Modes of a Pan-Pipe

Wind or human breath running over the top of a pan-pipe (a tube open at both ends) can drive resonances and make several harmonic tones. Suppose sound waves are generated in this manner in a narrow tube of length $L = 10$ cm open at both ends. Express your answers to the questions below in terms of $L$ and $v$ (the speed of sound in air) before substituting any values to get a numerical answer. You do not have to derive the answers to a-c, just show me that you know the answers.

a) Write down the wave equation (the differential equation of motion) for longitudinal displacement (sound) waves in air.

b) Write down the solution to this equation that describes resonant standing waves in a tube open at both ends, assuming that each mode is maximally displaced at $t = 0$.

c) Find $k_n, \omega_n, f_n, \lambda_n$ for the first four modes (resonant frequencies) supported by the tube. Sketch the displacement amplitudes in on the figure above the same way it was done in the book and lecture, labelling nodes and antinodes.