Journal zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Sound and Vibration (1984) 93(4), 473-480 zyxwvutsrqponmlkjihgfedcbaZYXWVUTS W A LL EFFECTS ON SOUND PROPAGATION IN TUBES N. W. PAGEANDD. J.MEE Department of M echanical Engineering, University of Queensland, St Lucia, Australia 4067 (Received 12 January 1983) Numerical solutions have been obtained for the exact equations describing the propaga- tion of periodic axisymmetric waves in a rigid cylindrical tube. Results were obtained for air over a range of conditions corresponding to shear wave numbers (s = Rm) from 0.2 to 5000 and reduced frequencies (k = oR/a) from 0.01 to 6. For conciseness and convenient application, the results for the attenuation and phase shift coefficients are given in the form of simple polynomials for the ranges 5 4 s s 5000 and 0.01 s k G 6. This range covers virtually all values of tube diameter and sound frequency likely to be met in practical situations that are consistent with a continuum gas model. 1. INTRODUCTION The effect of thermoviscous action at the wall of a rigid circular tube in which there are small amplitude oscillations of a fluid is one of the classical problems of acoustics. The attenuation and dispersion resulting has been of great interest to scientists and engineers for the last century or so. Kirchhoff was the first to provide a complete solution to the problem. A convenient full description of this has been given by Rayleigh [l]. Kirchhoff’s solution was, however, in the form of a complex transcendental equation which has been found difficult to interpret for practical situations. Much of the subsequent work on this subject has been aimed at providing simpler solutions that can be readily applied to practical applications. Many workers have developed approximate analytical solutions valid for limited ranges of tube dimensions, frequency or fluid properties. This work has been comprehensively reviewed and added to by Weston [2] and more recently by Tijdeman [3]. There have been a number of numerical solutions to the problem also, but only those by Shields et al. [4] and Tijdeman [3] have treated the full Kirchhoff equations. All previously published solutions have provided results for only a limited range of conditions of engineering interest. For example, in the most complete solution so far [3], it was shown that the propagation constant r could be completely specified by an equation of the form r (y, a, s, k) = 0, where y is the ratio of specific heats, (+ the square root of the Prandtl number, s the shear wave number, k the reduced frequency, r’, the real part of r corresponding to the attenuation per unit length along the tube and r’, the imaginary part of r corresponding to the phase-shift per unit length along the tube (a complete list of symbols is given in the Appendix). Tijdeman presented results obtained numerically for the following conditions: k << 1, 0.2 < s < 100; 0.02557 G k < 0*5On, 0.2 c s G 100. In physical terms, these ranges correspond to relatively low frequencies in tubes of relatively small diameter. Many practical applications correspond to much larger shear wave numbers and larger reduced frequencies. In the present work then, Tijdeman’s approach is extended to permit solution over the more useful range for practical purposes of 0.01 G k 6 6,0*2 < s c 5000. In common with previous solutions, the theoretical basis to the solution is Kirchhoff’s formulation. This is presented in summary form in section 2 where the underlying assumptions are 473 0022-460X/84/080473+08 %03.00/O @ 1984 Academic Press Inc. (London) Limited