- Email: [email protected]
Acoustic wave dispersion in a cylindrical elastic tube filled with a viscous liquid
Ultrasonics 37 (2000) 537–547 www.elsevier.nl/locate/ultras
Acoustic wave dispersion in a cylindrical elastic tube filled with a viscous liquid L. Elvira-Segura * Instituto de Acu´stica, C.S.I.C., Serrano 144, Madrid 28006, Spain Received 4 August 1999
Abstract This paper deals with the study of the velocity and the attenuation of an acoustic wave propagating inside a cylindrical elastic tube filled with a viscous liquid. A theory describing the propagation of the axisymmetrical modes in such waveguides is presented, with special attention given to the absorption produced by the viscous mechanisms in the liquid. One of these mechanisms is related to the momentum transfer between the compression and rarefaction regions of a propagating wave. The other viscous mechanism is due to the momentum transport inside the viscous boundary layer, close to the tube wall. Numerical calculations were carried out to investigate the influence of different parameters (frequency, tube radii, viscosity coefficient) on the propagation of acoustic waves. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Attenuation; Elastic tube; High modes; Velocity; Waveguide PACS: 43.20.Bi; 43.20.Hq; 43.20.Mv
1. Introduction The measurement of the velocity and the attenuation of an ultrasonic wave through a liquid is one method commonly used to obtain information about the physical properties of that liquid [1]. It has also proven to be a powerful tool to study chemical relaxation kinetics, specially for very short times (<1 ms) [2]. In many of these applications, only a small amount of liquid can be used for the ultrasonic analysis. This liquid sample is often contained inside a solid cavity where the acoustic wave propagates or a standing wave is established. In this paper, a theoretical model of the propagation of an acoustic wave inside a cylindrical elastic cavity filled with a viscous liquid is presented. Different approximations treating the tube as a thin shell have been done previously [3,4]. In contrast, Del Grosso [5] considered the exact longitudinal and shear wave equations for the acoustic propagation in tubes of arbitrary thickness, coupling these equations with those describing the propagation inside the liquid. From this study, the phase velocity of the different modes could * Tel.: +34-91-5618806; fax: +34-91-4117651. E-mail address: [email protected] (L. Elvira-Segura)
be obtained. Using Del Grosso’s theory, Lafleur and Shields [6 ] studied the propagation of the acoustic waves inside a cylindrical elastic tube, paying special attention to the lower modes. They pointed out that, contrary to the rigid-wall tube, where only one mode (the plane wave) can propagate under a certain frequency value (the cut-off frequency), at least two different modes can propagate at any frequency if the elastic properties of the tube are considered. Nevertheless, the attenuation of the wave was not considered in these works. A theoretical approach to the absorption of an acoustic plane wave inside a tube with rigid walls was done by Weston [7], taking as a starting point the previous works of Kirchhoff [8] and Rayleigh [9]. The most important conclusion of these works was that the frequency dependence of the attenuation can be different from the quadratic dependence found for a wave propagating in the free medium. A transition from a quadratic square frequency dependence for the case of a narrow tube to the quadratic dependence of the very wide tube was obtained. In the present paper, the propagation of the acoustic waves through the tube is studied following Del Grosso’s theory [5], which was completed, including the absorption taking place in the liquid. Thermal effects were not
0041-624X/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 7 - 9
538
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
taken into account as they are usually negligible compared to the viscous attenuation for many liquids. The absorption in the liquid was considered, retaining all the viscous terms of the Navier–Stokes equation, where only non-linear terms were ignored. It was assumed that outside the tube, there was a gas that did not affect the propagation of the wave. The set of equations obtained was numerically solved for different cases. The influence of the frequency, the thickness of the tube walls and the fluid viscosity on the velocity and the attenuation are discussed. The velocity cross-section profiles of several modes were also calculated. From these profiles, additional information about the propagation properties relating to the coupling between the liquid and the wall, and the effect of the viscous boundary layer is obtained.
attenuation of the wave is obtained from the real and imaginary components of the wavenumber k . z From the equations of the elasticity, the stress components can be expressed as a function of the velocity:
C
r c2 s = w s rr jv
A
B
∂n(w) ∂n(w) (rn(w) )+ z +2 r r r ∂r ∂z ∂r
1∂
2n 1−2n
D (3a)
r c2 s = w s rz jv
C
∂v(w) ∂n(w) r + z ∂z ∂r
D
(3b)
where n is the Poisson ratio and r is the density of w the wall. 2.2. Propagation in the liquid
2. Theory A monochromatic acoustic wave propagating in the positive z direction inside an elastic tube, with inner and outer radii R and R and filled with a viscous fluid, is 1 2 considered. In a first step, the wave equations for the solid and the fluid are studied separately. The application of the boundary conditions leads to the set of equations describing the propagation through the coupled system.
From the linearized Navier–Stokes equation, the conservation of mass equation and the state equation, r
A B
∂n(l) g V(Vn(l)) =−Vp+gDn(l)+ f+ l ∂t 3
∂r ∂t
(4a)
+r Vn(l)=0 0
(4b)
2.1. Propagation in the solid
p=c2 (r−r ) 0 l
Following the theoretical analysis carried out by Del Grosso [5], the r and z components of the vibrational velocity of the cylindrical elastic tube can be expressed as
the propagation of a monochromatic acoustic wave in the liquid can be described by the following differential equation:
vw =ej(vt−kzx){−jk [AJ (rP )+BN (rP )] z z 0 m 0 m +T [CJ (rT )+DN (rT )]} m 0 m 0 m
n(1)=−
vw =ej(vt−kzx){−P [AJ (rP )+BN (rP )] r m 1 m 1 m +jk [CJ (rT )+DN (rT )]} z 1 m 1 m where
(1a)
(1b)
(2a) P =앀k2 −k2 c z m T =앀k2 −k2 (2b) m s z v k= (2c) c c c v (2d) k= . s c s c and c are the compressional and shear phase velocities c s in the solid, and the superindex (w) denotes that the vibrational velocity is referred to the tube wall. The information of the propagation velocity and the
A
1
k2 l
(4c)
+
j vr l
A BB f+
g
V(Vn(l))−
3
jg vr l
Dn(l),
(5)
where k is the wavenumber in free liquid, and f and g l are the expansion and shear coefficients of viscosity, relatively. The velocity can be expressed as a sum of two components, the first is obtained from a scalar potential, and the second from a rotational potential: n(l)=n(l) +n(l) =Vw+V×y. l 2
(6)
Substituting this relation in Eq. (5), a set of two equations is obtained: w=−
y=
A
1 k2 l
jg vr l
+
j vr l
A
BB
4 f+ g 3
V×(V×y).
