Group Velocity and Phase Velocity

The rate at which a wave propagates in any medium is known as its phase velocity. This is the speed at which the phase of any one of the wave’s frequency components travels. Any specific phase of the wave (for example, the crest) will seem to travel at the phase velocity for such a component. This modulation depicts the signal content as group velocity. This speed is known as the group velocity because each amplitude envelope comprises a group of internal waves.

Group Velocity

The square of the amplitude is related to the energy of the waves. The speed of the troughs and crests, known as the phase speed, must be distinguished from the speed and direction of the wave’s associated energy or information transmission, known as the group velocity, according to mathematical analysis. The two are equivalent for nondispersive long waves, however the group velocity is only half the phase speed for surface gravity waves in deep water. The wave front moves at only half the speed of the crests, which appear to flow through the packet of waves and vanish at the front, in a train of waves expanding out over a pond after a rapid disturbance at a point. The group velocity is one-half the phase speed for capillary waves. When a stone is dropped into the middle of a still pond, a circular pattern of waves with a quiescent core, also known as a capillary wave, forms in the water. The wave group is the spreading ring of waves within which individual waves that travel faster than the group as a whole can be discerned. Individual wave amplitudes increase as they emerge from the group’s trailing edge and decrease as they approach the group’s leading edge.

The velocity of a group of waves is referred to as group velocity. A packet of waves is formed when the complete envelope of a wave moves rather than a single wave. The Group Velocity refers to the speed at which this packet moves. Sound waves, water waves, and other types of waves are only a few instances of a packet of waves travelling at the same time. As a result, Group Velocity is calculated at the same time. The formula of group velocity is given as:

 vg= dωdk

Here dω is the  rate of change in angular frequency of wave and

 dk is the rate of change in angular wavenumber

Phase Velocity

The rate at which the phase of a wave propagates through space is known as its phase velocity. When a wave train is modulated by an envelope, the group velocity represents the envelope’s speed, while the phase velocity represents the wave’s speed within the envelope.

The phase of any one frequency component of the wave moves at this velocity. Any phase of the wave (for example, the crest) will appear to travel at the phase velocity when such a component is present. The main focus of phase velocity is on a wave’s particular features. The crest and trough are the two most important aspects of a wave. The Phase Velocity is the velocity of any of these components as they propagate through space. Because the Phase Velocity is proportional to the wave’s phase, it is highly dependent on the wavelength and time period. The formula of phase velocity is given as:

 vp= ω/k

Here is the angular frequency and

 k is the angular wavenumber

Relation Between Group Velocity and Phase Velocity

The Group Velocity of a group of waves, according to experts, is always directly proportionate to the Phase Velocity of a wave. As a result, the waves’ Group Velocity and Phase Velocity have a direct link with one another. The Group Velocity and the Phase Velocity might be considered mutually exclusive. If the Group Velocity increases, for example, the Phase Velocity must likewise increase at the same amount. If the Phase Velocity increases, the Group Velocity will also increase at the same pace. If either of the two types of velocities (Group Velocity and Phase Velocity) drops, the same effect will be noticed on the other.

Conclusion

Group Velocity and Phase Velocity are two fundamental concepts in the movement of waves, wave packets, and groups of waves. Because the wave packet reflects the total of numerous waves, the velocity of this wave packet or group of waves’ movement is referred to as Group Velocity. The Phase Velocity, on the other hand, is the velocity of the movement of a wave phase. The ideas of Group Velocity and Phase Velocity are connected. These could also be interpreted in terms of their interrelationships. Other linked elements, like wavelength, frequency, or travel duration, have a significant impact on Phase Velocity. The mathematical derivation of Phase Velocity and Group Velocity shows that they are both directly related and have a beneficial impact on one another.