Show that phase velocity is greater than particle velocity
Answers
Answered by
0
energy, information, is carried by the group velocity.
Just look the general weavy maths:
an “atomic” cosine wave is A. cos(kx-wt) = A.cos(k(x-ct)).
But in many cases dispersion is possible, i.e. both w and k can variate.
Then phase velocity is c_phi = w/k, while group velocity c_g is dw/dk (this can be proven by interfering waves of close characteristics).
The order and behavior of these 2 entities can be pretty arbitrary. E.g. for water surface waves, gravity waves (the swell) have c_g = 1/2 c_phi while for capillary waves c_g = 3/2 c_phi .
In the first case the wave front seems to appear at the rear of the packet and die at the front. And the opposite in the second case.
For light in matter, you can produce varing index of refraction making the derivative do kinky things.
Just look the general weavy maths:
an “atomic” cosine wave is A. cos(kx-wt) = A.cos(k(x-ct)).
But in many cases dispersion is possible, i.e. both w and k can variate.
Then phase velocity is c_phi = w/k, while group velocity c_g is dw/dk (this can be proven by interfering waves of close characteristics).
The order and behavior of these 2 entities can be pretty arbitrary. E.g. for water surface waves, gravity waves (the swell) have c_g = 1/2 c_phi while for capillary waves c_g = 3/2 c_phi .
In the first case the wave front seems to appear at the rear of the packet and die at the front. And the opposite in the second case.
For light in matter, you can produce varing index of refraction making the derivative do kinky things.
Similar questions