# Damped wave equation

1. Nov 11, 2012

### Dustinsfl

$$u_{tt} + 3u_t = u_{xx}$$
$$u(0,t) = u(\pi,t) = 0$$
$$u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10.$$
\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*}