Group velocity and energy propagation for three‐dimensional waves

This paper is a review and development of recent ideas on group velocity and energy propagation by Lighthill (l, 21, Landau and Lifshitz (3, $5 66 and 671, Ursell (7), and the present writer (a). Some of the ideas seem to be new but it is not always clear how much is implicit in the above references. A simple but general derivation of group velocity and its properties was given for one-dimensional dispersive waves by Rossby ( 101 and it was included in (I, p. 2861 as an example of kinematic waves. The derivation applies to a general wave train with a full range of frequencies; it is not limited to a wave packet. The group velocity determines the variation with time of the kine- matic properties such as local frequency, wave number and phase velocity at different parts of the wave train. An energy argument then predicts the variation in amplitude as the wave train propagates. In this paper, the generalization to two and three-dimensional wave motions is considered. The results can be described as an analogue of simple geometrical optics. For a non-dispersive medium high frequency waves and discontinuities propagate in a certain sense along the "rays" which are the curves orthogonal to the surfaces of constant phase. The waves propagate dong these rays with the phase velocity, and the amplitude variation agrees with the law that the flux of energy down a ray tube remains constant. For a dispersive medium the analogous rays are based on the group velocity and would not in general be orthogonal to the wave surfaces of constant phase. The amplitude variation is determined by the flux of energy down these "group lines". The above description is written in physical terms referring to '(energy" and so on, but the method is primarily one of the determination of asymp- totic solutions of certain types of linear equations, and ('energy" is just the