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D'Alembert's Solution

# D'Alembert's Solution - DAlemberts Solution There is an...

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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 + - = + = + = + = =

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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 x ct x ct ψ φ ψ φ ψ
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D'Alembert's Solution - DAlemberts Solution There is an...

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