Full Text
ABSORPTION OF ULTRASONIC WAVES IN LIQUIDS
I. G. Mikhailov and S. B. Gurevich
INTRODUCTION
One of the most pressing problems of modern molecular physics is the problem of the structure of liquids. There is no doubt that among the phenomena whose study makes it possible to elucidate certain details of the structure of liquids is the phenomenon of absorption of ultrasonic waves.
At the present time there already exist many studies devoted to this question. It should be noted, however, that a certain scatteredness and, in part, randomness characterize a considerable portion of this work. It therefore seemed desirable to collect and systematize the available experimental material and to discuss it from the standpoint of contemporary theories of absorption.
The scope of the present article has not made it possible to cover all questions connected with the absorption of ultrasound in liquids. Thus, the methodology for measuring the absorption coefficient has not been considered at all; the question of the dispersion of ultrasonic waves has not been sufficiently treated, although the theory of this phenomenon is closely connected with the theory of absorption.
1. THE STOKES–KIRCHHOFF THEORY
In the propagation of plane sound waves, the intensity of the sound decreases with distance according to the law
\[ I = I_0 e^{-2\alpha x}, \]
where \(I_0\) is the intensity of the sound at \(x = 0\), \(I\) is the intensity of the sound at the point \(x\), and \(\alpha\) is the amplitude absorption coefficient.
The absorption coefficient \(\alpha\) is a parameter characteristic of the substance in which the sound waves propagate and depends on its physical properties.
In 1845 Stokes \(^{1}\) first advanced the assumption that the absorption of sound is connected with the forces of internal friction, or viscosity, which arise during the propagation of a sound wave.
Using this hypothesis, one can calculate the coefficient \(\alpha\), expressing it in terms of a number of basic constants of the substance. For this purpose Stokes, taking into account the forces of internal friction, made use of an analogy between certain equations of the theory of elasticity and hydrodynamics. Following Stokes, it is not difficult to obtain the equation for the propagation of longitudinal elastic waves in a viscous medium in the following form\(^2\):
\[ \frac{\partial^2 u}{\partial t^2} - a^2 \frac{\partial^2 u}{\partial x^2} - \frac{\frac{4}{3}\mu+\eta}{\rho}\cdot \frac{\partial^3 u}{\partial x^2 \partial t} =0 \tag{1.1} \]
where \(u\) is the velocity of displacement of particles in a viscous medium, \(a\) is the velocity of wave propagation, \(\mu\) is the coefficient of ordinary shear viscosity, and \(\eta\) is the coefficient of volume viscosity; \(\rho\) is the density.
If there is a source of harmonic elastic waves, then at some distance \(x\), owing to damping, the oscillatory velocity of the particles will be
\[ u=u_0 e^{-\alpha x}\cos(\omega t-\beta x), \]
where \(\omega=2\pi\nu\) is the cyclic frequency, \(\beta=\dfrac{\omega}{a}\) is the wave number, and \(\alpha\) is the absorption coefficient per unit length.
Solving equation (1.1), one can obtain the following expressions for \(\alpha\) and \(\beta\):
\[ \beta^2-\alpha^2= \frac{a^2\omega^2}{ a^4+\dfrac{\left(\frac{4}{3}\mu+\eta\right)^2\omega^2}{\rho^2} }; \qquad 2\alpha\beta= \frac{\left(\frac{4}{3}\mu+\eta\right)\dfrac{\omega^2}{\rho}}{ a^4+\dfrac{\left(\frac{4}{3}\mu+\eta\right)^2\omega^2}{\rho^2} }. \tag{1.2} \]
In deriving equation (1.1), Stokes considered it possible to set the second, “volume,” viscosity \(\eta=0\). Later we shall dwell on the validity of this assumption in considering acoustic processes.
Further, Stokes was interested in the question of the absorption of sound in gases and, in particular, in atmospheric air. It is not difficult to verify that in this case, at sound frequencies,
\[ a^4 \gg \frac{16\mu^2\omega^2}{9\rho^2}. \]
Taking this condition into account, and also assuming \(\eta=0\), from equations (1.2) one easily obtains the values of \(\alpha\) and \(\beta\) for gases
\[ \beta=\frac{\omega}{a}, \]
\[ \alpha=\frac{2\mu\omega^2}{3\rho a^3}. \tag{1.3} \]
It follows from equation (1.3) that the absorption coefficient is proportional to the coefficient of viscosity and to the square of the frequency.
Equation (1.3) is also applicable to liquids, of course under the condition that the inequality
\[ a^4 \gg \frac{16\mu^2\omega^2}{9\rho^2} \]
is preserved in this case as well. Eas-
to be convinced that, for ordinary low-viscosity liquids and at not very high frequencies, this inequality holds. In studying the propagation of sound waves of sufficiently high frequency in very viscous liquids, it may happen that \(\omega^4\) will be less than \(\dfrac{16\mu^2\omega^3}{9\rho^2}\). However, in this case the hydrodynamic equations from which Stokes proceeded in calculating \(\alpha\) will be inapplicable*).
Kirchhoff\(^{3}\) further showed that the absorption of acoustic waves will occur not only at the expense of viscosity, but also at the expense of thermal conductivity. Indeed, since the process of propagation of acoustic waves is adiabatic, the temperature of the liquid is not constant throughout its volume, and there will be a transfer of energy from places with higher temperature to places with lower temperature, not associated with macroscopic motion.
