Phase and group velocity are two important and related concepts in wave mechanics. They arise in quantum mechanics in the time development of the state function for the continuous case, i.e. wave packets.
A one-dimensional harmonic wave (Figure 1) is described by the equation,
where A0 is the wave amplitude, w is the circular frequency; k is the wave number; and j is an initial, constant phase. The argument for the sine function, q (x, t) = wt - kx + j is called the phase. Sometimes the wave number is referred to as the spatial frequency or propagation constant.
In general, these waves propagate without warping. That is, the phase q (x, t) is a constant:
From the point of view of sending information, these waves are not useful. They are the same throughout time and space. Something must therefore be modulated, such as frequency or amplitude, in order to convey information. The resulting wave may be a perturbation that acts over a short distance, i.e. a wave packet. This wave packet can be considered to be a superposition of a number of harmonic waves, in other words a Fourier series or integral.
In order to convey information, something more than a simple harmonic wave is needed. However, the superposition of many such waves of varying frequencies can result in an "envelope" wave and a carrier wave within the envelope. The envelope can transmit data. A simple example is the superposition of two harmonic waves with frequencies that are very close (w1 ~ w2) and of the same amplitude. The equations for the motion are,
The plot of such a wave is shown in Figure 2.
The envelope (the green line) is given by u1 and travels at the group velocity. The carrier wave (the blue line) travels at the phase velocity and is given by u2. The wave packet moves at the group velocity. It is the envelope which carries information. Group velocity and phase velocity are not necessarily the same. Group velocity is given by,
Wave Packet Explorer Add waves to get a wave packet.
Demonstration of Group Velocity Applet and description of the difference between group and phase velocity.
Group Velocity and Phase Velocity Another demonstration of phase and group velocity.
Superposition Principle of Waves Applet demonstrating the addition of two waves.
Liboff, R. L. Introductory Quantum Mechanics, Fourth Edition. San Francisco, CA: Addison Wesley, pp. 156-157 2003.
Ostrovsky, L.; Potapov, A. Modulated Waves: Theory and Applications. Baltimore and London: The Johns Hopkins University Press, pp. 1-9, 1999.
Menzel, D.; Mathematical Physics. New York, NY: Dover Publications Inc,pp.349-351, 1961.