V(Vw)
(7a)
(7b)
Solving these equations and retaining those solutions that give a finite value for r=0, the velocity components
539
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
are
(8b) and (10), a set of six equations is obtained:
n(l) =n(l) +n(l) =ej(vt−kzz)(−jk EJ (k r)+FJ (ar)) z z1 z2 z 0 r 0
A
A
jk n(l) =n(l) +n(l) =ej(vt−kzz) −k EJ (k r)+ z FJ (ar) r r1 r2 r 1 r 1 a
B
A
(8b)
k∞= l
S
S
a=
(9a) 1
1 k2 l
+
j vr
l
A
4
B
(9b)
(9c)
Finally, from Eqs. 4(b) and 4(c), the pressure is expressed as a function of the velocity: r c2 p=j l l Vn(l). v
(10)
−jk J (R P )A−jk N (R P )B+T J (R T )C z 0 1 m z 0 1 m m 0 1 m +T N (R T )D+jk J (k R )E−J (R a)F=0 m 0 1 m z 0 r 1 0 1 (12d )
2.3. Boundary conditions The boundary conditions are imposed at the inner and the outer radii. The continuity of the stresses in the wall and the pressures in the fluid, and the continuity of the vibrational velocity in both materials are imposed. It should be noted that the continuity of the radial velocity (normal to the wall ) is usually included in the propagation models, but that is not the case of the continuity of the axial or z velocity (tangential to the wall ). This ‘no slip’ condition gives rise to the viscous absorption in a fluid moving near a solid wall. In this model, the movement of this wall is also considered. The boundary conditions are as follows. $ For r=R 1
$
B B
A B
f+ g 3
vr l −k2 . z jg
B
A A
where k =앀k∞2−k2 l z r
B
P S J (R P )+ m J (R P ) A m 0 1 m R 1 1 m 1 P + S N (R P )+ m N (R P ) B m 0 1 m 1 1 m R 1 1 +jk T J (R T )− J (R T ) C z m 0 1 m R 1 1 m 1 1 +jk T N (R T )− N (R T ) D z m 0 1 m 1 1 m R 1 r k k∞ 2 s l + l J (k R )E=0 (12a) 0 r 1 2r k w l jk P J (R P )A+jk P N (R P )B z m 1 1 m z m 1 1 m +S J (R T )C+S N (R T )D=0 (12b) m 1 1 m m 1 1 m −P J (R P )A−P N (R P )B+jk J (R T )C m 1 1 m m 1 1 m z 1 1 m +jk N (R T )D+k J (k R )E z 1 1 m r 1 r 1 jk (12c) − z J (R a)F=0 a 1 1
(8a)
s =−p rr
(11a)
s =0 rz
(11b)
n(w) =n(l) r r
(11c)
n(w) =n(l) z z
(11d )
For r=R 2 s =0 rr
(11e)
s =0. rz
(11f )
Applying them to Eqs. (1a), (1b), (3a), (3b), (8a),
A
B
P S J (R P )+ m J (R P ) A m 0 2 m R 1 2 m 2 P + S N (R P )+ m N (R P ) B 1 2 m m 0 2 m R 2 1 +jk T J (R T )− J (R T ) C z m 0 2 m R 1 2 m 2 1 N (R T ) D=0 (12e) +jk T N (R T )− 1 2 m z m 0 2 m R 2 jk P J (R P )A+jk P N (R P )B+S J (R T )C z m 1 2 m z m 1 2 m m 1 2 m +S N (R T )D=0 (12f ) m 1 2 m where
A
B
A A
k2 S =k2 − s . m z 2
B B
(13)
This system is solved using the condition that the six constants A, B, C, D, E and F do not vanish. This condition requires the determinant of the coefficient matrix being equal to zero, giving a solution for the real and imaginary components of k for each frequency. z Due to the great complexity of this procedure, the solution has to be obtained numerically.
540
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
2.4. Rigid wall solution When the tube can be considered as rigid compared to the liquid inside, the system of Eqs. (12a)–(12f ) is simplified. For this case, only the equations in the fluid are considered. The boundary conditions are now n (R )=n (R )=0. r 1 z 1 The system is composed of two equations: jk k J (k R )E− z J (aR )F=0 1 r 1 r l a 1
(14)
(15a)
jk J (k R )E−J (aR )F=0. (15b) z 0 r l 0 1 The solution for k is obtained when the determinant z corresponding to these equations vanishes. This condition leads to k2 J (k R )J (aR )+ak J (k R )J (aR )=0. (16) z 0 r 1 1 1 r 1 r 1 0 1 From this equation, both the Kirchhoff propagation constant of a plane wave in a tube and the classical propagation constant of a plane wave in the free space can be obtained (see Appendix A).