Taking this into account, Kirchhoff obtained the following expression for \(\alpha\):
\[ \alpha=\frac{\omega^2}{2\rho a^3}\left[\frac{4}{3}\mu+\chi\left(\frac{1}{C_v}-\frac{1}{C_p}\right)\right], \tag{1.4} \]
where \(\chi\) is the coefficient of thermal conductivity, and \(C_p\) and \(C_v\) are the heat capacities at constant pressure and volume. From equation (1.4) it follows that the dependence of the absorption coefficient on thermal conductivity is the same as on viscosity. Calculations show, however, that the influence of thermal conductivity on absorption, in comparison with the influence of viscosity, is in most cases very small**), so that thermal conductivity may be neglected. For water, for example, calculation by (1.4) leads to the conclusion that the absorption due to viscosity is more than 1000 times greater than the absorption due to thermal conductivity; for benzene, thirty times greater.
Comparison of the values of the absorption coefficients calculated from (1.3) or (1.4) with the coefficients measured by ultrasonic methods indicates that the measured values, as a rule, exceed the calculated ones. If for some substances (mercury, water) this excess is insignificant, then for many others (such as, for example, benzene, acetic acid, carbon disulfide) the measured \(\alpha\) is greater than the \(\alpha\) calculated by hundreds and thousands of times. Moreover, the proportionality of the absorption coefficient to the square of the frequency (required by Stokes’s expression), which is observed for some liquids (such as water, benzene, etc.), is not observed for a number of other liquids (acetic acid, etc.).
As will be seen below, in a number of liquids the dependence of \(\alpha\) on \(\mu\) also does not agree with that obtained by Stokes. The same may be said as well with regard to the dependence of the absorption coefficient on concentration in binary mixtures.
*) See, for example, the article by P. A. Bazhulin and M. A. Leontovich, DAN, 57, 27, 1947.
**) Exceptions are liquids such as mercury, in which \(\alpha_{\text{therm}}\) is greater than \(\alpha_{\text{visc}}\) by almost a factor of four.
From what has been said follows the necessity of revising Stokes’ theory. The fact that in some cases this theory is not in qualitative (and, for monatomic mercury, also quantitative) contradiction with the experimental data indicates that Stokes’ theory should be regarded as a special case of a more general theory of absorption.
In recent years several attempts have been made to correct Stokes’ theory (see, for example, the works of Lucas \(^{4}\) and Biquard \(^{5}\)). However, we shall not dwell on these attempts, since they have not led to any interesting practical results.
At present the most satisfactory theory of the absorption of ultrasonic waves in liquids must be considered to be the relaxation theory developed by L. I. Mandelstam and M. A. Leontovich \(^{6}\), Ya. I. Frenkel and Yu. N. Obraztsov \(^{7}\), and M. A. Isakovich \(^{8}\); it is on this theory that we shall dwell below.
2. THE THEORY OF ABSORPTION OF L. I. MANDELSTAM AND M. A. LEONTOVICH
In deriving the expression for the absorption coefficient, Stokes, as was already indicated, assumed the coefficient of second viscosity to be equal to zero. Elsewhere, however, Stokes \(^{9}\) drew attention to the inadequacy of the assumption he had made. After Stokes, doubts were repeatedly expressed as to the validity of this hypothesis, especially in considering acoustic processes \(^{10}\). Only comparatively recently did L. I. Mandelstam and M. A. Leontovich \(^{6}\), in developing the theory of absorption of ultrasonic waves in liquids, subject this question to special investigation.
The solution of the hydrodynamic equations with allowance for volume viscosity leads to the following expression for the absorption coefficient*):
\[ \frac{\alpha}{\nu^{2}} = \frac{2\pi^{2}}{\rho a^{3}} \left[ \frac{\frac{4}{3}\mu}{\rho} + \frac{\eta}{\rho} \right]. \tag{2.1} \]
Expression [2.1] provides broader possibilities for explaining the experimental data on absorption.
Indeed, the discrepancy between the experimentally determined value of the absorption coefficient and that calculated from Stokes’ expression (1.3) in such liquids as water, benzene, and others can be attributed to the second viscosity, so that
\[ \left(\frac{\alpha}{\nu^{2}}\right)_{\mathrm{exp}} - \left(\frac{\alpha}{\nu^{2}}\right)_{\mathrm{Stokes}} = \frac{2\pi^{2}\eta}{\rho a^{3}}. \tag{2.2} \]
*) In the literature on absorption, instead of the absorption coefficient \(\alpha\), the quantity \(\alpha/\nu^{2}\) is often considered. In what follows we shall use the following notation: \(\alpha'/\nu^{2}\) is the value of the quantity \(\alpha/\nu^{2}\) taking into account only the coefficient of ordinary viscosity (the Stokes value); \(\alpha''/\nu^{2}\) is the value of the quantity \(\alpha/\nu^{2}\) taking into account only the coefficient of volume viscosity.
Since in the indicated liquids \(\left(\dfrac{\eta}{v^2}\right)_{\mathrm{exp}} \gg \left(\dfrac{\eta}{v^2}\right)_{\mathrm{stock}}\), one must assume that in them the main influence on absorption is exerted not by the shear viscosity, but by the “volume” viscosity.
If, however, one turns to liquids of the acetic-acid type, in which the proportionality of the absorption coefficient to the square of the frequency is not observed, then here a simple allowance for the second viscosity is insufficient. An explanation of the dependence of the absorption coefficient on frequency in these liquids was given by L. I. Mandelstam and M. A. Leontovich\(^{11,6}\).
