Lecture04

Lecture04 - 12.4 D’Alembert’s Solution of the Wave...

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Unformatted text preview: 12.4 D’Alembert’s Solution of the Wave Equation. Char- acteristics. Recall the series solution of the wave equation ( 61 ) with homogeneous Dirich- let boundary conditions ( 62 ) (value of u prescribed as 0) and initial data f , g : u ( x,t ) = ∞ summationdisplay n =1 parenleftBig B n cos parenleftBig nπ L ct parenrightBig + B ∗ n sin parenleftBig nπ L ct parenrightBigparenrightBig sin parenleftBig nπ L x parenrightBig , x ∈ [0 ,L ] , t ≥ , (106) where, for n ∈ N , λ n , B n , and B ∗ n are given by B n = 2 L L integraldisplay f ( x ) sin parenleftBig nπ L x parenrightBig dx, (107) B ∗ n = 2 cnπ L integraldisplay g ( x ) sin parenleftBig nπ L x parenrightBig dx. (108) We assume for simplicity that g ≡ 0, i. e. B ∗ n = 0, n ∈ N . With the addition theorem for the sine function, we write sin parenleftBig nπ L x parenrightBig cos parenleftBig nπ L ct parenrightBig = 1 2 parenleftBig sin parenleftBig nπ L ( x + ct ) parenrightBig + sin parenleftBig nπ L ( x − ct ) parenrightBigparenrightBig , (109) for n ∈ N . Therefore, u ( x,t ) = 1 2 ∞ summationdisplay n =1 B n sin parenleftBig nπ L ( x + ct ) parenrightBig + 1 2 ∞ summationdisplay n =1 B n sin parenleftBig nπ L ( x − ct ) parenrightBig (110) = 1 2 ( f ∗ ( x + ct ) + f ∗ ( x − ct )) , (111) where f ∗ is defined as the odd periodic extension of the initial deflection f with period 2 L , i. e. f ∗ ( x ) := braceleftbigg f ( x ) , x ∈ [0 ,L ) − f (2 L − x ) , x ∈ [ L, 2 L ) , x ∈ [0 , 2 L ) , (112) and extended periodically. Note that because of the compatibility condition f (0) = f ( L ) = 0, the function f ∗ is continuous. 24 Remark: In Problem Set 3, you will verify that the Fourier series of f ∗ is indeed given by f ∗ ( x ) = ∞ summationdisplay n =1 B n sin parenleftBig nπ L x parenrightBig , B n = 2 L L integraldisplay f ( x ) sin parenleftBig nπ L...
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.

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Lecture04 - 12.4 D’Alembert’s Solution of the Wave...

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