Group velocity and phase velocity

Lecture notes (Feb, 2009) : M. S. SanthanamIISER, Pune / PRL, Ahmedabad.

This notes is meant to be a supplement and not a substitute for standard text books.

1 Phase and group velocity of waves

To understand the difference between phase and group velocity of waves, consider the followinganalogy. A group of people, say city marathon runners, start from the starting at the same time.Initially it would appear that all of them are running at the same speed. As time passes, groupspreads out (disperses) simply because each runner in the group is running with different speed.If you think of phase velocity to be like the speed of individual runner, then the group velocityis the speed of the entire group as a whole. Obviously and most often, individual runners canrun faster than the group as a whole. To stretch this analogy, we note that the phase velocityof waves are typically larger than the group velocity of waves. However, this really depends onthe properties of the medium.

Briefly, phase velocity refers to a monochromatic wave, let’s say, the velocity of one of thepeaks of the wave. For example, a monochromatic wave with angular frequency ω = 2πν (ν isthe frequency) travelling in +ve x-direction is given by,

y = A sin(ωt − kx).

On the other hand, group velocity refers to a group composed of waves within a frequency band.It is the velocity with which the entire group of waves travel.

The following figure 1 shows y = sin(2+ t) and y = sin(2+ t)+sin(2+1.1t). The last form isthe sum of two waves whose frequencies differ by 0.1. Notice that the amplitude of the group ismodulated as a function of t. The example here shows the waves as a function of t, but similarscenario holds good for waves as a function of x.

5 10 15 20 25 30

-1.0

-0.5

0.5

1.0

50 100 150

-2

-1

1

2

Figure 1: (left) A single travelling wave with frequency ω = 1. (right) A group of waves composedof two waves with frequencies ω = 1 and ω = 1.1.

In quantum physics, a group of waves can be associated with a material particle (De Broglie’shypothesis). If the particle is moving with speed v << c, then the wavelength of wave groupassociated with the particle is λ = h/mv, where h is Planck’s constant. If y(x) represents onesuch wave group, then |y(x)|2 gives the probability density for finding the particle in the range xand x + dx. Further, it can also be shown that the group velocity vg = v. Since the informationon the position and velocity of the particle can be obtained from y(x), this represents the stateof the particle.