Question

5. A pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths, the modulated amplitude becomes a sinusoid with phase speed ( modulation speed)Δω/Δk, and called group velocity. Let w1 = w – Δω ; w2=w+ Δω ; k1 = k – Δk; k2= k + Δk show that a. D(x,t) = sin(k1x-w1t) + sin(k2x-w2t)= 2sin(kx-wt)cos(Δk*x-Δω*t) b. group velocity = Δω / Δk → dw/dk(when Δω and Δk << 1) Since Vp is the phase velocity and = w/k (as in sin(kx-wt)) in a medium with a refracion index n, n= C/Vp ; show w=Ck/n; C vel of light in free space. In a medium(not free space) w and n is function of k ; Calculate dw/dk = C d(k/n)/dk and show that group vel = phase vel in free space.

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