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

4.23. Wave equation for an infinite lossy tube

Assumptions

The circular tube of radius a shown in Fig. 4.40 has z as the axial ordinate and w as the radial ordinate. In the following discussion, it is assumed that the radial pressure distribution is uniform and that the pressure variations are purely axial. This has been shown to be valid provided that a(meters)   ≤   10⁴/f ^3/2 [11]. Also, it is assumed that the radial velocity is zero, but the axial velocity is allowed to vary radially because of laminar flow resulting from viscous losses. Thermal conduction through the tube wall is also taken into consideration, where the wall is at ambient temperature T ₀. However, boundary slip is allowed for, whereby the axial particle velocity adjacent ...