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Unformatted text preview: Physics 325, Fall 2010 Solutions to Homework #7 1) The two ways of writing the Fourier series specified in the question are equiavalent if one can find c n and φ n such that c n cos( nωt φ n ) = a n cos( nωt ) + b n sin( nωt ) Using a trigonometric identity, we have c n cos( nωt φ n ) = c n cos( nωt ) cos( φ n ) + c n sin( nωt ) sin( φ n ) We therefore seek c n and φ n such that c n cos φ n = a n and c n sin φ n = b n . [This calculation is like a conversion between rectangular and polar coordinates.] c n = q a 2 n + b 2 n φ n = tan 1 b a ! 2) a) This first part of the question involves finding the Fourier series of a periodic function. F ( t ) = F  sin ω f t  = 1 2 a + ∞ X n =1 a n cos( nω f t ) + b n sin( nω f t ) with a n = 2 τ Z τ/ 2 τ/ 2 F ( t ) cos( nω f t ) dt b n = 2 τ Z τ/ 2 τ/ 2 F ( t ) sin( nω f t ) dt The period τ = 2 π/ω f so τ/ 2 = π/ω f and 2 /τ = ω f /π . Actually, this is twice the period, but it is easier to apply the Fourier series formulas from class if we center the intervals on zero. Also, it is important to use this defintion of the period because theon zero....
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Physics, mechanics, Work

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