The dispersion equation in bounded viscothermal fluid: General formulation and applications

Abstract

In the last century, G. Kirchhoff proposed a very fertile description of sound propagation in viscothermal fluid. He calculated the attenuation factors and speeds of propagation of harmonic plane waves in rigid‐walled tubes, which are solutions of the dispersion equation, for the wide tube case and the upper wavelength range, making approximations to the lowest orders possible. The problem of extending these classical Kirchhoff results to the case of higher‐order propagating modes was given by Beatty (1950), but the theory is not valid near the adiabatic cutoff frequency and for the evanescent modes. The initial purpose of this work is to provide an exact and unique general dispersion equation for all kinds of modes in cylindrical tubes. The main feature of this work is a new set of equations, derived from the basic classical theory of acoustic propagation in viscothermal fluid and valid not only in the frequency domain but also in the time domain. Three particular solutions of the general dispersion equation are two well‐known results, i.e., the equivalent specific impedance of a rigid plane wall (equivalent to the viscosity and thermal conduction effects inside the boundary layers) and the complex resonant frequencies of the spherical acoustic resonator, and a new one, i.e., the complex propagation constant of waves for all kind of modes in a rigid‐wailed circular tube. These results greatly facilitate calculations that are currently being worked on to obtain solutions to boundary problems such as the inertial mode coupling in acoustic gyros.