PHYS 422 Lecture 11 Superposition III

# PHYS 422 Lecture 11 Superposition III - The Superposition...

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Unformatted text preview: The Superposition of Waves III Anharmonic Periodic Waves Superposition of harmonic waves leads periodic but anharmonic wave to periodic but anharmonic wave Fourier Series Fourier’s Theorem: any function f ( x ) that has a spatial period λ can be synthesized by a sum of harmonic functions whose avelengths are integral submultiples of wavelengths are integral submultiples of λ ( λ , λ /2, λ /3, … ) . 2 s 2 s ⎞ ⎛ ⎞ ⎛ = π π x x ( ) ... 2 / cos cos 2 2 1 1 + ⎟ ⎠ ⎜ ⎝ + + ⎟ ⎠ ⎜ ⎝ + + ξ λ ξ λ C C C x f represents component with λ = ∞ , i.e. series for λ /0 , λ /1, λ /2, … traveling wave: f ( x- v t ) k =2 π / λ ( ) ( ) ( ) ( ) ... cos ... 2 cos cos 2 2 1 1 + + + + + + + + = m m mkx C kx C kx C C x f ξ ξ ξ kx kx kx n s s ( ) ( ) ( ) mkx B mkx A mkx C m m m m sin cos cos + = +ξ Fourier Series ( ) ( ) ( ) ∑ ∑ ∞ = ∞ = + + = 1 1 sin cos 2 m m m m mkx B mkx A A x f Can be shown (page 304): ( ) ( ) ∫ = λ λ cos 2 dx mkx x f A m ( ) ( ) ∫ = λ λ sin 2 dx mkx x...
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## This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.

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PHYS 422 Lecture 11 Superposition III - The Superposition...

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