10hw10 - Physics 315: Oscillations and Waves Homework 10:...

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Physics 315: Oscillations and Waves Homework 10: Due in class on Wednesday, Dec. 2nd 1. (a) Consider the geometric series S = s n =0 ,N - 1 z n , where z is a complex number. Demonstrate that S = 1 - z N 1 - z . (b) Suppose that z = e i θ , where θ is real. Employing the well-known identity sin θ 1 2 i ( e i θ - e - i θ ) , show that S = e i ( N - 1) θ/ 2 sin( N θ/ 2) sin( θ/ 2) . (c) Finally, making use of de Moivre’s theorem, e i n θ cos( n θ ) + i sin( n θ ) , demonstrate that C = s n =1 ,N cos( α y n ) , where y n = [ n - ( N + 1) / 2] d, evaluates to C = sin( N α d/ 2) sin( α d/ 2) .
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2. Suppose that a monochromatic laser of wavelength 632 . 8 nm emits a diFrac- tion-limited beam of initial diameter 2 mm. Estimate how large a light spot the beam would produce on the surface of the Moon (which is a mean distance 3 . 76 × 10 5 km from the surface of the Earth). Neglect any eFects of the Earth’s atmosphere. 3. Estimate how far away an automobile is when you can only just barely
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This note was uploaded on 11/02/2010 for the course PHY 59372 taught by Professor Fitzpatrick during the Spring '10 term at University of Texas.

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10hw10 - Physics 315: Oscillations and Waves Homework 10:...

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