Refraction of Water Waves

An investigation is presented of the propagation of three-dimensional, harmonic waves of small amplitude through water of constant depth or gradually varying depth. Within the framework of linear theory for potential flow, the results are exact for propagation over horizontal bottoms, e.g., diffraction, and approximate for propagation over sloping bottoms, e.g., refraction combined with diffraction. An expression is given for the velocity potential of three-dimensional waves, harmonic in time. The velocity potential must fulfill the classical hydrodynamical conditions. An analysis of the resulting relationships shows that the magnitudes of phase speed and energy flux are affected by amplitude variations. The time-averaged energy flux is directed along wave rays, regardless of amplitude gradients along the wave crests. The expressions for phase speed and energy flux do not involve wave rays; they provide the basis of a computational scheme in which the wave characteristics are computed in grid points which have a predetermined location. Such method may be an alternative for existing methods which utilize wave rays.