3. Numerical study 3.1. Rigid wall solution The results obtained in the last section are compared in Figs. 1 and 2 with the theories developed by Weston and Kirchhoff for the propagation of a plane wave in a rigid wall tube filled with water (R =1 cm). It can be 1
Fig. 1. Phase velocity of a plane wave propagating inside a tube with rigid walls (R =1 cm). 1
Fig. 2. Attenuation of a plane wave propagating inside a tube with rigid walls (R =1 cm). 1
seen that a good agreement between the theory developed in this work and Weston’s theory is obtained. Although there is no analytical expression for the complete theory presented, it is more general: there is no need to apply a different approach depending on the size of the tube (wide and very wide tube) as is required for Weston’s theory. In Figs. 3 and 4 some results are numerically calculated for the propagation velocity and absorption of the first mode, the plane wave, and the next three modes, which have not a plane wave front. For this case, the absorption is mainly due to the friction between the
Fig. 3. Phase velocity of the first four modes propagating inside a tube with rigid walls (R =1 cm). 1
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 4. Attenuation of the first four modes propagating inside a tube with rigid walls (R =1 cm). 1
541
Fig. 5. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =2 cm). 1 2
walls and the liquid. It can be seen that both the velocity and the attenuation of all the propagating modes tend to converge as the frequency increases. For the modes higher than the first, the absorption is not a function of the quadratic root of the frequency any more, showing a more complicated frequency dependence. This behaviour of the wavenumber for the rigid wall tube as a function of the frequency has some similarities with the more realistic case of an elastic wall tube as will be seen in the following sections. 3.2. Frequency dependence of the acoustic propagation through an elastic tube filled with liquid In Figs. 5–8, the velocity and attenuation of the first modes as a function of the frequency are plotted. An aluminium tube with an inner radius of 1 cm was considered. Figs. 5 and 6 were obtained with a 2 cm outer radius and Figs. 7 and 8 with a 1.25 cm outer radius. The modal dispersion shown in the figures is similar to that presented by Del Grosso [5] and Lafleur and Shields [6 ]. As pointed out by these authors, two modes can propagate in the zero frequency limit. It can be seen that when the elastic properties of the tube are considered, the attenuation shows several peaks as a function of the frequency. The amplitude of these peaks depends on the mode and the tube under consideration, reaching apparent zero values for some cases. This behaviour is more evident when the interaction between the fluid and the tube is increased (R =1.25 cm, Figs. 7 and 8). At those frequencies for 2 which the attenuation predicted by this model vanishes, the attenuation in the elastic tube could become the
Fig. 6. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =2 cm). 1 2
main absorption mechanism. Therefore, the viscous properties of the solid should be included for an accurate determination of the attenuation at these particular frequencies (although it is not relevant for the rest of the frequencies). It follows from these results that there is a resonance effect for some modes even they are propagating modes. It can be also concluded from Figs. 5–8 that, in general, as the wall is thicker, the tube is more rigid, and the solutions tend to the limit case, which was studied in the last section. This effect is more evident in the absorption plots.
542
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 7. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =1.25 cm). 1 2
Fig. 9. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =2 cm, frequency=200 kHz). 2
Fig. 8. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =1.25 cm). 1 2
3.3. Wall thickness dependence of the acoustic propagation through an elastic tube filled with liquid In this section, some calculations were done, changing the value of the inner radius, R , for a 200 kHz excita1 tion. The elastic properties of the tube are those of aluminium, and the outer radius, R , was fixed at a 2 value of 2 cm. The results are presented in Figs. 9 and 10. Fig. 9 shows the inner radius dependence of the propagation velocity. It can be seen that the lower modes are less affected by the decreasing of R (and 1 consequent wall thickness increasing). In general, the viscous attenuation (Fig. 10) due to the friction between the wall and the liquid is diminished as the inner radius of the tube increases from R =5 mm. This behaviour is 1
Fig. 10. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =2 cm, frequency=200 kHz). 2
modified by the existing resonances, especially for the highest modes. For inner radii less than 5 mm, the attenuation suffers a sudden descent for all the modes except the first. It can also be seen that the peaks of the attenuation coefficient occur at those values of R for 1 which the negative slope of the propagation velocity is larger. 3.4. Liquid shear viscosity dependence of the acoustic propagation through an elastic tube filled with liquid To evaluate the effect of the viscosity, an aluminium tube with R =1 cm, and R =2 cm is supposed to be 1 2
543
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
excited with a 200 kHz acoustic wave. The liquid inside the tube is assumed to have the same properties as water, except by the shear viscosity coefficient that was changed. The velocity ( Fig. 11) is weakly affected by the variations of the shear viscosity. Only a small decrease in velocity is reported for a high viscosity increase. From Fig. 12 it can be seen that the attenuation depends on the power of the viscosity in the log–log plot. A careful
Table 1 Functional relation between the shear viscosity and the attenuation coefficient: fitting function log Y=A log X+B
A B
1st mode
2nd mode
3rd mode
4th mode
5th mode
0.526 −3.98
0.528 −4.29
0.580 −4.85
0.554 −4.76
0.719 −7.26
analysis of the numerical data shows that for the first mode, the dependence is almost a function of the square root (as the Kirchhoff ’s expression for a rigid-wall tube states). The rest of the modes tend to leave that functional relation ( Table 1). 3.5. Cross-section profile of the vibrational velocity in an elastic tube filled with liquid
Fig. 11. Phase velocity of the first five modes propagating inside an aluminium tube filled with a liquid as a function of the shear viscosity coefficient (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 12. Attenuation of the first five modes propagating inside an aluminium tube filled with a liquid as a function of the shear viscosity coefficient (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
In this section, the distribution of the axial and radial vibrational velocities in the elastic tube and the liquid is investigated. Several calculations were made for the same case of the last section (R =1 cm, R =2 cm, fre1 2 quency=200 kHz, aluminium tube filled with water). The results of the magnitude and phase of the first four modes are presented in Figs. 13–20. For this case, no plane propagation was found even for the first mode, which has a 40% axial velocity variation in its wavefront propagating through the liquid (although it has the same phase). From the radial and axial velocity distributions, it can be seen that there is more acoustic energy propagating through the walls for the lowest modes than for the highest modes, showing a decreasing trend with the increasing order of the modes. The cross-section profiles in the liquid of first three modes seem to be equal to the modes propagating in a
Fig. 13. Axial velocity of the 1st propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
544
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 14. Radial velocity of the 1st propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 16. Radial velocity of the 2nd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 15. Axial velocity of the 2nd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 17. Axial velocity of the 3rd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
tube with rigid walls. This is not true, in general, as can be deduced from the fourth mode: the differences between it and the third mode take place mainly in the solid wall, having similar propagation characteristics in the liquid region. As was found by other authors [5,6 ], the phase difference between the radial and the axial components of the velocity is always ±p/2 for the liquid and the walls. That is, the motion is elliptically polarized in a plane containing the tube axis. The apparent gap appearing in the axial velocity plots between the propagation in the liquid and the walls should not be interpreted as a discontinuity
between them. This can be appreciated better in Fig. 21, which is a ‘zoom’ of the axial propagation corresponding to the first mode in the neighbourhood of the solid– liquid interface. These sharp changes in axial velocity are due to the shear viscosity effect in the liquid that works in a very thin boundary layer. The thickness of this layer inversely depends on the magnitude of a, which is a function of the frequency, the shear viscosity coefficient and the propagation wavenumber [see Eq. 9(c)]. From a comparison of Figs. 6, 13, 15, 17 and 19, it can be seen that there is a direct relation between the absorption of a given mode and the slope of the axial velocity in the boundary layer.
