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Unformatted text preview: Physics 315: Oscillations and Waves Homework 10: Solutions 1. (a) Let S = summationdisplay n =0 ,N âˆ’ 1 z n = 1 + z + Â·Â·Â· + z N âˆ’ 2 + z N âˆ’ 1 . (1) It follows that z S = z + z 2 + Â·Â·Â· + z N âˆ’ 1 + z N . (2) Thus, S âˆ’ z S = 1 âˆ’ z N , (3) or S = 1 âˆ’ z N 1 âˆ’ z . (4) (b) Let z = e i Î¸ . It follows that S = 1 âˆ’ e i N Î¸ 1 âˆ’ e i Î¸ = e i N Î¸/ 2 bracketleftbig e âˆ’ i N Î¸/ 2 âˆ’ e i N Î¸/ 2 bracketrightbig e i Î¸/ 2 bracketleftbig e âˆ’ i Î¸/ 2 âˆ’ e i Î¸/ 2 bracketrightbig . (5) Thus, S = e i ( N âˆ’ 1) Î¸/ 2 sin( N Î¸/ 2) sin( Î¸/ 2) . (6) since sin Î¸ â‰¡ 1 2 i ( e i Î¸ âˆ’ e âˆ’ i Î¸ ) . (7) (c) Now e i n Î¸ â‰¡ cos( n Î¸ ) + i sin( n Î¸ ) , (8) so it follows from (1) that Re ( S ) = summationdisplay n =0 ,N âˆ’ 1 Re ( z n ) = summationdisplay n =0 ,N âˆ’ 1 Re (e i n Î¸ ) = summationdisplay n =0 ,N âˆ’ 1 cos( n Î¸ ) . (9) Likewise, Im( S ) = summationdisplay n =0 ,N âˆ’ 1 Im (e i n Î¸ ) = summationdisplay n =0 ,N âˆ’ 1 sin( n Î¸ ) . (10) Hence, from (6), summationdisplay n =0 ,N âˆ’ 1 cos( n Î¸ ) = cos[( N âˆ’ 1) Î¸/ 2] sin( N Î¸/ 2) sin( Î¸/ 2) , (11) summationdisplay n =0 ,N âˆ’ 1 sin( n Î¸ ) = sin[( N âˆ’ 1) Î¸/ 2] sin( N Î¸/ 2) sin( Î¸/ 2) . (12) Let C = summationdisplay n =1 ,N cos( Î± y n ) = summationdisplay n =1 ,N cos[( n âˆ’ 1) Î± d âˆ’ ( N âˆ’ 1) Î± d/ 2] , (13) where y n = [ n âˆ’ ( N + 1) / 2] d . It follows that C = cos[( N âˆ’ 1) Î± d/ 2] summationdisplay n â€²...
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 Spring '10
 Fitzpatrick
 Work, Light, Sin, angular resolution, diameter, Angular diameter

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