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Homework Help: Damped wave equation

  1. Nov 11, 2012 #1
    $$
    u_{tt} + 3u_t = u_{xx}
    $$
    $$
    u(0,t) = u(\pi,t) = 0
    $$
    $$
    u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10.
    $$
    \begin{alignat*}{3}
    u(x,t) & = & \exp\left[-\frac{3t}{2}\right]\sin x\left[A_1\cosh\frac{t\sqrt{5}}{2} + B_1\sinh\frac{t\sqrt{5}}{2}\right]\\
    & + & \exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}\sin nx\left[C_n\cos \left(t\frac{\sqrt{4n^2 - 9}}{2}\right) + D_n\sin\left(t\frac{\sqrt{4n^2 - 9}}{2}\right)\right]
    \end{alignat*}
    The hyperbolic part is when n = 1 which would be overdamped and the rest are the underdamped modes.
    I have solved for all the coefficients except ##B_1##.
    ##A_1 = C_n = 0## and ##D_n = \begin{cases} 0, & \text{of n is even}\\ \frac{80}{n\pi\sqrt{4n^2 - 9}}, & \text{if n is odd}\end{cases}##
    However, I haven't been able to solve for ##B_1##. Help would be much appreciated.
    \begin{alignat*}{3}
    u(x,t) & = & B_1\exp\left[-\frac{3t}{2}\right]\sin x \sinh\frac{t\sqrt{5}}{2}+ \exp\left[-\frac{3t}{2}\right]\sum_{n = 2}^{\infty}D_n\sin nx\sin\left(t\frac{\sqrt{4n^2 - 9}}{2}\right)
    \end{alignat*}
     
  2. jcsd
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