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Unformatted text preview: D’Alembert’s Solution There is an elegant approach to solve the wave equation by introducing new variables: { } , , ( , ) ( , ) The use of these variables is because that the solution of the wave equation behaves in specific fashion that its spatial movement is related to the temporal variation throu v x ct z x ct u x t u v z = + = = gh the constant . Using these new variables, the derivative w.r.t x & t can be rewritten as ( ) ( ) Label = , , , x v z c u u v u z u x ct u x ct u u x v x z x v x z x v z u u u u u u etc x v z ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = + = + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = = ∂ ∂ ∂ 2 Similarly, u u u u u u u ( ) ( ) [ ] t t t Continue to convert all derivatives in x & t into derivatives in & , the wave equation to obtain the following equation: v z c c c v z v z v z v z u u z v z v ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + = + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ 0, this equation can be integrated twice ( ), ( , ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) : D'Alembert's solution u f v v u v z f v dv z v z u x y...
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This note was uploaded on 11/27/2011 for the course EML 3050 taught by Professor Staff during the Fall '09 term at FSU.
 Fall '09
 staff
 Mechanical Engineering

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