Many musical instruments operate by the formation of plane waves inside a cylindrical tube. To describe them we seek periodic solutions to the wave equation of the form

\[\delta_{\lambda}=f(x) \cos (n t)\]

These solutions represent stationary waves in the tube.

a. Show that the stationary wave solutions to the wave equation must obey the following ordinary differential equation.

\[\frac{d^{2} f}{d x^{2}}+\frac{n^{2}}{c_{\lambda}^{2}} f=0\]

where \(c_{\lambda}\) is the wave velocity.

b. The ends of our tube may be open or closed. If closed, the velocity vanishes there \(\left(\partial \delta_{\lambda} / \partial x=0\right)\). Consider a tube of length, \(L\), closed at both ends. Determine the generic solution for such a case. What is the lowest frequency note (fundamental) that can be sounded in such a configuration?

c. If a tube end is open, the pressure must equal atmospheric pressure at the end and so \(\partial \delta_{\lambda} / \partial t=0\). Consider a tube of length \(L\), whose end at \(x=0\) is closed and whose end at \(x=L\) is open. Determine the generic solution for this case. What is the fundamental frequency?

d. Is there any difference in fundamental frequency for a tube with both ends closed or both ends open?