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Unformatted text preview: Physics 2214 Fall 2008 Solutions to Problem Set #6 1  1. Beats (a) To put the given expression s(x,t) = A exp i(k 1 x + ω 1 t) + A exp i(k 2 x + ω 2 t) into the desired form, we use the following substitutions: k 1 = k − + Δ k 2 ; ω 1 = ω − + Δω 2 ; k 2 = k − Δ k 2 ; ω 1 = ω − Δω 2 . This gives s(x,t) = A exp i ⎩⎪ ⎨ ⎪ ⎧ ⎭ ⎪ ⎬ ⎪ ⎫ ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ k − + Δ k 2 x + ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ ω − + Δω 2 t + A exp i ⎩ ⎪ ⎨ ⎪ ⎧ ⎭ ⎪ ⎬ ⎪ ⎫ ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ k − Δ k 2 x + ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ ω − Δω 2 t . Now we can factor out the common terms: s(x,t) = A ⎩⎪ ⎨ ⎪ ⎧ ⎭ ⎪ ⎬ ⎪ ⎫ exp i ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ Δ k 2 x + Δω 2 t + exp i ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ Δ k 2 x + Δω 2 t exp i(k − x + ω − t). We use Euler's identity to simplify the part in curly brackets, s(x,t) = 2A cos ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ Δ k 2 x + Δω 2 t exp i(k − x + ω − t), and then we obtain A(x,t) = 2A cos ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ Δ k 2 x + Δω 2 t . (b) For a point of constant phase on the carrier wave, φ carrier (x,t) = k − x + ω − t = constant. The phase speed is then given by v ph = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ x ∂ t = ω − /k − .  2  (c) The envelope of the signal is A(x,t). For a point of constant phase on the envelope, we have φ envelope (x,t) = Δ k 2 x + Δω 2 t = constant. The group speed is then given by v gr = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ x ∂ t = Δω Δ k . (d) In a dispersionless medium, v is independent of k , so that v = ω /k for all k. Then ω 1 = vk 1 and ω 2 = vk 2 ; and the phase and group speeds become v ph = ω − / k − = ½( ω 1 + ω 2 ) ½(k 1 + k 2 ) = vk 1 + vk 2 k 1 + k 2 = v , and v gr = Δω Δ k = ω 1 ω 2 k 1 k 2 = vk 1 vk 2 k 1 k 2 = v . So we have shown that v ph = v gr in a dispersionless medium. When v ph = v gr , the speed of the envelope is the same as that of the carrier. Since they travel along at the same speed, individual peaks in the carrier have fixed amplitude. If v ph ≠ v gr , the carrier wave and the envelope have different velocities, so that the carrier moves relative to the envelope. Individual peaks in the carrier then increase and decrease in amplitude as the envelope passes around them. peaks in the carrier then increase and decrease in amplitude as the envelope passes around them....
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This note was uploaded on 12/24/2008 for the course PHYS 2214 taught by Professor Giambattista,a during the Fall '07 term at Cornell University (Engineering School).
 Fall '07
 GIAMBATTISTA,A
 Physics

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