L25 - The Wave Equation t u ( t, L ) = 0 u ( t, 0) = 0 u tt...

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Unformatted text preview: The Wave Equation t u ( t, L ) = 0 u ( t, 0) = 0 u tt = 2 u xx L x u (0 , x ) = f ( x ) u t (0 , x ) = g ( x ) For u ( t, x ) = T ( t ) X ( x the differential equation imp T ( t ) X ( x ) = 2 T ( t ) X ( x ) = T / 2 T ( t ) = X ( x ) /X ( x ) = The boundary conditions im X (0) = X ( L ) = 0 To start with, find all funct X ( x ) and constants with X ( x ) = X ( x ) X (0) = X ( L ) = 0 The situation is very similar to the first heat conduction problem we studied: We have a differential equation u tt = 2 u xx , ( E ) two homogeneous boundary conditions u ( t, 0) = u ( t,L ) = 0 , ( BC ) and two initial conditions: u (0 ,x ) = f ( x ) , ( IP ) and u t (0 ,x ) = g ( x ) . ( IV ) As in the heat conduction case we go first after the three homogeneous Green con- ditions ( E ) , ( BC ) . 2 Again we try first for solutions of them that have their variables separated: u ( t,x ) = T ( t ) X ( x ) . ( SV ) Any linear combination of those will still satisfy ( E ) and...
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This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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L25 - The Wave Equation t u ( t, L ) = 0 u ( t, 0) = 0 u tt...

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