As was shown by M. A. Leontovich\(^{12}\), from the kinetic theory of the propagation of sound waves in gases it follows that \(\eta\) is different from zero only for polyatomic gases. The term containing \(\eta\) takes into account phenomena associated with the transition of the external energy of the molecule (the energy of translational and rotational degrees of freedom) into internal energy (vibrational degrees of freedom), and conversely.
L. I. Mandelstam and M. A. Leontovich proposed for liquids a theory of absorption analogous to the theory for polyatomic gases, but considerably more general. This theory takes into account not only absorption caused by the exchange of energy between external and internal degrees of freedom, but also absorption due to any processes accompanying changes in the thermodynamic state. The extent to which the term containing the second viscosity will influence absorption depends on how rapidly the equilibrium state is established (i.e. on the relaxation time of the process).
Mandelstam and Leontovich assumed that the state of a liquid at a given point is determined not only by specifying its temperature \(T\) and density \(\rho\), but also by specifying some quantity \(\xi\) (one or several), which determines the internal state of the liquid.
The quantities \(\xi\) may be, for example, the concentrations of mutually reacting components of liquid mixtures, the concentrations of excited or associated molecules, as well as other parameters determining the internal structure of the liquid.
Using an equation for \(\xi\) (analogous to the “excitation reaction equation” in the theory of absorption for gases) and carrying out the corresponding calculations, L. I. Mandelstam and M. A. Leontovich obtained the following expressions for dispersion and absorption:
\[ \frac{a_0}{a} = 1-\frac{1}{2} \frac{\omega^2\tau^2\left(\dfrac{a_\infty^2}{a_0^2}-1\right)} {1+\omega^2\tau^2}; \tag{2.3} \]
\[ \alpha = \frac{1}{2a_0} \frac{\omega^2\tau\left(\dfrac{a_\infty^2}{a_0^2}-1\right)} {1+\omega^2\tau^2}, \tag{2.4} \]
where
\[ \frac{a_{\infty}^{2}}{a_{0}^{2}}-1 = \frac{1}{\rho^{2}a_{0}^{2}} \left[ \frac{P_{\xi}^{2}}{\Psi_{\xi\xi}} + \frac{ T\left(P_{T}-\dfrac{P_{\xi}E_{\xi}}{T\Psi_{\xi\xi}}\right) }{ C_{v}-\dfrac{E_{\xi}^{2}}{T\Psi_{\xi\xi}} } - \frac{TP_{T}^{2}}{C_{v}} \right]. \tag{2.5} \]
Here \(E_{\xi}\) is the derivative of the energy per unit mass with respect to the parameter \(\xi\), \(\Psi_{\xi\xi}\) is the second derivative with respect to the same parameter of the free energy per unit mass, \(P_{T}\) is the temperature derivative of the equilibrium pressure, and \(P_{\xi}\) is the derivative with respect to the parameter \(\xi\) of the nonequilibrium pressure, which depends, in addition to \(T\) and \(\rho\), also on \(\xi\).
It follows from (2.5) that the factor entering (2.3) and (2.4),
\[ \frac{a_{\infty}^{2}}{a_{0}^{2}}-1, \]
depends, first, on quantities pertaining to the equilibrium state, \(P_{T}\) and \(C_{v}\) (their values may be regarded as known for liquids), and, second, on the quantities \(E_{\xi}\) and \(P_{\xi}\), which determine the change in energy and pressure when \(\xi\) changes. The values of the latter two quantities could be obtained only by making certain assumptions about the nature of the processes in liquids. Finally, the factor (2.5) also depends on the quantity \(\Psi_{\xi\xi}\), which can be determined if the magnitude of the thermal fluctuations of the parameter \(\xi\) is known, since
\[ \overline{\Delta \xi^{2}}=\frac{kT}{\Psi_{\xi\xi}}. \]
As for the quantity \(\tau\), which also enters expressions (2.3) and (2.4), it can be determined, for example, from the rate of the reaction occurring in the liquid. Thus, the expressions for the absorption and dispersion coefficients obtained by Mandelstam and Leontovich make it possible to calculate these quantities under definite assumptions about the processes occurring in the liquid. At the same time, and this is very important, an experimental investigation of the absorption coefficient makes it possible to draw certain conclusions about the nature of these processes.
Mandelstam and Leontovich further prove that the quantities \(a_{0}^{2}\), \(a_{\infty}^{2}\), and \(\tau\) can be expressed through the coefficient of volume viscosity, whose complex value is
\[ \eta^{*}=\frac{\tau\rho}{1+i\omega\tau}\left(a_{\infty}^{2}-a_{0}^{2}\right). \tag{2.6} \]
Taking (2.6) into account, it is easy to obtain the expression for the absorption coefficient in the following form:
\[ \alpha=\frac{\omega^{2}}{2\rho a^{3}}\, \frac{\eta_{0}}{1+\omega^{2}\tau^{2}}, \tag{2.7} \]
where \(\eta_0\) is the coefficient of bulk viscosity at \(\omega=0\). For the relaxation time \(\tau\) we obtain
\[ \tau=-\frac{\eta_0}{\rho(a_\infty^2-a_0^2)\eta}. \tag{2.8} \]
It is not difficult to see that for \(\omega\tau \ll 1\) expression (2.7) coincides with (2.1), without taking into account the term depending on \(\mu\). For \(\omega\tau \gg 1\), according to (2.7), \(\alpha\) no longer depends on the frequency. The frequency dependence of the absorption coefficient in the region \(\omega\tau \sim 1\) is shown in Fig. 1, in which, for comparison, the frequency dependence of the magnitude of the absorption coefficient due to ordinary viscosity is also given.
