Problem #3

You may be wondering how a vibrating string would behave if the right termination were non-inverting instead of inverting. This could be realized physically by attaching the end of the string to a frictionless, vertical guide rod so that

$$y_l(t,L) = y_r(t,L)$$ (4)

Figure 7: Triangular initial condition for traveling waves in a vibrating string with a frictionless, vertical guide rod termination on the right

It is actually easier to think of a realistic example involving columns of air. Consider pressure waves in a pipe with the left end open and the right end closed. At the open end on the left, the pressure will be relatively small since the ambient pressure outside the pipe is low. This means that the open end behaves like a node for air pressure waves. Conversely, the closed right end behaves like an anti-node for pressure waves. This is because the closed, confined end allows the pressure to become large.

Determine how the waves specified by this initial condition will propagate over time. Sketch the traveling-wave decomposition for at least four key points in time during one period.