What is meant by phase and group velocity differentiate it?
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.
Is the group velocity the derivative of phase velocity?
Phase and group velocity are related through Rayleigh’s formula, If the derivative term is zero, group velocity equals phase velocity. In this case, there is no dispersion. Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to “spread out” over time.
How do you derive group velocity?
Derivation of Group Velocity Formula u ≡ d x d t = Δ ω Δ k .
How do you find phase velocity and group velocity?
In a given medium, the frequency is some function ω(k) of the wave number, so in general, the phase velocity vp= ω/k and the group velocity vg = dω/dk depend on the frequency and on the medium. The ratio between the speed of light c and the phase velocity vp is known as the refractive index, n = c/vp = ck/ω.
What is the meaning of phase velocity?
The phase velocity is defined as the velocity for a single-wavelength wave, whereas the group velocity is defined as the velocity for a packet of waves in which the waves vary in wavelength.
What is the difference between particle velocity and phase velocity?
The particle velocity has to depend on the wave amplitude, while the phase speed-wavelength relationship is for linear waves (independent of amplitude).
What is phase of a wave?
Phase is the same frequency, same cycle, same wavelength, but are 2 or more wave forms not exactly aligned together. The phase involves the relationship between the position of the amplitude crests and troughs of two waveforms.
What do you understand by phase velocity?
Definition of phase velocity : the velocity of a wave motion as determined by the product of the wavelength and frequency.
What do you mean by group velocity?
The group velocity of a wave is the velocity with which the overall envelope shape of the wave’s amplitudes—known as the modulation or envelope of the wave—propagates through space.
What is phase velocity formula?
For this reason, phase velocity is defined as v p = ω / k . This relation is generalized to three dimensions in Eq. (59). Given the phase velocity, the phase refractive index np is the factor describing how much slower the phase is propagating than the speed of light c0.
What is group velocity and particle velocity?
While each individual wave of the set that now describes the particle still travels with its individual phase velocity viP, the maximum of the total amplitude of all waves – signifying the most likely place to find the particle – travels with a velocity that must be identified with the particle velocity v and that is …
What is phase and group velocity?
Introduction Phase and group velocity are two important and related concepts in wave mechanics. They arise in quantum mechanics in the time development of the state function for the continuous case, i.e. wave packets. Discussion Harmonic Waves and Phase Velocity A one-dimensional harmonic wave (Figure 1) is described by the equation,
What is the partial derivative of ω?
So Vg is the partial derivative of ω. Based on the definition Vp = ω/k, we can replace ω with k*Vp, then we get When these two monochrome waves are propagating in vacuum, they have the same phase velocity c (the speed of light in vacuum), and the superposed wave’s phase velocity equals its group velocity (both are c).
How to find phase velocity and group velocity of superposed wave?
Phase velocity Vp can be concluded by keeping the phase a constant: Then by doing derivative of z we get the Phase Velocity Vp of the superposed wave: Similarly we can get the Group Velocity Vg by keeping the amplitude a constant:
What is the phase velocity of a wave in a vacuum?
In a vacuum, the wave clearly moves with velocity c = ω/k. Now, if that wave passes into a medium with refractive index n , then some of its properties will become Simple enough. The group velocity is a bit more complicated. We can with a bit of work derive a relationship between the phase velocity and group velocity.