Example 3.  Consider the wave equation where  .  The string at rest has length  .  Assume that the initial position

          

Use the finite difference method to solve the wave equation over the rectangle  .  

Solution 3.

Remark.  We will be using   .  This forces  .

Using the same functions as in Example 2, we increase the interval width ever so slightly to be  .  Notice the instability this produces which was predicted in advance in our mathematical analysis discussed in the text.

Now set up the table of solutions.  Solve it.  Plot it.  

(c) John H. Mathews 2004