Two Energy Types in Wave Motion and Their Relation to Group and Wave Velocity

Abstract

The energy of an element in a wave motion can be of two kinds: first, non-interactive like the energy of a set of independent pendulum bobs, and second, interactive like the potential energy of an element of a stretched string which is dependent only on the relative position of the neighboring elements. In the former case the group velocity is zero, and in the latter it equals the wave velocity. The slowing down of energy transmission in intermediate cases is discussed qualitatively and the ratio of group to wave velocity is calculated quantitatively in a number of cases on the assumption that the energy propagation is given not only by the product of energy density and group velocity, but also by the product of twice the interactive energy density and wave velocity. A general relation is suggested without proof for the connection between energy types and the ratio of group to wave velocity. The same ideas are applied to de Broglie's phase waves, the group velocity of which is the particle velocity.