Phase Velocity vs Group Velocity
Phase velocity and group velocity are two very important concepts in physics. They play a vital role in fields such as wave mechanics, optics, quantum mechanics and even sound engineering. It is vital to have a solid understanding in both phase velocity and group velocity in order to excel in such fields. In this article, we are going to discuss what phase velocity and group velocity are, definitions of group velocity and phase velocity, their applications, their similarities and finally the difference between these two.
What is Phase Velocity?
Phase velocity is a concept discussed in propagation of waves. The phase velocity of a wave is the velocity of a “phase” which propagates. For clarification, assume a crest of a wave, which is travelling in the x direction of the axis. The phase velocity is the x component of the velocity of the selected point at the crest. This also can be obtained by dividing the wavelength by the time taken for a single wavelength to pass a selected point. This time is equal to the period of the oscillation, which is causing the wave. Now consider a standard sine wave A sin (wt – kx), where w is the angular velocity of the source, t is the time, k is the wave number (number of complete wavelengths per length of 2π), and x is the position on the x-axis. At the crest, wt – kx is equal to zero. Therefore, the phase velocity (x/t) is equal to w / k. mathematically, the value p=wt – kx is the phase of the wave.
What is Group Velocity?
Group velocity is discussed under superposition of waves. To understand group velocity one must first understand the concept of superposition. When two waves intercept each other in space, the resultant oscillation is somewhat complex than the sine behavior. Particle at a point oscillates with varying amplitudes. The maximum amplitude is the unison of the two amplitudes of the original waves. The minimum amplitude is the minimum difference between the two original amplitudes. If the two amplitudes are equal, the maximum is twice the amplitude and the minimum is zero. For the sake of clarity, let us assume that the two modulated waves are of the same amplitude and different frequencies. This causes the wave with the higher frequency to be enveloped in the wave with the lower frequency. This causes a group of waves packed in an envelope. The velocity of this envelope is the group velocity of the wave. It must be noted that, for a standing wave, the group velocity is zero. For the group velocity to be zero both of the waves have to be of the same frequency and they must have opposite directions of travel.
|
What is the difference between group velocity and phase velocity? • Phase velocity is defined for both, the single waves and superimposed waves. • The group velocity is defined only to the superimposed waves. • The group velocity is the velocity of the wave with lower frequency, but the phase velocity is the velocity of the wave with higher frequency.
|