Learn the Relation Between Group Velocity and Phase Velocity

Wave is a very interesting topic of Physics because you are surrounded by waves from all sides. The sound you hear, the light which enables us to see things around, and the flow of water all are kinds of waves. When two or more waves interact together they superimpose each other and can be grouped as a wave packet. The rate at which these wave packets travels is referred to as the Group velocity and the rate of change of phase of the wave is termed Phase Velocity. In this Physics article, we are going to discuss the relation between group velocity and phase velocity.

Before that, let’s understand in brief what these two velocities are.

What is Group Velocity?

Group Velocity is a concept that we discuss while studying the superposition principles of the waves. So before moving directly to the Group velocity, let’s first develop an idea of what Superposition of waves is?

Let’s consider two waves propagating through a medium and intersecting each other in space, and producing a resultant oscillation and this resultant oscillation is far more complex than the behaviour of original sine or cosine waves. Particles at any point in the wave oscillate with different amplitude values. The most extreme amplitude is the integral sum of the two amplitudes of the original waves. The minimum amplitude is the difference between the two original amplitudes. Assuming the two amplitudes are equivalent, then the most extreme or maximum value is two times the original amplitude and the minimum value is zero.

For sake of convenience, let us accept that the two balanced or modulated waves are of similar amplitude and have different frequencies. This makes the wave with the higher frequency get enveloped in the wave with the lower frequency. This causes a group of waves to get pressed or packed in an envelope and the velocity of this envelope is referred to as the group velocity of the wave. For the group velocity to be zero both of the waves must be of similar frequency and they should have an inverse direction of motion to each other.

The group velocity of the wave can be expressed mathematically as,

Here, = group Velocity of the wave

= Angular Velocity (usually expressed in radian per second)

k = Angular wave number ( usually expressed in radian per metre)

What is Phase Velocity?

As the name itself suggests, the phase velocity of the wave is actually the velocity of its phase, which propagates along with the wave. Phase velocity is defined as the rate of propagation of phase of waves through the space. It can be referred to as the position of wave at a given point at that instant of time.

Phase velocity of wave can be calculated using the expression,

Here, = wavelength

T =Time period

We can also express it in terms of angular frequency and wave numbers as,

Here, = Angular frequency, specifies the number of oscillations per units of time.

and, k = wave number, specifies the number of oscillations per units of space

Relation Between Group Velocity and Phase Velocity

The phase velocity of any wave can be expressed mathematically as,

—–(1)

On rearranging the above equation, we obtained;

, let’s call it equation (2)

Now, differentiate the newly obtained, equation (2), with respect to, Angular wave number, k we get,

——– (3)

As, we know that the expression for group velocity is,

On substituting this value, in equation (3)

We obtained,

This is the required expression that signifies the relationship between group velocity and phase velocity.

Here, from the above-mentioned relation, you can easily figure out that, both the phase velocity and group velocity of the wave are directly proportional to each other. That means, if the group velocity increases, the phase velocity also increases and vice versa.

Here, we have,

= measure of angular velocity

k = Angular wave number

t = time

x = position of the wave

= group velocity of the wave

= phase velocity of the wave

Group Velocity and Phase Velocity Relation for Dispersive Wave Non-Dispersive Wave

The relation between Group Velocity and Phase velocity can be mathematically expressed as follows: 

The formula for phase velocity can be written as, 

Where, 

is the phase velocity.

is the wavelength.

T is the time period of the wave.

The expression for group velocity is, 

Where, 

is the group velocity 

is the angular frequency of the wave

is the angular wave number

The relation between phase velocity and group velocity can be expressed in the following manner, 

Group velocity and Phase velocity relation for dispersive and non dispersive waves is as follows:

• For a dispersive wave, phase velocity is not equal to group velocity and the ratio of phase velocity with respect to the wave number is not equal to zero either.

and 

• For a non-dispersive wave, the phase velocity is equal to group velocity, hence, ratio of phase velocity with respect to the wave number is equal to zero.

And 

Group Velocity And Phase Velocity Relation

Phase velocity and group velocity are directly proportional. This means that 

• Phase velocity increases with increase in group velocity.
• Group velocity increases with increase in phase velocity.

Difference Between Phase Velocity and Group Velocity

The difference between phase velocity and group velocity are listed in the table below.

Parameters	Phase Velocity	Group Velocity
Definition	Speed at which the phase of a wave propagates in space.	Speed at which the energy or information of a wave packet propagates in space.
Formula
Relation with Frequency	Phase velocity is determined by the frequency of the wave and its wavelength.	Group velocity depends on the dispersion properties of the medium and the wave.
Relation with Velocity	Phase velocity may not represent the actual velocity of energy transfer.	Group velocity represents the actual velocity of energy or information transfer.
Wave packets & superposition	Phase velocity is not relevant for wave packets and superposition of multiple waves.	Group velocity is relevant for analyzing wave packets and their behavior, such as dispersion and transmission characteristics.
Wave distortion	Phase velocity does not cause waveform distortion.	Group velocity affects the distortion of waveforms in a wave packet.
Applications	Used to analyze wave interference, diffraction, and wave propagation.	Used in signal processing, communication, waveguide design, and dispersion analysis.

Facts to Remember about Group Velocity and Phase Velocity

Here are a few things to remember about the concepts of Group velocity and Phase velocity: 

• Phase velocity is the velocity at which the phase of a wave travels.
• Group velocity is equal to the energy at which the wave travels. 
• In a dispersive medium, the phase velocity depends on the frequency of the wave.
• In a non-dispersive medium, the phase velocity of a wave is independent of the frequency of the given wave. 
• In a dispersive medium, the group velocity can vary depending on the properties of the medium of propagation but never equal to zero. 
• In a non dispersive medium, the group velocity is always equal to the phase velocity. 

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