Numerical Solution for Sound Velocity and Absorption in Cylindrical Tubes

Abstract

A numerical solution of the Kirchhoff equation for the propagation constant of longitudinal sound waves in infinitely long cylindrical tubes has been obtained. The solution, which avoids the wide‐tube approximations, shows that the percentage errors in the von Helmholtz‐Kirchhoff tube velocity correction and tube absorption are both roughly equal to the percentage the velocity correction is of the free‐space velocity. The error in the von Helmholtz‐Kirchhoff equations can be plotted as a function of fD/a, pD/ηa, and γ. (f is the sound frequency, D the tube diameter, a the free‐space velocity,p the gas pressure, η the viscosity, and γ the ratio of specific heats.) Recent absorptionmeasurements in Ar are in agreement with values calculated numerically, but measuredvelocities indicate the need for considering molecular slip at the tube wall. Thermal relaxation is introduced into Kirchhoff's basic equation by using the Eucken relation k/c o η − (9 γ −5)/4 and considering γ to be the ratio of complex relaxing specific heats. Viscothermal and relaxation effects are found to be additive only if the frequency is near the cutoff frequency for the first unsymmetric mode and the f/p values do not extend to the megacycle/second atmospheres range.