A time-domain derivation of the telegraph equation to describe sound propagation in rigid tubes

Starting from first principles, we describe the propagation of sound waves in rigid tubes using a simple approach that derives a telegraph equation with frequency-independent parameters from the lossless wave equation by relaxing one assumption: that the dynamical variables are constant over the entire cross-sectional area of the tube. We do this by introducing a relatively narrow boundary layer at the wall of the tube, over which the dynamical variables decrease linearly to zero. This allows us to make very simple corrections to the lossless case, and to express them in terms of two unorthodox parameters, the viscous diffusion time constant and the thermal diffusion time constant. A comparison with the telegraph equation for the electrical transmission line establishes precise relationships between the electrical circuit elements and the physical properties of the fluid, which are thus proven a posteriori rather than asserted a priori. We thus arrive at an instructive and useful derivation of the acoustic telegraph equation, which takes viscous damping and thermal dissipation into account but does not resort to the combined heavy machinery of fluid dynamics and thermodynamics, does not assume that the waveforms are sinusoidal, and does not assume any particular cross sectional shape of the tube. The approach is novel in that a comparable treatment appears neither in the standard physics and acoustics texts nor in the literature.