Group velocity

A short and largely traditional review of the concept of group velocity, prepared for the forthcoming Encyclopedia of Nonlinear Science Waves often propagate as a packet, or group, within which there are several wave crests. A common illustration is given by the wave pattern formed when a stone is thrown into a pond. An axially symmetric ring of waves on the water surface propagates outwards as a wave group; within the group, however, one can see that the wave crests propagate through the group, apparently from the rear to the front. The speed of the waves crests, called the phase velocity, is different from that of the group as a whole, this speed being called the group velocity. For water waves, the phase velocity is greater than the group velocity, except for the very short waves dominated by surface tension. This important distinction between phase and group velocity arises in all physical systems which support waves. The concept of group velocity appears to have been first formulated by Hamilton in 1839, quoted by Havelock (1914). The first recorded observation of the group velocity of a (water) wave is due to Russell in 1844 (note also his remark that “the sound of a cannon travels faster than the command to fire it” , Russell (1885)).However, our present understanding of group velocity is usually attributed to Stokes, who used it as the topic of a Smith’s Prize examination paper in 1876. For simplicity, first consider a linearised system, with just a single active spatial variable x, and with time represented by t. Then a sinusoidal wave has the representation, u(x, t) = a cos (kx− ωt+ φ) . (1)