Fig. 1. Frequency dependence of the coefficient of absorption of ultrasonic waves according to Stokes and according to Mandelstam and Leontovich.
Thus, the expressions following from the theory of Mandelstam and Leontovich make it possible to explain the experimental course of the dependence of \(\alpha\) on \(\nu\) in liquids of the acetic-acid type, in which \(\alpha/\nu^2\) decreases with increasing \(\nu\). This will be discussed in more detail below.
In 1938, i.e. one year after the appearance of the works of Mandelstam and Leontovich, Kneser\(^{13,14}\) published an article in which he attempts to extend his relaxation theory\(^{15}\) to liquids, assuming that in this case as well the anomalous absorption is caused by the same effects as in polyatomic gases. However, in liquids, owing to the much stronger interaction of molecules, the mechanism of relaxation processes is far more complex than in polyatomic gases.
If one does not specify a concrete relaxation mechanism and considers Kneser’s theory in its most general form, for any relaxation process, then it is very easy to convince oneself*) that this theory—
*) See, for example, the article by A. Anselm\(^{70}\).
... theory gives nothing new in comparison with the theory of Mandelstam and Leontovich. Moreover, the equations of Kneser’s theory contain empirical quantities which make it possible to investigate qualitatively only the frequency dependence, and make it impossible to say anything about the dependence of the absorption coefficient on temperature, concentration, and other parameters. At the same time, using the theory of Mandelstam and Leontovich, it is in principle possible, as we saw above, to calculate the magnitude of the absorption coefficient and the dispersion.
Considerably later, namely in 1942, Tisza[^16] published a paper in which he repeats all the main propositions and conclusions of the theory of Mandelstam and Leontovich. It is characteristic that this author, as well as some other foreign authors studying the absorption of ultrasonic waves in liquids, do not mention the works of L. I. Mandelstam and M. A. Leontovich.
3. GENERAL RELAXATION THEORY
Experimental investigations established that in a number of liquids (glycerin, oils) the measured absorption coefficient agrees with that calculated by Stokes’ formula (1.3). This means that in these liquids the influence of shear viscosity on absorption masks the influence of volume viscosity and, consequently, the term with $\eta$ in equation (2.1) may be neglected.
In addition, it was established that, for example, in castor oil $(\alpha/\nu^2)_{\text{Stokes}}$ decreases with increasing frequency, which indicates relaxation of the shear viscosity. The theory of Mandelstam and Leontovich does not make it possible to explain these facts.
The further development of relaxation theory, taking into account also the influence of shear viscosity, was carried out by Ya. I. Frenkel and Yu. N. Obraztsov[^7] and by M. A. Isakovich[^8].
The theory of shear-viscosity relaxation is based on the hypothesis, confirmed by experiment[^17], that if an external force acts during a sufficiently short interval of time, then the liquid behaves like an elastic solid. Consequently, when an ultrasonic wave passes through it, the liquid will represent a medium in which both viscous and elastic properties are manifested. Formally, the simultaneous accounting of viscous and elastic properties was introduced as early as 1867 by Maxwell[^18], who obtained the following equation characterizing the viscoelastic properties of a medium:
\[ \frac{\partial u}{\partial y} = \frac{1}{G}\cdot \frac{dP}{dt} + \frac{1}{\mu}P, \tag{3.1} \]
where $P$ is the stress, $G$ is the shear modulus.
Correcting the original Stokes equation by taking into account Maxwell’s equation and the relaxation of volume viscosity, it is not difficult to obtain the following...
...expressions for the velocity and the absorption coefficient:
\[ a^2=a_0^2+\frac{\frac{4}{3}\mu_0\omega^2\tau_1}{\rho\left(1+\omega^2\tau_1^2\right)} +\frac{\eta_0\omega^2\tau_2}{\rho\left(1+\omega^2\tau_2^2\right)}, \tag{3.2} \]
\[ \alpha=\frac{\omega^2}{2a^3}\left[ \frac{\frac{4}{3}\mu_0}{\rho\left(1+\omega^2\tau_1^2\right)} +\frac{\eta_0}{\rho\left(1+\omega^2\tau_2^2\right)} \right], \tag{3.3} \]
where \(\tau_2\) is the relaxation time of the bulk viscosity, taken into account in the theory of Mandelstam and Leontovich, and \(\tau_1\) is the relaxation time of the shear viscosity, the value of which can be obtained in solving equation (3.1), putting \(\dfrac{\partial u}{\partial y}=0\), whence
\[ \tau=\frac{\mu_0}{G}. \tag{3.4} \]
It is not difficult to see that, if the quantities \(\omega^2\tau_1^2\) and \(\omega^2\tau_2^2\) are considerably smaller than unity, then (3.3) passes into (2.1), and \(a^2\) turns out to be approximately equal to \(a_0^2\).
