A didactically novel derivation of the telegraph equation to describe sound propagation in rigid tubes

Abstract

Starting from first principles, we derive the telegraph equation to describe the propagation of sound waves in rigid tubes by using a simple approach that yields a lossy transmission line model with frequency-independent parameters. The approach is novel in the sense that it has not been found in the literature or textbooks. To derive the lossy acoustic telegraph equation from the lossless wave equation, we need only to relax the assumption that the dynamical variables are constant over the entire cross-sectional area of the tube. In this paper, we do this by introducing a relatively narrow boundary layer at the wall of the tube, over which the dynamical variables decrease linearly from the constant value to zero. This allows us to make very simple corrections to the lossless case, and to express them in terms of two parameters, namely the viscous diffusion time constant and the thermal diffusion time constant. The coefficients of the resulting telegraph equation are frequency-independent. 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. These relationships are thus proven a posteriori rather than asserted a priori. In this way, we arrive at an instructive and useful derivation of the acoustic telegraph equation, which takes viscous damping and thermal dissipation into account, and is accessible to students at the undergraduate level. This derivation 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. Surprisingly, we have been unable to find a comparable treatment in the standard introductory physics and acoustics texts, or in the literature.

Impedance modelling of pipes

“…Nodes and antinodes in pipes are dependent on the length of the pipe and boundary conditions (open ended, closed ended, or opened with a flange). The frequencies of the resonant standing waves can be found using acoustic impedance models [5][6][7][8] with circuit analysis tools to combine the impedances of the individual pipe system components into an equivalent circuit pipe model (ECPM). In this equivalent model, the impedances provide equations for the system that can be used to approximate the natural frequencies and the acoustic pressure in the pipe.…”

Bilateral Spontaneous Otoacoustic Emissions Show Coupling between Active Oscillators in the Two Ears

“…1 , C and D ). Propagation of the sound wave was governed by the damped-acoustic-telegraph equation ( 38 ). The ensuing displacements of each eardrum were then used to calculate the corresponding SOAEs.…”

Bilateral spontaneous otoacoustic emissions show coupling between active oscillators in the two ears

“…We assumed that the vibrations of an eardrum created not only ipsilateral SOAEs but also sound waves that traversed the mouth and caused a pressure difference across the contralateral tympanum; this provided acoustic coupling between the eardrums ( Figure 1C, D). Propagation of the sound wave was governed by the dampedacoustic-telegraph equation 26 . The ensuing displacements of each eardrum were then used to calculate the corresponding SOAEs.…”