Relation Between Group Velocity and Phase Velocity

Ans. Group velocity and Phase Velocity can be understood only in the context of waves. Both are different types of velocity which can be understood as different characteristics of the waves. 

When the waves travel in the group or packet, their velocity is called the Group Velocity. Whereas, the movement of the waves in phase is measured in the Phase Velocity. The latter is also influenced by multiple specific factors like a crest or the wavelength of the concerned waves.

Ans. According to various experiments, the Phase Velocity is mainly greater than the Group Velocity. However, at the same time, the Phase Velocity can be greater than that of the speed of light and at times it can also be less than that. Simultaneously, if we consider the example of the wave propagation in the deep water, then it could be seen that its Phase Velocity is twice that of the Group Velocity. 

Another example could be taken in the case of the velocity of the surface particles. Here also, the velocity of the group of waves in the surface water shall be lesser than that of the Phase Velocity. Most of the time, if the propagation of the packet of waves takes place without any disturbances, then the Group Velocity always remains less than that of the Phase Velocity.

Ans. The dispersive and the non – dispersive sinusoidal waves are linked to the concepts of Phase Velocity and Group Velocity, respectively. Dispersion simply refers to the concept that the Phase Velocity of a wave is influenced by the frequency. Therefore, the dispersive waves represent the phenomenon when the waves of different frequencies travel with different speeds and the wave changes its shape, referring to the Phase Velocity. However, if the same medium becomes non – dispersive, then the waves of varying frequencies travel at the same speed, making the wave function remain uniform throughout.

Ans. The dispersive medium is one in which the waves having varying frequencies travel with varying speeds. 

The main reason behind this condition is that due to the electromagnetic radiations, the index of refraction becomes dependent on the frequency of the waves, that is, the waves and their velocity gets heavily influenced by their components like frequency.

Ans. The Phase Velocity of the matter waves is known for not carrying any influential information with itself. Rather, the significant details like the momentum, velocity, etc. of a matter-wave are mainly understood through the Group Velocity. Therefore, in the case of the matter waves, the Group Velocity is concerned and not the Phase Velocity.

Ans. The total relativistic energy and the relativistic momentum of any matter are derived which is known to always remain as equal to that of the Group Velocity of that particular particle. 

As a result, it has been found that the space velocity of the photons is equal to that of the speed of light. However, this phenomenon takes place only in the case of relativistic freedom.

Ans. Phase velocity is highly influenced by the significant elements of a wave, that is the crest and trough. Therefore, the Phase Velocity is mainly concerned with the specific properties of a wave. And the velocity of any of these components while propagating into the space is measured as the Phase Velocity. However, we can mathematically represent the Phase Velocity as Vp = λ / T

• Vp is equal to the Phase Velocity,
• λ is equal to the Wavelength,
• And T is equal to the time period which also positively impacts the Phase Velocity

Ans. The relation between the Phase Velocity and the Group Velocity be mathematically derived as the following.

In order to find out the amplitude of wave packet, let us assume,

• ω as the angular velocity given by ω = 2πf
• k as the angular wavenumber given by – k = 2π / λ
• t as time
• x as the position
• Vp (Phase Velocity) 
• Vg (the Group Velocity)

Vp = ω / k……. (equation 1)

Rewriting the above equation, 
ω = kVp…… (equation 2)

Differentiating (equation 2) with respect to k we obtain,
dw / dk = Vp + k (dVp / dk)……( equation 3)

Since, Vg = dw / dk,
Therefore, Vg = Vp + k (dVp / dk)

Ans. When an entire envelope of waves moves, rather than a single wave, it forms a packet of waves. The velocity of the movement of this packet is called the Group Velocity.

Ans. The condition for non dispersive waves is \(\begin{array}{l}\frac{dV_{p}}{dk}=0\end{array}\)

Ans. In a dispersive medium, the group velocity is lesser than the phase velocity.

Ans. A dispersive medium is a medium in which waves of different frequencies travel at varied velocities. For example – water and lenses.

Ans. A medium in which the speed of a wave is independent of the frequency of the wave is known as a non-dispersive medium. For example, the air is a non-dispersive medium for sound waves.

Ans. The phase velocity in waveguides is always greater than the speed of light. This further implies that the group velocity is small.

Ans. The phase velocity is given by the equation Vp = c cos θ. By substituting for cos θ = 2.5 and the value of the speed of light, we get the phase velocity as 7.5 x 10⁸ m/s.

Ans. The cut off wavelength and the guided wavelength are related as (1/λ)² = (1/λc)² + (1/λg)².

On substituting for the value of λ_c = 0.5 and λ_g = 2,

we get λ = 0.48 units.

Ans. The product of the phase and the group velocities is given by the square of the speed of the light. Therefore, Vp x Vg = c².

Ans. We know that the phase and the group velocities are related as Vp x Vg = c².

On substituting the value for Vg = 6 x 10⁶ and the speed of light,

we get Vp = 150 x 10⁸ m/s.

Ans. The guided wavelength and the phase constant are related by the equation 2π/βg = λg,

where βg is the guided phase constant

λg is the guided wavelength.

Ans. The phase velocity is the velocity of a single wave while the group velocity is the velocity of a group of waves.