On the relation between the wavefront speed and the group velocity concept

Abstract

The relation between the wavefront speed and the group velocity concept is studied in this work. The relationship between the more well-known velocity concept named as the phase velocity and the speed of propagation of a front of an acoustic pulse is discussed. This is of interest since it concerns transient wave propagation and is, in general, not well known. The form and properties of a pulse can be obtained by means of a Fourier integral and estimates based on quantities derived for monochromatic waves, such as the phase velocity, can be severely misleading and confusing. The wavefront velocity is defined as the high-frequency limit of the phase velocity. This quantity can be far less than the value of the phase velocity for finite frequencies which for example is the case for bubbly fluids. Then the group velocity concept is discussed, which was introduced in order to characterize the propagation of water waves of essentially the same wavelength. However, more confusion occurs in that it is sometimes believed that a wavefront is propagating with the group velocity (a limit process not mentioned) since it can be related to the propagation of energy. This interpretation of energy propagation is based on sinusoidal waves and involves time as well as space averages and is not applicable for pulses. However, by means of the expression for the group velocity given by Stokes it is shown that the speed of a wavefront can be found from the group velocity at a limiting high frequency. This result can be understood geometrically from the definition of the group velocity given by Lamb which is conservation of wavelength. A wavefront is a discontinuity and limiting short wavelengths will be found there.