Conversely, if \(\omega^2\tau_1^2\gg 1\) and \(\omega^2\tau_2^2\gg 1\), then (3.3) passes into
\[ \alpha=\frac{1}{2a^3}\left[\frac{4}{3}\frac{\mu_0}{\rho\tau_1^2} +\frac{\eta_0}{\rho\tau_2^2}\right], \tag{3.5} \]
i.e. in this case the absorption coefficient does not depend on frequency. For the square of the sound velocity we obtain
\[ a^2=a_0^2+\frac{4}{3}\frac{\mu_0}{\rho\tau_1} +\frac{\eta_0}{\rho\tau_2}. \tag{3.6} \]
Next, putting, for \(\omega\to\infty\), first \(\mu_0\) and then \(\eta_0\) equal to zero, we obtain the values of the relaxation times of the shear and bulk viscosities
\[ \tau_1=\frac{\frac{4}{3}\mu_0}{\rho\left(a_\infty^2-a_0^2\right)_\mu}; \tag{3.7} \]
\[ \tau_2=\frac{\eta_0}{\rho\left(a_\infty^2-a_0^2\right)_\eta}. \tag{3.8} \]
Here \(\left(a_\infty^2-a_0^2\right)_\mu\) and \(\left(a_\infty^2-a_0^2\right)_\eta\) are the values of the dispersion caused, respectively, by the relaxations of the ordinary and bulk viscosities.
It is interesting to note that, comparing (3.7) with (3.4), one can obtain the relation
\[ G=\frac{3}{4}\rho\left(a_\infty^2-a_0^2\right)_\mu, \]
which makes it possible to determine the shear modulus of a liquid from measurements either of the magnitude of the dispersion or of the relaxation time.
Table 1
Absorption coefficients in certain liquids
| No. | Substance | Chemical formula | \(t^\circ\) C | \(\rho\), g/cm³ | \(a\), m/sec | \(\mu\), poise | Frequency range, MHz | \(a/\gamma^2 \cdot 10^{17}\) (exper.) | Theoretical \(a'/\gamma^2 \cdot 10^{17}\) (Stokes) | Source |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Mercury | Hg | 20 | 13,6 | 1407 | 0,0155 | 21—996 | 12—13 | 10,3 | 19, 20, 21 |
| 2 | Helium (liquid) | He | 270 | — | 220 | — | 15 | 110 | 97 | 22 |
| 3 | Water | H₂O | 20 | 0,998 | 1485 | 0,0109 | 1—250 | 23 | 8,5 | 5,23—28 |
| 4 | Acetone | CH₃COCH₃ | 20 | 0,790 | 1200 | 0,0032 | 5—70 | 27 | 7 | 26,29 |
| 5 | Methyl alcohol | CH₃OH | 20 | 0,792 | 1130 | 0,0059 | 7—250 | 30 | 14,5 | 30—32 |
| 6 | Ethyl ether | (C₂H₅)₂O | 17 | 0,736 | 1020 | 0,0023 | 5—39 | 39 | 9 | 23 |
| 7 | Ethyl alcohol | C₂H₅OH | 20 | 0,789 | 1205 | 0,0172 | 13—220 | 55 | 22 | 32—34 |
| 8 | Isopropyl alcohol | C₃H₇OH | 20 | 0,784 | 1170 | 0,0222 | 75—280 | 75 | 42 | 33,34 |
| 9 | Xylene | C₆H₄(CH₃)₂ | 20 | 0,861 | 1328 | 0,0081 | 4—11 | 68 | 8,4 | 5 |
| 10 | Toluene | C₆H₅CH₃ | 20 | 0,863 | 1320 | 0,0059 | 5—37 | 80 | 7,7 | 5,23 |
| 11 | Cyclohexane | C₆H₁₂ | 20 | 0,772 | 1257 | 7—24 | 195 | 16 | 23 | |
| 11 | Cyclohexane | C₆H₁₂ | 20 | 0,772 | 1257 | 0,0096 | 4—7 | 475 | 10 | 5 |
| 12 | Chloroform | CHCl₃ | 20 | 1,526 | 983 | 0,0056 | ||||
| 13 | Carbon tetrachloride | CCl₄ | 21 | 1,596 | 928 | 0,0095 | 4—13 | 660 | 20 | 23 |
| 14 | Benzene | C₆H₆ | 20 | 0,872 | 1310 | 0,0065 | 75—105 | 520 | 8,6 | 34 |
| 14 | Benzene | C₆H₆ | 20 | 0,872 | 1310 | 0,0065 | 1—165 | 850 | 8,6 | 5,19, 23—26; 30, 31, 33—37 |
| 15 | Methyl acetate | CH₃COOCH₃ | 20 | 0,959 | 1140 | 0,0037 | 1—70 | 550—34 | 6,8 | 5,24 |
| 16 | Ethyl acetate | C₄H₈O₂ | 20 | 0,924 | 920 | 0,0044 | 3—40 | 800—87 | 16,5 | 5,38 |
| 17 | Formic acid | HCOOH | 20 | 1,244 | 1287 | 0,0173 | 4—10 | 2270—1170 | 17 | 23 |
| 18 | Acetic acid | CH₃COOH | 20 | 1,072 | 1194 | 0,0122 | 3,7—31 | 2880—189 | 17 | 23 |
| 19 | Carbon disulfide | CS₂ | 20 | 1,258 | 1150 | 0,0037 | 2—6 | 6500—5100 | 5 | 23 |
| 19 | Carbon disulfide | CS₂ | 20 | 1,258 | 1150 | 0,0037 | 75—105 | 1440 | 5 | 34 |
| 20 | Glycerin | C₃H₅(OH)₃ | 22 | 1,26 | 1910 | 8,5 | 3,15 | 2600—2500 | — | 26,39 |
| 21 | Olive oil | — | 22 | 0,91 | 1440 | 0,808 | 3,15 | 1350 | 780 | 39 |
| 22 | Linseed oil | — | 20 | 0,93 | 1470 | 0,516 | 3,15 | 680 | 460 | 39 |
| 23 | Castor oil | — | 19 | 0,96 | 1500 | 0,9 | 3,15 | 9500 | 8100 | 39 |
| 24 | Castor oil | — | 21 | 0,96 | 1500 | 10,3 | 4—16 | 4500—2100 | 8100 | 40 |
It follows from what has been said above that the decrease in the quantity $\alpha/\nu^2$ and the increase in the velocity $c$ with increasing frequency may be attributed to relaxation processes connected not only with volume viscosity, but also with shear viscosity. In both cases the dependences of absorption and velocity on frequency have one and the same character. It should also be noted that the frequency dependence of the absorption coefficient (and likewise of the velocity of sound) may in fact prove to be considerably more complicated, owing to the fact that in liquids there may exist a whole series of relaxation processes having different relaxation times.
