Question

A train of plane light waves propagates in the medium where the phase velocity `v` is a linear function of wavelength: `v = 1 + blambda`, where `a` and `b` are some positive constants. Demonstrate that in such a medium the shape of an arbitary train of light waves is restroed after the time interval `tau = 1//b`.

Answer 1

We write
`v = (omega)/(k)= a+ b lambda`
so `omega = k(a+b lambda) = 2pi b + ak`.
(since `k = (2pi)/(lambda)`). Suppose a wavetrain at time `t = 0` has the form
`F(x, 0) = int f(k) e^(ikx) dk`
Then at time `t` it will have the form
`F(x,t) = int f(k) e^(ikx-i omegat) dk`
`=int f(k)e^(ikx -i(2pib + ak)t) = int f(k)e^(ik(x-at)e^(-2pi bt) dk`
At `t = (1)/(b) = tau`
`F(x, tau) = F(x - a tau, 0)`
so at time `t = tau` the wave train has regained its shape through it has advanced by `a tau`.