10soln10 - Physics 315 Oscillations and Waves Homework 10...

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Unformatted text preview: Physics 315: Oscillations and Waves Homework 10: Solutions 1. (a) Let S = summationdisplay n =0 ,N − 1 z n = 1 + z + ··· + z N − 2 + z N − 1 . (1) It follows that z S = z + z 2 + ··· + z N − 1 + z N . (2) Thus, S − z S = 1 − z N , (3) or S = 1 − z N 1 − z . (4) (b) Let z = e i θ . It follows that S = 1 − e i N θ 1 − e i θ = e i N θ/ 2 bracketleftbig e − i N θ/ 2 − e i N θ/ 2 bracketrightbig e i θ/ 2 bracketleftbig e − i θ/ 2 − e i θ/ 2 bracketrightbig . (5) Thus, S = e i ( N − 1) θ/ 2 sin( N θ/ 2) sin( θ/ 2) . (6) since sin θ ≡ 1 2 i ( e i θ − e − i θ ) . (7) (c) Now e i n θ ≡ cos( n θ ) + i sin( n θ ) , (8) so it follows from (1) that Re ( S ) = summationdisplay n =0 ,N − 1 Re ( z n ) = summationdisplay n =0 ,N − 1 Re (e i n θ ) = summationdisplay n =0 ,N − 1 cos( n θ ) . (9) Likewise, Im( S ) = summationdisplay n =0 ,N − 1 Im (e i n θ ) = summationdisplay n =0 ,N − 1 sin( n θ ) . (10) Hence, from (6), summationdisplay n =0 ,N − 1 cos( n θ ) = cos[( N − 1) θ/ 2] sin( N θ/ 2) sin( θ/ 2) , (11) summationdisplay n =0 ,N − 1 sin( n θ ) = sin[( N − 1) θ/ 2] sin( N θ/ 2) sin( θ/ 2) . (12) Let C = summationdisplay n =1 ,N cos( α y n ) = summationdisplay n =1 ,N cos[( n − 1) α d − ( N − 1) α d/ 2] , (13) where y n = [ n − ( N + 1) / 2] d . It follows that C = cos[( N − 1) α d/ 2] summationdisplay n ′...
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10soln10 - Physics 315 Oscillations and Waves Homework 10...

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