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Unformatted text preview: 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 and t 0 which satisfies the following initial value problem of the wave equation: utt = c2 uxx u(0, t) = u(, t) = 0, u(x, 0) = x( - x), ut (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 ut , utt , ux , uxx , and find boundary conditions for f (x). (c) Plug u = f g back into the PDE and separate variables by dividing both sides of the equation by c2 f g. Set this equal to a constant , and write down two ODEs: one for f (x) and one for g(t). Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 7b (d) Solve the ODE for f (x) (including the boundary conditions), and determine . You may assume that = - 2 < 0. (e) Now that you know what is, solve the ODE for g(t). (f) Find the general solution of the PDE. As before, it will be a superposition (infinite sum) of solutions un (x, t) = fn (x)gn (t). Written by M. Macauley 2 MthSc 208 (Spring 2011) Worksheet 7b (g) Find the particular solution to the initial value problem by using the initial conditions. The following information is useful: 4 (1 - (-1)n ) sin nx. The Fourier sine series of x( - x) is n3
n=1 (h) What is the long-term behavior of the system? Give a mathematical, and physical, justification. Written by M. Macauley 3 ...
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