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 18. Radial velocity of the 3rd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 19. Axial velocity of the 4th propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
It has been shown that the propagating characteristics (velocity and attenuation) of a given mode change with the frequency. The same situation occurs with the crosssection profiles due to the frequency-dependent interaction between the liquid and the walls of the tube. This effect is shown in Figs. 22 and 23, where the second propagating mode is plotted for three different frequencies. A log plot was chosen because of the great differences existing between them. This second mode, which has an almost plane profile at 20 kHz, presents a totally different distribution velocity at 200 kHz, where a Bessel function behaviour is evident. These changes in crosssection profiles represent a particular characteristic of
545
Fig. 20. Radial velocity of the 4th propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 21. Magnitude of the axial velocity of the 1st propagating mode inside an aluminium tube filled with water in the neighbourhood of the inner wall (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
the propagation through an elastic tube, which does not appear if the walls are rigid.
4. Conclusions A theoretical model describing the propagation of an acoustic wave through a cylindrical elastic tube filled with liquid was presented. This model completes the theory presented in previous works, including the viscous losses related to the propagation of the acoustic wave
546
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
gated, indicating the relation existing between these profiles and the attenuation of the wave. It was shown that the propagation characteristics of the acoustic wave through an elastic tube filled with liquid could not be described, in general, by a rigid-wall tube model due to the interaction between the walls of the tube and the liquid inside it. Therefore, an accurate analysis of the acoustic propagation in real tubes filled with liquid should include the vibration of the walls.
Acknowledgements I would like to thank Dr. E. Riera and Dra. C. Campos for their interesting comments and suggestions.
Fig. 22. Axial velocity of the 2nd propagating mode inside an aluminium tube filled with water for 20, 50 and 200 kHz (R =1 cm, 1 R =2 cm). 2
Appendix A For most of the practical cases, the product aR is 1 large enough to use the asymptotic expressions of the Bessel functions:
S S
J (aR )# 0 1
A
B
A
B
p cos aR − 1 paR 4 1 2
(A1)
p cos aR −3 . (A2) 1 paR 4 1 On the contrary, for a plane wave solution, the product k R must be small, and the following expresr 1 sions can be used instead of the Bessel functions: 2
J (aR )# 1 1
A B
J (k R )=1− 0 r 1
kR 2 r 1 2
(A3)
kR J (k R )= r 1 . 1 r 1 2
Fig. 23. Radial velocity of the 2nd propagating mode inside an aluminium tube filled with water for 20, 50 and 200 kHz (R =1 cm, 1 R =2 cm). 2
in free space and viscous losses due to the friction between the liquid and the walls of the tube. The model was compared with some analytical results obtained for the case of a tube with rigid walls, showing a good agreement. For the general case of a tube with elastic walls, the influence of the frequency, wall thickness and viscosity on the propagation velocity and the attenuation of several modes was studied. The frequency dependence of the cross-section profiles was also investi-
(A4)
When these are substituted into Eq. (16) and k is r written as a function of k , a quadratic equation is z obtained:
A
B
2a 4 2a −k∞2− j k2 + jk∞2=0. (A5) k4 + l l z z R R2 R 1 1 1 If only terms of the first order of (aR )−1 are retained, 1 the solution of this equation is:
A
B A
j jk2 k =k∞ 1− #k 1− l z l l aR 2vr 1 l
A
4 3
g+f
BBA
1−
j
B
. aR 1 (A6)
When the absorption caused by the friction with the walls is large compared to the absorption of a plane
547
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
wave in free space,
A
j
k =k 1− z l aR 1
and then
B
(A7)
A
1 a =−k z l 2R 1
S B 2g
S
vr 0
A
A
4
l
3
,
g+f
3
B
g+f ,
(A12)
References (A9)
which are the Kirchhoff ’s expressions when the thermal absorption is neglected. However, if the dominant viscous mechanism is the absorption due to the compressions and expansions of the acoustic wave, jk2 k =k 1− l z l 2vr
A
4
(A8)
vr 0
2g
jk3 a =− l z 2vr
(A11)
l which are the classical expressions for the propagation of a plane wave in the free space.
and 1 c =c 1− z l 2R 1
c =c z l
BB
(A10)
[1] E.P. Papadakis, R.N. Thurston, A.D. Pierce ( Eds.), Physical Acoustics Vol. XIX, Academic Press, New York, 1990, pp. 81–155. Chapters 2 and 3. [2] F. Eggers, Th. Funck, Rev. Sci. Instrum. 44 (8) (1973) 969–977. [3] J.C.F. Chow, J.T. Apter, J. Acoust. Soc. Am. 44 (2) (1968) 437–443. [4] M. El-Raheb, J. Acoust. Soc. Am. 71 (2) (1982) 296–306. [5] V.A. Del Grosso, Acustica 24 (1971) 299–311. [6 ] L.D. Lafleur, F.D. Shields, J. Acoust. Soc. Am. 97 (3) (1995) 1435–1445. [7] D.E. Weston, Proc. Phys. Soc. Lond. B 66 (1953) 695–709. [8] G. Kirchhoff, Anallen der Physik and Chemie 6 (134) (1868) 177–193. [9] L. Rayleigh, The Theory of Sound, Macmillan, London, 1901.
