Fourier series pt 3

Fourier Series Pt 3
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Unformatted text preview: University of Waterloo CivE 331 Determine the Fourier Series representation of the following function using the half-range cosine expansion. Graph the expansion for n = 10. By inspection, the domain length ( L ) is 10. In the period of interest (0 < x < 10): ( 29 ( 29 < < = = < < + =---- 10 4 ; 62 . 14 5 5 4 ; 1 4 4 x e e e e x x x f kx kx k x k Substitute 10 = L into the HRC expression: ( 29 10 ; 10 cos 1 < < + = = x x n a a x f n n π Substitute 10 = L and f ( x ) into the coefficient expressions: ( 29 ( 29 + + = = - 10 4 4 10 62 . 14 1 10 1 10 1 dx e dx x dx x f a kx ( 29 ( 29 + + = = - 10 4 4 10 10 cos 62 . 14 10 cos 1 10 2 10 cos 10 2 dx x n e dx x n x dx x n x f a kx n π π π Now evaluate a and a n . The following relationships will be useful (and would be provided in an exam or quiz context): ( 29 ( 29 ( 29 a ax x a ax dx ax x sin cos cos 2 + = ( 29 ( 29 ( 29 [ ] 2 2 sin cos cos b a bx b bx a e dx bx e ax ax + + = ( 29 ( 29 A A 2 sin 2 1 2 cos- = 0.00 1.00 2.00 3.00 4.00 5.00 6.00 1 2 3 4 5 6 7 8 9 10 ( 29 x x f + = 1 ( 29 ( 29 26824 . ; 5 4 = =-- k e x f x k University of Waterloo CivE 331 Evaluate a : ( 29 691 . 2 62 . 14 12 10 1 62 . 14 2 10 1 10 4 10 4 4 2 = - + = - + ...
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