential energy is null, while the surface-tension energy travels forward along the wave surface everywhere at twice the wave velocity, c. The flux of kinetic energy, when viewed traveling with a wave, provides a pattern of steady flux lines which originate and end on the free surface after making vertical excursions into the wave, even to the bottom, and these are calculated. The pictures produced in this way provide immediate insight into the basic mechanisms of wave motion. In Sect. 3 the modulated gravity wave is considered in deep water and the balance of terms involved in the propagation of the energy in the wave group is determined; it is shown again that vertical transport of kinetic energy to the surface is fundamental in allowing the propagation of the modulation, and in determining the well-known speed of the modulation envelope, c(g).