In Section 2.4, we examined the propagation of one-dimensional waves in a loss-free tube. To be able to neglect viscous losses inside the tube, the radius of the tube must not be too small. Also, to be able to neglect transverse resonances in the tube, the radius must not be too large. Here we shall rederive the one-dimensional wave equation using a slightly different procedure than before, taking into account the viscous and thermal losses that take place at the boundary wall, using what are known as the Navier–Stokes equations. In accordance with the continuum theory of gases, traditional models have assumed that the axial velocity at the wall of the tube is zero and that the temperature there is ambient because of the sheer number of collisions occurring between air molecules and the wall. However, as the diameter of the tube is reduced relative to the mean free path of the molecules, fewer collisions occur so that the axial velocity and temperature both increase at the wall. This is generally known as a slip boundary condition. Formulation is now available [10] that models this slip, thus allowing us to model tubes of much smaller diameter than was previously possible. The resulting wave number is complex, and from this new wave number we shall derive two new parameters called the dynamic density and dynamic compressibility which replace the density and inverse bulk modulus, respectively, in the expressions for wave number and characteristic impedance. These result from the average flow over the cross section of the tube as if the losses were homogeneous throughout the bulk of the acoustic medium, although they are actually localized near the tube wall. We will also define a viscous boundary thickness to define the region within which most of the viscous losses occur. For those readers who are not interested in the full derivation but only wish to apply the results to practical uses, you may skip on to the results shown in Figs. 4.46–4.51.