 # Relation Between Group Velocity And Phase Velocity

A wave, in simple words, can be described as a mere disturbance along any plane surface. For instance, consider an ocean, where the waves move along the ocean surface from one place to another. The crest of a wave moves at intervals and therefore travels a specific distance over a specific time period. In this chapter, you are dealing with phase and group velocity, which are properties of waves. When a wave moves in groups, the speed or velocity at which it moves is known as the group velocity.

Furthermore, you should also be able to differentiate and establish the relation between phase and group velocity in physics. This will help you in clearing your concept while preparing you for advanced understanding.

### What is Group Velocity?

The group velocity of a wave is defined as the velocity at which an entire envelope of waves moves through a medium. A most common example, in this case, can be that of throwing stones in a water body which causes multiple waves on the surface of water.

On throwing the stone, a ripple is created around the point where the stone drops. The ripple is formed of small wavelets which propagate away from the dropping point in multiple directions. Here, a wavelet having the shortest wavelength propagates faster than others.

However, to understand what is group velocity, you should also have an idea of simple harmonic motion too. This will help you in understanding the concepts better and in a more natural way.

### What is Phase Velocity?

Considering the fact that a wave consists of two significant parts crest and trough, its phase velocity is also dependent on the same. Students should have prior knowledge of it to understand what phase velocity is.

It is the velocity at which a specific component of a wave, say crest, propagates in space. This feature or velocity is directly dependent on the time period and wavelength. Alongside, you should also be clear about the relation between phase velocity and group velocity. For such understanding, knowing the mathematical formula or representation is highly beneficial.

### What is the Relation Between Phase Velocity and Group Velocity?

To dig deeper into the concept of the relation between group velocity and phase velocity, take a look at the mathematical expression explained below.

The expression for phase velocity is presented below -

Vp = ƛ/T

Here, Vp is the phase velocity, ƛ (read lambda) is the wavelength, and T is the time period.

The expression for group velocity is -

Vg = 𝜹w/𝛅k

Here, Vg is the group velocity, 𝜹w is the angular frequency of the wave, and 𝛅k is the angular wavenumber.

Therefore, students should first define group velocity and phase velocity in their respective mathematical formats. Consequently, they can relate both these velocities in the following manner -

Vg = Vp + k (dVp/dk)

Among the valid points to note here while you define phase velocity and group velocity, most crucial is that both are directly proportional to each other.

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1. What do you mean by Group Velocity?

Ans. The velocity at which a group of waves propagate in the space is called group velocity. It is directly proportional to phase velocity of a wave.

2. How does Group Velocity Differ from Phase Velocity?

Ans. Both phase and group velocity are the characteristics of a wave. The former is specific to a particular phase of a wave, say crest or trough and its propagation. Contrarily, the latter considers a group of wavelets propagating in a medium.

3. Is There a Relation Between Phase and Group Velocity?

Ans. The vital relation between phase and group velocity is that they are proportional to each other. It signifies that a change in one velocity leads to a difference in the other.