Problem 3 [20 pts]
wave equation.
We will study the vibration behaviour of a string. Consider a string with two end fixed on the wall as shown in Fig.2. $u$ is the displacement of the string, $x$ is the horizontal axis. The string stand still until a perturbation make it start to vibrate. At $t = 0$ the displacement $u(x, t = 0) = 0$, but the initial velocity of the string $\frac{\partial u(x, t = 0)}{\partial t} = v_0$ here $v_0$ is a non zero constant. The displacement $u$ can be described by following equation:
$u_{tt} = u_{xx}$ (2)
(i). write down the boundary conditions of the string system for $u(x, t)$
(ii). use the initial condition and the boundary condition, solve for the solution of $u(x, t)$ using separation of variable
