Problem #1

Look carefully at the initial conditions shown in Figure 5. Determine how the waves propagate over time and sketch the traveling-wave decomposition for at least four key points in time during one period. For each decomposition, you will probably find it easiest to first sketch what happens to $y_l(t,x)$ and $y_r(t,x)$, and then to sketch $y(t,x)$ as the sum of the other two due to (1).

Figure 5: Wide, rectangular initial condition for traveling waves in a vibrating string with rigid terminations

It would admittedly be hard to start a string from this particular initial condition in real life, but the exercise will help you understand how the rectangular pulses are summed and differenced at different points in time during the period.