Acoustic wave dispersion in a cylindrical elastic tube filled with a viscous liquid L. Elvira-Segura * Instituto de Acu´stica, C.S.I.C., Serrano 144, Madrid 28006, Spain Received 4 August 1999
Abstract This paper deals with the study of the velocity and the attenuation of an acoustic wave propagating inside a cylindrical elastic tube filled with a viscous liquid. A theory describing the propagation of the axisymmetrical modes in such waveguides is presented, with special attention given to the absorption produced by the viscous mechanisms in the liquid. One of these mechanisms is related to the momentum transfer between the compression and rarefaction regions of a propagating wave. The other viscous mechanism is due to the momentum transport inside the viscous boundary layer, close to the tube wall. Numerical calculations were carried out to investigate the influence of different parameters (frequency, tube radii, viscosity coefficient) on the propagation of acoustic waves. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Attenuation; Elastic tube; High modes; Velocity; Waveguide PACS: 43.20.Bi; 43.20.Hq; 43.20.Mv
1. Introduction The measurement of the velocity and the attenuation of an ultrasonic wave through a liquid is one method commonly used to obtain information about the physical properties of that liquid [1]. It has also proven to be a powerful tool to study chemical relaxation kinetics, specially for very short times (<1 ms) [2]. In many of these applications, only a small amount of liquid can be used for the ultrasonic analysis. This liquid sample is often contained inside a solid cavity where the acoustic wave propagates or a standing wave is established. In this paper, a theoretical model of the propagation of an acoustic wave inside a cylindrical elastic cavity filled with a viscous liquid is presented. Different approximations treating the tube as a thin shell have been done previously [3,4]. In contrast, Del Grosso [5] considered the exact longitudinal and shear wave equations for the acoustic propagation in tubes of arbitrary thickness, coupling these equations with those describing the propagation inside the liquid. From this study, the phase velocity of the different modes could * Tel.: +34-91-5618806; fax: +34-91-4117651. E-mail address: [email protected] (L. Elvira-Segura)
be obtained. Using Del Grosso’s theory, Lafleur and Shields [6 ] studied the propagation of the acoustic waves inside a cylindrical elastic tube, paying special attention to the lower modes. They pointed out that, contrary to the rigid-wall tube, where only one mode (the plane wave) can propagate under a certain frequency value (the cut-off frequency), at least two different modes can propagate at any frequency if the elastic properties of the tube are considered. Nevertheless, the attenuation of the wave was not considered in these works. A theoretical approach to the absorption of an acoustic plane wave inside a tube with rigid walls was done by Weston [7], taking as a starting point the previous works of Kirchhoff [8] and Rayleigh [9]. The most important conclusion of these works was that the frequency dependence of the attenuation can be different from the quadratic dependence found for a wave propagating in the free medium. A transition from a quadratic square frequency dependence for the case of a narrow tube to the quadratic dependence of the very wide tube was obtained. In the present paper, the propagation of the acoustic waves through the tube is studied following Del Grosso’s theory [5], which was completed, including the absorption taking place in the liquid. Thermal effects were not
0041-624X/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 7 - 9
538
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
taken into account as they are usually negligible compared to the viscous attenuation for many liquids. The absorption in the liquid was considered, retaining all the viscous terms of the Navier–Stokes equation, where only non-linear terms were ignored. It was assumed that outside the tube, there was a gas that did not affect the propagation of the wave. The set of equations obtained was numerically solved for different cases. The influence of the frequency, the thickness of the tube walls and the fluid viscosity on the velocity and the attenuation are discussed. The velocity cross-section profiles of several modes were also calculated. From these profiles, additional information about the propagation properties relating to the coupling between the liquid and the wall, and the effect of the viscous boundary layer is obtained.
attenuation of the wave is obtained from the real and imaginary components of the wavenumber k . z From the equations of the elasticity, the stress components can be expressed as a function of the velocity:
C
r c2 s = w s rr jv
A
B
∂n(w) ∂n(w) (rn(w) )+ z +2 r r r ∂r ∂z ∂r
1∂
2n 1−2n
D (3a)
r c2 s = w s rz jv
C
∂v(w) ∂n(w) r + z ∂z ∂r
D
(3b)
where n is the Poisson ratio and r is the density of w the wall. 2.2. Propagation in the liquid
2. Theory A monochromatic acoustic wave propagating in the positive z direction inside an elastic tube, with inner and outer radii R and R and filled with a viscous fluid, is 1 2 considered. In a first step, the wave equations for the solid and the fluid are studied separately. The application of the boundary conditions leads to the set of equations describing the propagation through the coupled system.
From the linearized Navier–Stokes equation, the conservation of mass equation and the state equation, r
A B
∂n(l) g V(Vn(l)) =−Vp+gDn(l)+ f+ l ∂t 3
∂r ∂t
(4a)
+r Vn(l)=0 0
(4b)
2.1. Propagation in the solid
p=c2 (r−r ) 0 l
Following the theoretical analysis carried out by Del Grosso [5], the r and z components of the vibrational velocity of the cylindrical elastic tube can be expressed as
the propagation of a monochromatic acoustic wave in the liquid can be described by the following differential equation:
vw =ej(vt−kzx){−jk [AJ (rP )+BN (rP )] z z 0 m 0 m +T [CJ (rT )+DN (rT )]} m 0 m 0 m
n(1)=−
vw =ej(vt−kzx){−P [AJ (rP )+BN (rP )] r m 1 m 1 m +jk [CJ (rT )+DN (rT )]} z 1 m 1 m where
(1a)
(1b)
(2a) P =앀k2 −k2 c z m T =앀k2 −k2 (2b) m s z v k= (2c) c c c v (2d) k= . s c s c and c are the compressional and shear phase velocities c s in the solid, and the superindex (w) denotes that the vibrational velocity is referred to the tube wall. The information of the propagation velocity and the
A
1
k2 l
(4c)
+
j vr l
A BB f+
g
V(Vn(l))−
3
jg vr l
Dn(l),
(5)
where k is the wavenumber in free liquid, and f and g l are the expansion and shear coefficients of viscosity, relatively. The velocity can be expressed as a sum of two components, the first is obtained from a scalar potential, and the second from a rotational potential: n(l)=n(l) +n(l) =Vw+V×y. l 2
(6)
Substituting this relation in Eq. (5), a set of two equations is obtained: w=−
y=
A
1 k2 l
jg vr l
+
j vr l
A
BB
4 f+ g 3
V×(V×y).
