## Abstract

The propagation of sound in hollow tubes is a fundamental theme common to many areas of classical acoustics. Kirchhoff's theory explaining the propagation of sound in a circular tube is now playing an important role as a starting point in studying sound in porous media. This paper reports on measurements of the phase velocity and attenuation coefficient in the narrow regions of tubes, where the sound undergoes anomalous dispersion and is seen to slow down remarkably to the extent that a runner can pass ahead of it. Kirchhoff's theory can be verified by experiment over a wide range of thermodynamical conditions, from isentropic to isothermal.

## 1. Introduction

The speed of sound in free air was first calculated by Newton in the seventeenth century. His calculation was further developed by Laplace and Poisson, and this became the first step in showing the importance of thermodynamics in sound. Until now, many research papers have been published with respect to the accurate measurement of the speed of sound in free space (e.g. Benedetto *et al*. 1999). The study of sound in hollow tubes also has a long history; notably Helmholtz, Kirchhoff and Rayleigh did some theoretical work on this topic, in which sound exhibits a rich variety of visco-thermal effects through interactions with the solid walls of the tubes. The inherent nature of sound might be revealed when it propagates through porous media such as a layer of snow and the surface of the earth rather than through free space. The propagation of sound in hollow tubes is indeed a fundamental theme common to many areas of classical acoustics. Kirchhoff's theory explaining the propagation of sound in a circular tube is now used as the starting point for the acoustical analysis and for developing a theoretical model for the propagation of sound in porous media (e.g. Attenborough 1982; Stinson 1991; Cummings 1993).

We report on measurements of the phase velocity and attenuation coefficient for sound in circular tubes under widely different conditions varying values for parameters, tube radius, mean pressure and frequency. This paper gives an assurance of the validity of Kirchhoff's theory over a wide range of thermodynamical conditions, from isentropic to isothermal.

## 2. Propagation constant

The full solution to the propagation constant for sound in a rigid circular tube was given by Kirchhoff (1868) in the form of a complex transcendental equation, which was later reproduced by Rayleigh with a very detailed account in his book (1945, §348–350). Almost a century later, Tijdeman (1975) showed that for a given gas the solution can be completely rewritten with only two parameters: one is the shear wave number , being the ratio of the inner radius *R* of the tube to the viscous boundary layer thickness ( times smaller than the commonly used definition), and the other is the reduced frequency *K*=*Rω*/*c*, which is a measure of whether the sound travels through the tube with a plane wavefront or not, where *ω* is the angular frequency of oscillation of the gas, *ν* the kinematic viscosity of the gas and *c* the free space speed of sound. Most of the practical applications of engineering interest fulfil both *K*≪1 and *K*/*S*≪1. In such cases, the Kirchhoff solution to the propagation constant *Γ* depends only on *S* and can be written as(2.1)where *σ* is the Prandtl number, *γ* the ratio of specific heats and *J*_{n} the *n*-th order complex Bessel function. Thus, a pressure wave travelling in the +*x* direction in a half infinite tube can be written in the form *p*=*p*_{a} exp(i*ωt*−*k*_{0}*Γx*), where *k*_{0} is the free space wave number given by *k*_{0}=*ω*/*c* and *p*_{a} is the pressure constant, so that the theoretical phase velocity *v* and attenuation coefficient *β* are given by *v*=*c*/Im[*Γ*] and *β*=*k*_{0}Re[*Γ*], respectively. Any acoustic field in a cylindrical tube is governed by *Γ*.

The parameter *S* represents the viscous shearing effects of the wall, whereas is the ratio of *R* to the thermal boundary layer thickness (*α* is the thermal diffusivity of a gas and ). The parameter , independent of viscosity, controls the heat exchange between the reciprocating gas parcel and the solid tube walls; the second square root in (2.1) takes an asymptotic value of or 1 according to whether approaches zero (isothermal) or infinity (isentropic). Equation (2.1) completely agrees with the theoretical *low reduced frequency solution* proposed by Zwikker & Kosten (1949) and Rott (1969) who started from the basic equations using some assumptions.

Although the propagation constant *Γ* as well as Laplace's speed of sound, *c*, is one of the most important physical quantities in classical acoustics and thermodynamics, only a few measurements (e.g. Lawley 1952; Shields 1965; Wilen 1998; Petculescu & Wilen 2001) have been made to verify the relevance of the Kirchhoff solution to practical situations. We report on measurements of the phase velocity and attenuation coefficient over four orders of magnitude in that covers all areas from adiabatic to isothermal thermodynamical processes.

