Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

2.4. Solution of wave equation for air in a tube terminated by an impedance

For this example of wave propagation, we shall consider a hollow cylindrical tube, terminated at one end (x   =   0) by an impedance Z _T and at the other (x   =   l) end by a flat vibrating piston (see Fig. 2.4). Alternatively, we could have interchanged the positions of the piston and termination impedance, but the arrangement shown has been chosen because it simplifies the equations. For example, in the case of a rigid termination the particle velocity is shown to be proportional to sin x as opposed to sin (l   −   x). However, care needs to be taken when calculating the impedance where the velocity has to be taken as that in the negative x direction. The ...