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s11_mthsc208_ws7b-WaveEqn

# s11_mthsc208_ws7b-WaveEqn - MthSc 208(Spring 2011 Worksheet...

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MthSc 208 (Spring 2011) Worksheet 7b MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 7b: The Wave Equation NAME: We will solve for the function u ( x, t ) defined for 0 x π and t 0 which satisfies the following initial value problem of the wave equation: u tt = c 2 u xx u (0 , t ) = u ( π, t ) = 0 , u ( x, 0) = x ( π - x ) , u t ( x, 0) = 1 . (a) Carefully descsribe (and sketch) a physical situation that this models. (b) Assume that u ( x, t ) = f ( x ) g ( t ). Compute u t , u tt , u x , u xx , and find boundary conditions for f ( x ). (c) Plug u = fg back into the PDE and separate variables by dividing both sides of the equation by c 2 fg . Set this equal to a constant λ , and write down two ODEs: one for f ( x ) and one for g ( t ). Written by M. Macauley 1

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MthSc 208 (Spring 2011) Worksheet 7b (d) Solve the ODE for f ( x ) (including the boundary conditions), and determine
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s11_mthsc208_ws7b-WaveEqn - MthSc 208(Spring 2011 Worksheet...

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