In what follows we shall consider the relaxation theory in its simplest form, assuming that there is only one relaxation time of the shear viscosity $\tau_1$ and one—“volume”—$\tau_2$.
4. EXPERIMENTAL DATA ON ABSORPTION AND THEORY
At the present time there is a sufficiently large body of experimental material on the absorption of ultrasonic waves in liquids. However, this material must be treated with a certain caution, since in a number of cases the absorption coefficients measured by different authors in the same liquids differ greatly from one another.
In this connection we considered it necessary to carry out a careful analysis of all known experimental works on absorption and to select the results only of those works that deserve the greatest confidence.
In Table I we give data for liquids obtained as a result of such a selection. In it, for comparison, are indicated the values of $\alpha'/\nu^2$ (i.e. the part of $\alpha/\nu^2$ which takes into account only the ordinary viscosity, and for mercury also the thermal conductivity), as well as certain constants of the liquids. The liquids are arranged in order of increasing $\alpha/\nu^2$.
Comparison of the quantities $\alpha/\nu^2$ and of the course of their dependence on frequency makes it possible to divide these liquids into the following groups.
To the first group may be assigned mercury and liquid helium. The absorption coefficients measured in them almost correspond to those calculated from Stokes’ theory. On the basis of these data we may suppose that the coefficient of volume viscosity in monatomic mercury and helium is close to zero, or at any rate is smaller than the coefficient of ordinary viscosity.
To the second group may be assigned liquids whose absorption coefficient considerably exceeds the absorption coefficient calculated according to Stokes, but for which in the frequency range considered (of the order of $10^6$–$10^8$) the proportionality of the absorption coefficient to the square of the frequency is obeyed fairly well.
This group includes the liquids placed in the table under numbers 3–14. The coefficient of volume viscosity for them is considerably
but exceeds the coefficient of ordinary viscosity. This is seen, for example, from Table II, which gives the values of the coefficient of volume viscosity calculated from absorption data.
Table II
Coefficients of volume viscosity (at 20° C)
| No. | Substance | $\alpha''/\nu^2 \cdot 10^{17}$ | $\eta_0$, poise |
|---|---|---|---|
| 1 | Methyl alcohol | 15.5 | 0.006 |
| 2 | Ethyl ether | 30 | 0.008 |
| 3 | Acetone | 20 | 0.009 |
| 4 | Water | 14.5 | 0.018 |
| 5 | Ethyl alcohol | 34 | 0.026 |
| 6 | Toluene | 72 | 0.054 |
| 7 | $m$-Xylene | 60 | 0.057 |
| 8 | Cyclohexane | 179 | 0.11 |
| 9 | Chloroform | 465 | 0.23 |
| 10 | Carbon tetrachloride | 540 | 0.27 |
| 11 | Benzene | 840 | 0.64 |
The third group is formed by liquids for which $\alpha/\nu^2$ decreases with frequency, and at the beginning of the decrease its value is several orders of magnitude greater than $\alpha'/\nu^2$. These liquids are acetic acid, formic acid, carbon disulfide, methyl acetate, and ethyl acetate (Fig. 2).
The fourth group should include such liquids, still little studied, as glycerin, olive oil, and linseed oil, in which the absorption is large, but $\alpha/\nu^2$ does not exceed $\alpha'/\nu^2$ by very much.
A separate group may conditionally be made of castor oil, for which, according to Bazhulin’s data^40, for frequencies of 4–16 MHz, $\alpha/\nu^2$ is less than $\alpha'/\nu^2$ and, moreover, decreases with increasing frequency. According to Gunter’s measurements^39, $\alpha/\nu^2$ only slightly exceeds $\alpha'/\nu^2$ at a frequency of 3.15 MHz.
Fig. 2. Dependence of $\alpha/\nu^2$ on frequency in ethyl acetate.
Such a division of liquids in which the absorption coefficient has been measured into groups is justified by the relaxation theory. Indeed, if we turn to the simplest scheme of this theory, taking into account only one relaxation time of the volume viscosity and one relaxation time of the shear viscosity, then we obtain the course of the depend-
ABSORPTION OF ULTRASONIC WAVES IN LIQUIDS
dependence of $\alpha/\nu^2$ on frequency, shown in Fig. 3. As is seen from this graph, the curve of the dependence of $\alpha/\nu^2$ on frequency can be divided into five separate regions. The division into such regions is due to differences in the absolute magnitude and in the course of the dependence of $\alpha/\nu^2$ on frequency.
In the first region, $\alpha/\nu^2$ exceeds $\alpha'/\nu^2$ by the amount $\alpha''/\nu^2$ and does not depend on frequency. In the second region there is a decrease of $\alpha/\nu^2$ from $\alpha/\nu^2=\alpha'/\nu^2+\alpha''/\nu^2$ to $\alpha/\nu^2=\alpha'/\nu^2$ owing to the decrease with frequency of $\alpha''/\nu^2$.