## 3. Experiments

The schematic in figure 1 shows the apparatus used in our experiments. The dry air column, which is at a mean pressure, *P*_{m}, and ambient air temperature (equal to 298 K), is confined in a thick-walled copper tube to avoid an axial temperature gradient due to thermoacoustic effects (see Swift 1988, 2002; Yazaki *et al.* 2002). Both ends of the tube are closed, one end with a small pressure sensor to measure *P*_{m} and the other with stainless steel dynamic bellows. A rectangular voltage pulse (with a pulse width of 10 ms) consisting of many wave components of different frequencies is generated by a function generator. This is applied to the loudspeaker (Fostex FW108) attached to the bottom plate of the bellows, resulting in the emission of a forward travelling pulse (with a pulse height of about 10^{2} Pa) from the bellows. The speaker and the bellows are hermetically sealed in a brass vessel. Mean pressure in the vessel is always kept at that in the copper tube through a capillary tube. The experiments were done under the desired mean pressure by using a rotary pump and a compressed dry air tank. The tube length *L* is so long (more than 40 m, depending on *R*) that the pressure pulse has been attenuated completely by the time it reaches the right-hand end of the tube. Thus, no standing wave component is formed, except only a pure forward travelling wave. Our experimental apparatus meets all the assumptions used in the derivation of (2.1).

To determine experimentally the universal parameter governing sound propagation, we used three different tube radii (*R*=0.6, 1.0 and 2.0 mm) at five different mean pressures (*P*_{m}=2.5×10^{3}, 5.0×10^{3}, 1.02×10^{4}, 4.06×10^{4} and 1.01×10^{5} Pa). By changing *R* and *P*_{m}, we were able to cover four orders of magnitude of the parameter, , with large overlap, because *α* (or *ν*) are inversely proportional to *P*_{m}.

Spatio-temporal evolution of the pressure pulse along the tube axis was measured by small and identical pressure sensors (Toyoda Koki, model no. DD102-1F) labelled *n* (*n*=0, 1, 2, …, 7) as shown in figure 1. These are periodically flush-mounted on the inner tube wall with axial spacing *d* (equal to 1.51 m). The sensor labelled *n*=0 is used for reference and is placed at a point 1.8 m away from the bottom plate of the bellows. Each sensor consists of a cavity and a throat that has a dead volume (approx. 2×10^{−8} m^{3}), which is negligibly small compared with *πR*^{2}*d*. However as pointed out by Sugimoto (1999), even for such a small dead volume a periodic array of sensors causes undesirable Bloch wave-type dispersion, especially for *S*≤1. We obtained the true sound propagation constant only after filling up the dead volume of each sensor with viscous silicon oil.

To determine the experimental propagation constants *v* and *β*, the only quantity that needs to be measured is the transfer function between the pressure signals at any two different positions. We measured the transfer function *H*_{n}(*ω*)=*P*_{n}(*ω*)/*P*_{0}(*ω*) given by the ratio of the Fourier transform of the pressure signal *p*_{n}(*t*) at the *n*-th position to the Fourier transform of *p*_{0}(*t*) at the 0-th as the reference. The amplitude *G*_{n}(*ω*) and phase angle *Φ*_{n}(*ω*) of *H*_{n}(*ω*) show the gain and phase shift between the two signals, respectively, so the amplitude at the *n*-th position is attenuated by *G*_{n}(*ω*) and the phase is shifted by *Φ*_{n}(*ω*) relative to the reference. Thus, using the measured *G*_{n}(*ω*) in decibels and *Φ*_{n}(*ω*) in radians, which are shown in figure 2*a*,*b* as typical examples, we are able to determine experimentally the sound propagation constants from *v*=*ωnd*/*Φ*_{n}(*ω*) and *β*=−*G*_{n}(*ω*)log e/20*nd*, while confirming *G*_{n}(*ω*)=*nG*_{1}(*ω*) and *Φ*_{n}(*ω*)=*nΦ*_{1}(*ω*).

## 4. Results and discussion

The experimental results are shown in figure 3, where the dimensionless phase velocity *v*/*c* and attenuation coefficient *β*/*k*_{0} are plotted as functions of *ωτ* instead of ; *ωτ*=*σS*^{2}/2, where the thermal relaxation time *τ* given by *R*^{2}/2*α* is the time required to establish thermal equilibrium across the section of the flow channel (see Tominaga 1995; Yazaki 1993; Yazaki *et al*. 1998). In figure 3, we used *c*=346 m s^{−1} as the adiabatic sound speed and *α*=2.18×10^{−5} m^{2} s^{−1} at atmospheric pressure. The measurements were made under widely different conditions with widely varying values for the physical parameters *R*, *P*_{m}, and *ω*, and all of the results obtained fall on a single curve when plotted in terms of *ωτ*. This is conclusive evidence that *ωτ* (or *S*) is the universal parameter governing sound propagation in circular tubes over a range of seven orders of magnitude. Therefore, as Tijdeman (1975) has said, the terms *narrow tube* or *wide tube*, which were once used, may be misleading. Additionally, the results are in full accord with the theoretical phase velocity and attenuation coefficient based on (2.1), which are shown in figure 3 by the solid lines representing 1/Im[*Γ*] and Re[*Γ*], respectively. The physical constants used in calculations are *γ*=1.4 and *σ*=0.71. The values of *K* and *K*/*S* in all the data are less than 7×10^{−2} and 6×10^{−3}, respectively, fully satisfying the conditions *K*≪1 and *K/S*≪1, so that equation (2.1) is applicable.

