section 19.2

section 19.2 - 640:527: Solutions Text, 19.2; 5(b) The pde...

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Unformatted text preview: 640:527: Solutions Text, 19.2; 5(b) The pde is the wave equation c 2 y xx = y tt with 0 < x < L , t > 0 and boundary conditions y (0 ,t ) = 0, y x ( L,t ) = 0. By separation of variables y ( x,t ) = X ( x ) T ( t ) is a solution if X 00 + X = 0, X (0) = 0 and X ( L ) = 0, and T 00 + c 2 T = 0. The Sturm-Liouville problem defined by the equation for X has eigenvalues n = ((2 n- 1) / 2 L ) 2 , n = 1 , 2 ,... with corresponding eigenfunctions X n ( x ) = sin((2 n- 1) x/ 2 L ). The solution T corresponding to n is T ( t ) = A cos((2 n- 1) ct/ 2 L )+ B sin((2 n- 1) ct/ 2 L ). Hence the general solution to this wave equation with boundary conditions is y ( x,t ) = X n =1 a n cos( (2 n- 1) ct 2 L ) + b n sin( (2 n- 1) ct 2 L ) sin( (2 n- 1) x 2 L ) The initial conditions y ( x, 0) = f ( x ) and y t ( x, 0) require f ( x ) = X n =1 a n sin( (2 n- 1) x 2 L ) 0 = X n =1 b n (2 n- 1) c 2 L sin( (2 n- 1) x 2 L ) These are quarter-range sine expansions and are achieved setting...
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section 19.2 - 640:527: Solutions Text, 19.2; 5(b) The pde...

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