V(Vw)
(7a)
(7b)
Solving these equations and retaining those solutions that give a finite value for r=0, the velocity components
539
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
are
(8b) and (10), a set of six equations is obtained:
n(l) =n(l) +n(l) =ej(vt−kzz)(−jk EJ (k r)+FJ (ar)) z z1 z2 z 0 r 0
A
A
jk n(l) =n(l) +n(l) =ej(vt−kzz) −k EJ (k r)+ z FJ (ar) r r1 r2 r 1 r 1 a
B
A
(8b)
k∞= l
S
S
a=
(9a) 1
1 k2 l
+
j vr
l
A
4
B
(9b)
(9c)
Finally, from Eqs. 4(b) and 4(c), the pressure is expressed as a function of the velocity: r c2 p=j l l Vn(l). v
(10)
−jk J (R P )A−jk N (R P )B+T J (R T )C z 0 1 m z 0 1 m m 0 1 m +T N (R T )D+jk J (k R )E−J (R a)F=0 m 0 1 m z 0 r 1 0 1 (12d )
2.3. Boundary conditions The boundary conditions are imposed at the inner and the outer radii. The continuity of the stresses in the wall and the pressures in the fluid, and the continuity of the vibrational velocity in both materials are imposed. It should be noted that the continuity of the radial velocity (normal to the wall ) is usually included in the propagation models, but that is not the case of the continuity of the axial or z velocity (tangential to the wall ). This ‘no slip’ condition gives rise to the viscous absorption in a fluid moving near a solid wall. In this model, the movement of this wall is also considered. The boundary conditions are as follows. $ For r=R 1
$
B B
A B
f+ g 3
vr l −k2 . z jg
B
A A
where k =앀k∞2−k2 l z r
B
P S J (R P )+ m J (R P ) A m 0 1 m R 1 1 m 1 P + S N (R P )+ m N (R P ) B m 0 1 m 1 1 m R 1 1 +jk T J (R T )− J (R T ) C z m 0 1 m R 1 1 m 1 1 +jk T N (R T )− N (R T ) D z m 0 1 m 1 1 m R 1 r k k∞ 2 s l + l J (k R )E=0 (12a) 0 r 1 2r k w l jk P J (R P )A+jk P N (R P )B z m 1 1 m z m 1 1 m +S J (R T )C+S N (R T )D=0 (12b) m 1 1 m m 1 1 m −P J (R P )A−P N (R P )B+jk J (R T )C m 1 1 m m 1 1 m z 1 1 m +jk N (R T )D+k J (k R )E z 1 1 m r 1 r 1 jk (12c) − z J (R a)F=0 a 1 1
(8a)
s =−p rr
(11a)
s =0 rz
(11b)
n(w) =n(l) r r
(11c)
n(w) =n(l) z z
(11d )
For r=R 2 s =0 rr
(11e)
s =0. rz
(11f )
Applying them to Eqs. (1a), (1b), (3a), (3b), (8a),
A
B
P S J (R P )+ m J (R P ) A m 0 2 m R 1 2 m 2 P + S N (R P )+ m N (R P ) B 1 2 m m 0 2 m R 2 1 +jk T J (R T )− J (R T ) C z m 0 2 m R 1 2 m 2 1 N (R T ) D=0 (12e) +jk T N (R T )− 1 2 m z m 0 2 m R 2 jk P J (R P )A+jk P N (R P )B+S J (R T )C z m 1 2 m z m 1 2 m m 1 2 m +S N (R T )D=0 (12f ) m 1 2 m where
A
B
A A
k2 S =k2 − s . m z 2
B B
(13)
This system is solved using the condition that the six constants A, B, C, D, E and F do not vanish. This condition requires the determinant of the coefficient matrix being equal to zero, giving a solution for the real and imaginary components of k for each frequency. z Due to the great complexity of this procedure, the solution has to be obtained numerically.
540
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
2.4. Rigid wall solution When the tube can be considered as rigid compared to the liquid inside, the system of Eqs. (12a)–(12f ) is simplified. For this case, only the equations in the fluid are considered. The boundary conditions are now n (R )=n (R )=0. r 1 z 1 The system is composed of two equations: jk k J (k R )E− z J (aR )F=0 1 r 1 r l a 1
(14)
(15a)
jk J (k R )E−J (aR )F=0. (15b) z 0 r l 0 1 The solution for k is obtained when the determinant z corresponding to these equations vanishes. This condition leads to k2 J (k R )J (aR )+ak J (k R )J (aR )=0. (16) z 0 r 1 1 1 r 1 r 1 0 1 From this equation, both the Kirchhoff propagation constant of a plane wave in a tube and the classical propagation constant of a plane wave in the free space can be obtained (see Appendix A).