Fig. 3. Dependence of $\alpha/\nu^2$ on frequency according to the relaxation theory in the case of the simplest scheme with two relaxation times.
The angular frequency $\omega$ at which $\alpha''/\nu^2$ will have one half of its initial value will be equal to the reciprocal of the relaxation time of the volume viscosity,
$$ \omega=\frac{1}{\tau_2}. $$
In the third region $\alpha/\nu^2=\alpha'/\nu^2$ and does not depend on frequency up to the fourth region, in which there is a decrease with frequency of $\alpha'/\nu^2$.
Just as in the second region, the frequency at which $\alpha'/\nu^2$ will have one half of its initial value determines the relaxation time of the ordinary viscosity $\left(\omega=\frac{1}{\tau_1}\right)$.
Finally, in the fifth region both viscosities have relaxed ($\omega\tau_1\gg1$ and $\omega\tau_2\gg1$), and the value of $\alpha/\nu^2$ will be inversely proportional to the square of the frequency (the absorption coefficient does not depend on frequency).
According to this scheme, the data for the individual groups of liquids presented in Table I can be assigned to different sections of the curve in Fig. 3.
For the monatomic liquids belonging to the first group, there will be almost no distinction between the first and third regions, and the decrease of \(\alpha/\nu^2\) with frequency, caused by volume viscosity, will be hardly noticeable. In this connection, the results of Rykman\(^{20}\) and Berg\(^{19}\) are understandable: they did not find, for mercury, any change in the quantity \(\alpha/\nu^2\) with frequency over a wide range of frequencies.
For liquids of the second group, absorption is to a significant degree due to volume viscosity. It is easy to see that the frequencies at which the absorption coefficient was measured in them lie entirely in the first region. It should therefore be expected that at higher frequencies \(\alpha/\nu^2\) will begin to fall with frequency. This in fact does occur, as we shall see somewhat below.
Liquids of the third group may be of especially great interest. The frequencies at which measurements were made in these liquids must be assigned to the second region. In this region the coefficient of volume viscosity decreases with frequency, thereby causing a lowering of \(\alpha''/\nu^2\), and at the same time of \(\alpha/\nu^2\). It is of interest to note that, knowing the dependence of the absorption coefficient on frequency throughout this region, one could determine \(\eta_0\) (the static value of the coefficient of volume viscosity), the relaxation time of volume viscosity, and also the magnitude of the dispersion caused by the ordinary viscosity. Unfortunately, in all the liquids of the third group listed in Table I the absorption coefficient was measured only in part of the second region. Nevertheless, by selecting such a theoretical curve for the dependence of \(\alpha/\nu^2\) on frequency on which the experimental points would lie in the best way, one can approximately determine the values of \((\alpha''/\nu^2)_0\), \(\eta\), \(\tau_2\), and \((a_\infty^2-a_0^2)_\eta\). The values of these quantities for a number of liquids are given in Table III. In the third
Table III
Coefficients and relaxation times of volume viscosity and the magnitude of dispersion caused by it
| No. | Substance | \(\alpha''/\nu^2 \cdot 10^{17}\) | \(\eta_0\) poise | \(\tau_2 \cdot 10^9\) | Dispersion \((a^2-a_0^2)_\eta 10^{-8}\) | Dispersion \(\dfrac{a_\infty-a_0}{a_0}\,100\) |
|---|---|---|---|---|---|---|
| 1 | Acetic acid | 6200 | 6.2 | 4.5 | 1.3 | 0.5 |
| 2 | Formic acid | 3500 | 3.5 | 2.5 | 1.1 | 0.4 |
| 3 | Carbon disulfide | 7500 | 7.2 | 1.5 | 3.8 | 1.4 |
| 4 | Methyl acetate | 1000 | 0.7 | 7 | 0.1 | 0.04 |
| 5 | Ethyl acetate | 1200 | 0.45 | 5 | 0.1 | 0.06 |
in the column are given the values of \(\alpha''/\nu^2\) at frequencies so low that they do not depend on frequency. These quantities are determined from the curves constructed by the method indicated above. In the fourth column are given the values \(\eta_0\), calculated from \((\alpha''/\nu^2)_0\) by formula (2.7) for \(\omega\tau \ll 1\). Next, in the fifth column are indicated the relaxation times obtained from the value of the cyclic frequency \(\omega\), at which \(\alpha''/\nu^2\) is half as large as \((\alpha''/\nu^2)_0\). In the sixth column are given the values \((a_\infty^2 - a_0^2)\eta\), calculated by formula (3.8). Finally, in the seventh column is indicated the expected percentage increase in velocity. The latter, as is seen from the data presented, for all liquids, with the exception of carbon disulfide, does not exceed \(0.5\%\).
It should be noted that, since there are no complete investigations of the dependence of \(\alpha''/\nu^2\) on \(\nu\), all the data presented are approximate in character.
