The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape

TLDR

Kirchhoff theory of sound propagation in circular tubes simplifies for both narrow and wide tubes, reducing to the Zwikker–Kosten approximation when tube radii exceed 10⁻³ cm and frequencies satisfy rw f³⁄² < 10⁶ cm s⁻³⁄², with sound pressure nearly uniform across the cross‑section and excess density and pressure of comparable magnitude. The study aims to generalize sound propagation theory to uniform tubes of arbitrary cross‑sectional shape by extending the simplified Kirchhoff framework. The authors treat viscosity and thermal conductivity separately using complex density and compressibility functions, derive a two‑dimensional wave equation for particle velocity or excess temperature, and compute complex density, compressibility, propagation constants, and characteristic impedances. The method was applied to a rectangular‑cross‑section tube, yielding its propagation characteristics.

Abstract

The general Kirchhoff theory of sound propagation in a circular tube is shown to take a considerably simpler form in a regime that includes both narrow and wide tubes. For tube radii greater than rw=10−3 cm and sound frequencies f such that rwf3/2&amp;lt;106 cm s−3/2, the Kirchhoff solution reduces to the approximate solution suggested by Zwikker and Kosten. In this regime, viscosity and thermal conductivity effects are treated separately, within complex density and complex compressibility functions. The sound pressure is essentially constant through each cross section, and the excess density and sound pressure (when scaled by the equilibrium density and pressure of air, respectively) are comparable in magnitude. These last two observations are assumed to apply to uniform tubes having arbitrary cross-sectional shape, and a generalized theory of sound propagation in narrow and wide tubes is derived. The two-dimensional wave equation that results can be used to describe the variation of either particle velocity or excess temperature over a cross section. Complex density and compressibility functions, propagation constants, and characteristic impedances may then be calculated. As an example, this procedure has been used to determine the propagation characteristics for a tube of rectangular cross section.