Exercise 5.9: Finite differences for the wave equation

Write a program to solve the wave equation using finite differences. Assume that the string has a length $l=1$m, a linear density $\rho =0.01$kg/m, and tension $k=40$N.

1. Assume as initial conditions that the string is ``plucked'':
   
   $$u(x,t=0)=\left\{ \begin{array}{ll} 1.25x/l, & \mathrm{for ... ... \\ 5.0(1-x/l), & \mathrm{for }x>0.8l, \end{array}\right.$$
   
   
   Plot the displacement $u$ as a function of $(x,t)$ using a space step $% \Delta x=0.01$m, and choosing the time step such that the solution is stable.

2. Explore the use of different steps $\Delta x$ and $\Delta t$ and determine at which values the solution becomes unstable. Does your assessment agree with the condition (60)?

3. Change the initial conditions to
   
   $$u(x,t=0)=\left\{ \begin{array}{ll} x/l, & \mathrm{for }0... ...-x/l, & \mathrm{for }0.5\leq x\leq 0.1, \end{array}\right.$$
   
   
   and compare results with the previous simulation.

4. Consider a string plucked at two points:
   
   $$u(x,t=0)/0.005=\left\{ \begin{array}{ll} 0, & 0.0\leq x\leq... ...\leq x\leq 0.9, \\ 0 & 0.9\leq x\leq 1.0. \end{array}\right.$$
   
   
   Solve and observe wether the pulses move or oscillate up and down.

5. Explore what happens when the string is places in a ``normal mode'', for example:

   
   

   $$u(x,t=0)=0.001\sin (2\pi x).$$
   
   

   Try other modes. See if the sum of two modes gives ``beating''.