3. Numerical study 3.1. Rigid wall solution The results obtained in the last section are compared in Figs. 1 and 2 with the theories developed by Weston and Kirchhoff for the propagation of a plane wave in a rigid wall tube filled with water (R =1 cm). It can be 1
Fig. 1. Phase velocity of a plane wave propagating inside a tube with rigid walls (R =1 cm). 1
Fig. 2. Attenuation of a plane wave propagating inside a tube with rigid walls (R =1 cm). 1
seen that a good agreement between the theory developed in this work and Weston’s theory is obtained. Although there is no analytical expression for the complete theory presented, it is more general: there is no need to apply a different approach depending on the size of the tube (wide and very wide tube) as is required for Weston’s theory. In Figs. 3 and 4 some results are numerically calculated for the propagation velocity and absorption of the first mode, the plane wave, and the next three modes, which have not a plane wave front. For this case, the absorption is mainly due to the friction between the
Fig. 3. Phase velocity of the first four modes propagating inside a tube with rigid walls (R =1 cm). 1
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 4. Attenuation of the first four modes propagating inside a tube with rigid walls (R =1 cm). 1
541
Fig. 5. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =2 cm). 1 2
walls and the liquid. It can be seen that both the velocity and the attenuation of all the propagating modes tend to converge as the frequency increases. For the modes higher than the first, the absorption is not a function of the quadratic root of the frequency any more, showing a more complicated frequency dependence. This behaviour of the wavenumber for the rigid wall tube as a function of the frequency has some similarities with the more realistic case of an elastic wall tube as will be seen in the following sections. 3.2. Frequency dependence of the acoustic propagation through an elastic tube filled with liquid In Figs. 5–8, the velocity and attenuation of the first modes as a function of the frequency are plotted. An aluminium tube with an inner radius of 1 cm was considered. Figs. 5 and 6 were obtained with a 2 cm outer radius and Figs. 7 and 8 with a 1.25 cm outer radius. The modal dispersion shown in the figures is similar to that presented by Del Grosso [5] and Lafleur and Shields [6 ]. As pointed out by these authors, two modes can propagate in the zero frequency limit. It can be seen that when the elastic properties of the tube are considered, the attenuation shows several peaks as a function of the frequency. The amplitude of these peaks depends on the mode and the tube under consideration, reaching apparent zero values for some cases. This behaviour is more evident when the interaction between the fluid and the tube is increased (R =1.25 cm, Figs. 7 and 8). At those frequencies for 2 which the attenuation predicted by this model vanishes, the attenuation in the elastic tube could become the
Fig. 6. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =2 cm). 1 2
main absorption mechanism. Therefore, the viscous properties of the solid should be included for an accurate determination of the attenuation at these particular frequencies (although it is not relevant for the rest of the frequencies). It follows from these results that there is a resonance effect for some modes even they are propagating modes. It can be also concluded from Figs. 5–8 that, in general, as the wall is thicker, the tube is more rigid, and the solutions tend to the limit case, which was studied in the last section. This effect is more evident in the absorption plots.
542
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 7. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =1.25 cm). 1 2
Fig. 9. Phase velocity of the first five modes propagating inside an aluminium tube filled with water (R =2 cm, frequency=200 kHz). 2
Fig. 8. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =1 cm, R =1.25 cm). 1 2
3.3. Wall thickness dependence of the acoustic propagation through an elastic tube filled with liquid In this section, some calculations were done, changing the value of the inner radius, R , for a 200 kHz excita1 tion. The elastic properties of the tube are those of aluminium, and the outer radius, R , was fixed at a 2 value of 2 cm. The results are presented in Figs. 9 and 10. Fig. 9 shows the inner radius dependence of the propagation velocity. It can be seen that the lower modes are less affected by the decreasing of R (and 1 consequent wall thickness increasing). In general, the viscous attenuation (Fig. 10) due to the friction between the wall and the liquid is diminished as the inner radius of the tube increases from R =5 mm. This behaviour is 1
Fig. 10. Attenuation of the first five modes propagating inside an aluminium tube filled with water (R =2 cm, frequency=200 kHz). 2
modified by the existing resonances, especially for the highest modes. For inner radii less than 5 mm, the attenuation suffers a sudden descent for all the modes except the first. It can also be seen that the peaks of the attenuation coefficient occur at those values of R for 1 which the negative slope of the propagation velocity is larger. 3.4. Liquid shear viscosity dependence of the acoustic propagation through an elastic tube filled with liquid To evaluate the effect of the viscosity, an aluminium tube with R =1 cm, and R =2 cm is supposed to be 1 2
543
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
excited with a 200 kHz acoustic wave. The liquid inside the tube is assumed to have the same properties as water, except by the shear viscosity coefficient that was changed. The velocity ( Fig. 11) is weakly affected by the variations of the shear viscosity. Only a small decrease in velocity is reported for a high viscosity increase. From Fig. 12 it can be seen that the attenuation depends on the power of the viscosity in the log–log plot. A careful
Table 1 Functional relation between the shear viscosity and the attenuation coefficient: fitting function log Y=A log X+B
A B
1st mode
2nd mode
3rd mode
4th mode
5th mode
0.526 −3.98
0.528 −4.29
0.580 −4.85
0.554 −4.76
0.719 −7.26
analysis of the numerical data shows that for the first mode, the dependence is almost a function of the square root (as the Kirchhoff ’s expression for a rigid-wall tube states). The rest of the modes tend to leave that functional relation ( Table 1). 3.5. Cross-section profile of the vibrational velocity in an elastic tube filled with liquid
Fig. 11. Phase velocity of the first five modes propagating inside an aluminium tube filled with a liquid as a function of the shear viscosity coefficient (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 12. Attenuation of the first five modes propagating inside an aluminium tube filled with a liquid as a function of the shear viscosity coefficient (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
In this section, the distribution of the axial and radial vibrational velocities in the elastic tube and the liquid is investigated. Several calculations were made for the same case of the last section (R =1 cm, R =2 cm, fre1 2 quency=200 kHz, aluminium tube filled with water). The results of the magnitude and phase of the first four modes are presented in Figs. 13–20. For this case, no plane propagation was found even for the first mode, which has a 40% axial velocity variation in its wavefront propagating through the liquid (although it has the same phase). From the radial and axial velocity distributions, it can be seen that there is more acoustic energy propagating through the walls for the lowest modes than for the highest modes, showing a decreasing trend with the increasing order of the modes. The cross-section profiles in the liquid of first three modes seem to be equal to the modes propagating in a
Fig. 13. Axial velocity of the 1st propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
544
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 14. Radial velocity of the 1st propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 16. Radial velocity of the 2nd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 15. Axial velocity of the 2nd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 17. Axial velocity of the 3rd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
tube with rigid walls. This is not true, in general, as can be deduced from the fourth mode: the differences between it and the third mode take place mainly in the solid wall, having similar propagation characteristics in the liquid region. As was found by other authors [5,6 ], the phase difference between the radial and the axial components of the velocity is always ±p/2 for the liquid and the walls. That is, the motion is elliptically polarized in a plane containing the tube axis. The apparent gap appearing in the axial velocity plots between the propagation in the liquid and the walls should not be interpreted as a discontinuity
between them. This can be appreciated better in Fig. 21, which is a ‘zoom’ of the axial propagation corresponding to the first mode in the neighbourhood of the solid– liquid interface. These sharp changes in axial velocity are due to the shear viscosity effect in the liquid that works in a very thin boundary layer. The thickness of this layer inversely depends on the magnitude of a, which is a function of the frequency, the shear viscosity coefficient and the propagation wavenumber [see Eq. 9(c)]. From a comparison of Figs. 6, 13, 15, 17 and 19, it can be seen that there is a direct relation between the absorption of a given mode and the slope of the axial velocity in the boundary layer.
