InterviewSolution
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InterviewSolution
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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`. |
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Answer» 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`. |
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