We concentrate on the asymptotic behaviour of the sound propagation which holds when *ωτ* is either very small or very large. Now, we write the complex wave number in the form *k*=Re[*k*]−iIm[*k*], where Re[*k*] is equal to *ω*/*v* and Im[*k*] to *β*. It can be seen from figure 3 that for *ωτ*≫2 where the oscillating gas approaches an adiabatic process, the asymptotic expressions become Im[*k*]/*k*_{0}≈1/ and Re[*k*]/*k*_{0} (equal to *c*/*v*)≈1, showing that the sound is lightly damped due to viscous and thermal attenuation at the surfaces. However, the dispersion is negligible, as in the case of sound in free space.

At the other branch (*ωτ*≪2), where the gas motion approaches an isothermal process, and the viscous boundary layer *δ*_{v} simultaneously fills up the flow channel, the data obtained follow asymptotic expressions such as Im[*k*]/*k*_{0}≈ and Re[*k*]/*k*_{0} ≈. The acoustic pressure is observed to drop to 1/e of its initial value *p*_{a} within the attenuation length that is given by 1/*β*=, and for a given value of *D* (≡*c*^{2}*τ*/4) it is hard to disturb the pressure with increasing *ω* in a fashion similar to high frequency electric currents in a conductor. Such an attenuated wave is found to be associated with *anomalous dispersion* (such as light passing through a prism), the relation for which is given by *ω*=2*D*(Re[*k*])^{2}, thereby indicating that a high frequency wave travels faster than a low frequency wave and the phase velocity is always smaller than the group velocity in all frequency regions. The features appearing in this branch agree with those given by the solution to the diffusion equation with a diffusion constant *D* (e.g. Pierce 1989). For *ωτ*≪2, the gas parcel in the tube might instantaneously attain thermal equilibrium with the tube wall temperature, without any energy dissipation due to thermal conduction. Moreover, the speed of sound should approach as predicted by Newton if the viscous effects of the gas are negligibly small. These mean that the strong attenuation and acute reduction in the speed of sound observed in this branch can undoubtedly be attributed to the unavoidable viscous effects of the gas.

The asymptotic expression for the complex wave number obtained as a result of the experiments for *ωτ*≪2, , is in excellent agreement with the *narrow tube solution*, , which Rayleigh (1945) and Tijdeman (1975) predicted by assuming an isothermal sound wave (shown by the dotted lines in figure 3). The percentage difference between (2.1) and Rayleigh's expression is 1.3% for *v*/*c* and 1.9% for *β*/*k*_{0} at *ωτ*=0.1. In practice, we can use Rayleigh's asymptotic expression in place of (2.1), as long as *ωτ* remains at least below 0.1. On the other hand, the real part of the wide tube solution proposed by Kirchhoff himself,(4.1)which has subsequently been used for various regions, is found to give a reasonable prediction for the phase velocity for *ωτ*>2, while the imaginary part of (4.1) is significantly smaller than that given by the experimental data even at *ωτ*=20 (shown by the dash-dotted lines in figure 3). These simple asymptotic solutions cover most of the experimental data for at least the phase velocity. However, many acousticians are now turning their attention to areas in which neither of these apply, such as energy conversion in thermoacoustic engines (e.g. Yazaki & Tominaga 1998; Backhaus & Swift 1999, 2000; Garrett 1999; Biwa *et al*. 2004) and silencing of catalytic converters (Peat 1997). Kirchhoff's theory can undoubtedly be used as the starting point for acoustical analysis and for developing a theoretical model for the propagation of sound in such devices utilizing porous media.

## 5. Conclusions

We have measured the propagation constants, the phase velocity and attenuation coefficient, for sound in circular tubes under widely different conditions varying values for the parameters, the tube radius *R*, the mean pressure *P*_{m} and the angular frequency *ω*. Experimental results were compared with Kirchhoff's theory. The applicability of his theory to sound waves passing through narrow tubes was verified by experiment over a wide range of *ωτ* from 10^{−4} to 10^{2}.

## Acknowledgments

This research was partially supported by a Grant-in-Aid for Scientific Research (no. 16540340) from Ministry of Education, Culture, Sports, Science and Technology of Japan. The authors would like to thank Prof. Tominaga for his helpful comments.

## Footnotes

- Received March 28, 2007.
- Accepted July 5, 2007.

- © 2007 The Royal Society