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
Fig. 18. Radial velocity of the 3rd propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 19. Axial velocity of the 4th propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
It has been shown that the propagating characteristics (velocity and attenuation) of a given mode change with the frequency. The same situation occurs with the crosssection profiles due to the frequency-dependent interaction between the liquid and the walls of the tube. This effect is shown in Figs. 22 and 23, where the second propagating mode is plotted for three different frequencies. A log plot was chosen because of the great differences existing between them. This second mode, which has an almost plane profile at 20 kHz, presents a totally different distribution velocity at 200 kHz, where a Bessel function behaviour is evident. These changes in crosssection profiles represent a particular characteristic of
545
Fig. 20. Radial velocity of the 4th propagating mode inside an aluminium tube filled with water (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
Fig. 21. Magnitude of the axial velocity of the 1st propagating mode inside an aluminium tube filled with water in the neighbourhood of the inner wall (R =1 cm, R =2 cm, frequency=200 kHz). 1 2
the propagation through an elastic tube, which does not appear if the walls are rigid.
4. Conclusions A theoretical model describing the propagation of an acoustic wave through a cylindrical elastic tube filled with liquid was presented. This model completes the theory presented in previous works, including the viscous losses related to the propagation of the acoustic wave
546
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
gated, indicating the relation existing between these profiles and the attenuation of the wave. It was shown that the propagation characteristics of the acoustic wave through an elastic tube filled with liquid could not be described, in general, by a rigid-wall tube model due to the interaction between the walls of the tube and the liquid inside it. Therefore, an accurate analysis of the acoustic propagation in real tubes filled with liquid should include the vibration of the walls.
Acknowledgements I would like to thank Dr. E. Riera and Dra. C. Campos for their interesting comments and suggestions.
Fig. 22. Axial velocity of the 2nd propagating mode inside an aluminium tube filled with water for 20, 50 and 200 kHz (R =1 cm, 1 R =2 cm). 2
Appendix A For most of the practical cases, the product aR is 1 large enough to use the asymptotic expressions of the Bessel functions:
S S
J (aR )# 0 1
A
B
A
B
p cos aR − 1 paR 4 1 2
(A1)
p cos aR −3 . (A2) 1 paR 4 1 On the contrary, for a plane wave solution, the product k R must be small, and the following expresr 1 sions can be used instead of the Bessel functions: 2
J (aR )# 1 1
A B
J (k R )=1− 0 r 1
kR 2 r 1 2
(A3)
kR J (k R )= r 1 . 1 r 1 2
Fig. 23. Radial velocity of the 2nd propagating mode inside an aluminium tube filled with water for 20, 50 and 200 kHz (R =1 cm, 1 R =2 cm). 2
in free space and viscous losses due to the friction between the liquid and the walls of the tube. The model was compared with some analytical results obtained for the case of a tube with rigid walls, showing a good agreement. For the general case of a tube with elastic walls, the influence of the frequency, wall thickness and viscosity on the propagation velocity and the attenuation of several modes was studied. The frequency dependence of the cross-section profiles was also investi-
(A4)
When these are substituted into Eq. (16) and k is r written as a function of k , a quadratic equation is z obtained:
A
B
2a 4 2a −k∞2− j k2 + jk∞2=0. (A5) k4 + l l z z R R2 R 1 1 1 If only terms of the first order of (aR )−1 are retained, 1 the solution of this equation is:
A
B A
j jk2 k =k∞ 1− #k 1− l z l l aR 2vr 1 l
A
4 3
g+f
BBA
1−
j
B
. aR 1 (A6)
When the absorption caused by the friction with the walls is large compared to the absorption of a plane
547
L. Elvira-Segura / Ultrasonics 37 (2000) 537–547
wave in free space,
A
j
k =k 1− z l aR 1
and then
B
(A7)
A
1 a =−k z l 2R 1
S B 2g
S
vr 0
A
A
4
l
3
,
g+f
3
B
g+f ,
(A12)
References (A9)
which are the Kirchhoff ’s expressions when the thermal absorption is neglected. However, if the dominant viscous mechanism is the absorption due to the compressions and expansions of the acoustic wave, jk2 k =k 1− l z l 2vr
A
4
(A8)
vr 0
2g
jk3 a =− l z 2vr
(A11)
l which are the classical expressions for the propagation of a plane wave in the free space.
and 1 c =c 1− z l 2R 1
c =c z l
BB
(A10)
[1] E.P. Papadakis, R.N. Thurston, A.D. Pierce ( Eds.), Physical Acoustics Vol. XIX, Academic Press, New York, 1990, pp. 81–155. Chapters 2 and 3. [2] F. Eggers, Th. Funck, Rev. Sci. Instrum. 44 (8) (1973) 969–977. [3] J.C.F. Chow, J.T. Apter, J. Acoust. Soc. Am. 44 (2) (1968) 437–443. [4] M. El-Raheb, J. Acoust. Soc. Am. 71 (2) (1982) 296–306. [5] V.A. Del Grosso, Acustica 24 (1971) 299–311. [6 ] L.D. Lafleur, F.D. Shields, J. Acoust. Soc. Am. 97 (3) (1995) 1435–1445. [7] D.E. Weston, Proc. Phys. Soc. Lond. B 66 (1953) 695–709. [8] G. Kirchhoff, Anallen der Physik and Chemie 6 (134) (1868) 177–193. [9] L. Rayleigh, The Theory of Sound, Macmillan, London, 1901.