s11_mthsc208_ws7b-WaveEqn - MthSc 208 (Spring 2011)...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

Page1 / 3

s11_mthsc208_ws7b-WaveEqn - MthSc 208 (Spring 2011)...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online