1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Damped wave equation
  1. Wave equation (Replies: 0)

  2. PDE wave equation. (Replies: 0)

Loading...