On the calculation of the transmission line parameters for long tubes using the method of multiple scales

Abstract

The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Linear and nonlinear propagation of higher order modes in hard-walled circular ducts containing a real gas

“…The thus obtained results extend the solutions for the radial modes given in Ref. 6 with the corresponding expressions for the asymmetrical modes and, furthermore, also account for the effects arising from the adoption of a general equation of state.…”

“…The influence of the viscothermal damping mechanisms in the core region on ⌫ 00 and the propagation parameters of the radial modes has already been discussed in Ref. 6. In extension to that, in Fig.…”

“…As already shown in Ref. 6 for the symmetrical modes, a higherorder perturbation analysis also incorporating the dissipative losses in the core region turns out to be necessary if the length of the tube becomes so large that the exponentially growing effects due to viscosity and heat conduction in the wall layer do not only affect the second-order correction terms of the sound pressure but also the leading order terms. The thus obtained results extend the solutions for the radial modes given in Ref.…”

“…As shown in Ref. 6, the investigation of the most significant features of the leading order and second-order terms generated by acoustic waves emerging over spatial ranges in the longitudinal direction of the order O(␣ Ϫ1 ) requires the introduction of the length scales z, z 1 ϭ␣z, and z 2 ϭ␣ 2 z according to…”

“…A detailed investigation on that topic within the framework of a linear multiple scales perturbation theory is presented in Ref. 6. However, this study is confined to the case of axisymmetric wave motion.…”