Question

show that group velocity (vg) = c when and vg = 0 when , where ω, c, k, m and vg have their usual meanings.

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Answer 1

Answer:

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Answer 2

The phase velocity of a wave is the rate at which any one frequency component of the wave travels. On the other hand, the group velocity of a wave is the rate with which modulations of the wave's amplitudes travel through space.

Explanation:

Given a refractive medium, the ratio between c, the speed of light, and the phase velocity vp is called the refractive index:

n=c/vp=ck/ω

Rearranging,

vp=c/n and ω=ck/n

If we assume n to be a function of k, and we take the derivative of ω with respect to k, we get, for the group velocity vg,

vg=dω/dk

=c/n−(ck/n^2)(dn/dk)

= vp−(ck/n^2)(dn/dk)

From the above, we see that group velocity is equal to phase velocity if and only if dn/dk=0, i.e. when the refractive index is constant and there is no dispersion.