The meaning of negative group velocity and amplitude modulated signals

Abstract

Negative group velocities have been reported in calculations for waves generated on elastic objects. It is usual to interpret a group velocity with the travel time of a wave front or the time that the energy of a wave generated travels along some trajectory. One may look at the group velocity as the rate at which an envelope of a wave train travels. More generally, one may exploit the principle of stationary phase along with the Fourier transform in frequency and derive the concept of group velocity from the first order asymptotic term. This derivation requires that the modulus of the integral is changing slowly relative to the phase term. In addition it is required that the group velocity is not near a stationary point in frequency. In that case first order asymptotics is no longer valid and one must employ higher asymptotics. Here, the standard concept of group velocity is no longer meaningful. Further, the signal becomes amplitude modulated rather than frequency modulated. We examine this for some examples, assuming losslessness. In the event of loss the concept of group velocity has already been reported to be of dubious value.