Relating Pressure and Velocity in Acoustic Tubes

Do Equations C.8 and C.9 look familiar They should, because the form is exactly the same traveling wave equation we derived for the plucked string in Appendix B. Note that the area term has cancelled out in both traveling wave equations. 

Just as we proved the D Alembert solution to the ideal string (Section B. 1.2), we know that a solution to Equation C.9 for the acoustic tube is  

We can relate pressure and velocity in the acoustic tube by going back to Equation C.5  

We can also express this using the D Alembert solution form  

These state that the spatial derivative of pressure is equal to a constant multiplied by the time derivative of flow. The constant pc/a is called the characteristic impedance, which we will use in the next section. 

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