Of considerable interest also are the data for liquids of the fourth group. For olive and linseed oils there are only Hunter’s measurements \(^{39}\) at one frequency, \(3.15\) MHz, but at different temperatures. For dehydrated glycerin there are Hunter’s measurements \(^{39}\) at \(\nu = 3.15\) MHz and Willard’s \(^{26}\) at \(\nu = 6.57\) MHz. Bazhulin \(^{40}\) measured the frequency dependence of the absorption coefficient in glycerin containing about \(50\%\) water (with a coefficient of ordinary viscosity \(\mu = 4.2\)) in the frequency range \(6\)—\(20\) MHz. For castor oil we have the most interesting data. Absorption in castor oil was measured by Bazhulin \(^{40}\) in the frequency range \(4\)—\(16\) MHz and by Hunter \(^{39}\) at a frequency of \(3.15\) MHz (as a function of temperature). Concerning the frequency dependence of the absorption coefficient for these liquids, the following may be noted: for glycerin the measurements of Hunter and Willard give approximately the same value for \(\alpha/\nu^2\), whereas Willard’s frequency is twice the frequency at which Hunter made his measurements. This permits the conclusion that \(\alpha/\nu^2\) for glycerin in this region does not change with frequency or changes only slightly.
Bazhulin also found that in 50-percent glycerin over a wide frequency range (\(6\)—\(20\) MHz) \(\alpha/\nu^2\) does not depend on frequency. As for the absolute magnitude of \(\alpha/\nu^2\), according to the data of Hunter and Willard it is almost equal to \(\alpha'/\nu^2\). In Bazhulin’s case, for glycerin containing water, \(\alpha/\nu^2\) somewhat exceeds \(\alpha'/\nu^2\) (\(\alpha/\nu^2 \sim 1700\), \(\alpha'/\nu^2 \sim 1300\)). Thus, for glycerin in the indicated frequency region the following proves to be characteristic: 1) \(\alpha/\nu^2\) does not depend on frequency; 2) \(\alpha/\nu^2\) differs little from \(\alpha'/\nu^2\). The same holds, as was already indicated, for mercury, although the absolute value of \(\alpha/\nu^2\) for mercury is considerably lower than for glycerin. But unlike mercury, for glycerin there is no basis for regarding \(\eta_0\) as very small, since here we are dealing not with a monatomic liquid and therefore bulk viscosity must manifest itself to one degree or another. The fact,
that \(\alpha/\nu^2\) is almost equal to \(\alpha'/\nu^2\), indicates that the frequencies at which the measurements were made belong to the third region. Apparently, in this same region (or at the end of the second) lies the frequency at which the measurements in linseed and olive oils were made, since \(\alpha/\nu^2\) in these liquids only slightly exceeds \(\alpha'/\nu^2\).
For castor oil, however, Bazhulin’s data (Fig. 4) quite definitely indicate a decrease of \(\alpha/\nu^2\) with frequency; moreover, throughout the entire frequency range in which the measurements were made, \(\alpha/\nu^2\) is less than \(\alpha'/\nu^2\). This indicates that the decrease with frequency of the quantity \(\alpha/\nu^2\) here is due to relaxation phenomena associated with ordinary, and not bulk, viscosity. It follows from Fig. 4 that \(\tau_1\) (the relaxation time of ordinary viscosity, equal to the reciprocal of the cyclic frequency at which \(\alpha/\nu^2\) is equal to one half of \(\alpha'/\nu^2\), i.e. \(\sim 2\pi\cdot 6\cdot 10^6\)) has approximately the following value:
Fig. 4. Dependence of \(\alpha/\nu^2\) on frequency in castor oil (according to Bazhulin’s data).
\[ \tau_1 \cong \frac{1}{40\cdot 10^6} \cong 2.5\cdot 10^{-8}\ \text{sec}. \]
Using (3.7), we can also determine the magnitude of the dispersion
\[ (a_\infty^2-a_0^2)_\mu=\frac{4}{3}\frac{\mu_0}{\rho\tau} =\frac{4}{3}\frac{10}{1\cdot 2.5\cdot 10^{-8}} \cong 5\cdot 10^8, \]
i.e. the total excess of \(a_\infty\) over \(a_0\), due to relaxation of ordinary viscosity, is of the order of \(1.5\%\).
Shakhovad \(^{41}\), measuring the speed of sound in castor oil in the frequency range from 2 to 6 MHz, found a dispersion of \(\sim 1\%\). Thus, the frequencies at which Bazhulin’s measurements were made in castor oil may be assigned to the fourth region.
Until recently, measurements relating to the fifth region were entirely lacking. Recently this region too was investigated by the authors of the present paper. The results obtained will be considered below.
Let us now turn to the dependence of the absorption coefficient on temperature (viscosity).
Of unquestionable interest is the dependence of \(\alpha/\nu^2\) on temperature in water, investigated by Fox and Rock \(^{42}\), and also earlier by Baumgart \(^{43}\). It is seen from Fig. 5 that \(\alpha/\nu^2\) decreases with temperature,
that follows also from Stokes (the curve \(\alpha'/\nu^2\)), but falls significantly faster than \(\alpha/\nu^2\).
The latter can be explained by the circumstance that the coefficient of volume viscosity decreases with increasing temperature more strongly than the coefficient of ordinary viscosity. This is seen from Fig. 6. The curve of the dependence of \(\mu\) on temperature here is constructed from tabular data, and the curve of the dependence of \(\eta\) on temperature from experimental absorption data.
Fig. 5. Dependence of \(\alpha/\nu^2\) on temperature in water.
Fig. 6. Dependences of the coefficients of ordinary and volume viscosity of water on temperature.
From Fig. 6 it also follows that the fall of the coefficient of volume viscosity with temperature becomes less noticeable as the latter increases.
In Fig. 7 the dependence of \(a/y^2\) on temperature in benzene is presented, and here, for comparison, on another scale the dependence of \(a'/y^2\) on temperature is also given. Since \(a/y^2 \gg a'/y^2\), it may be approximately assumed that the dependence of \(a''/y^2\) coincides with the dependence of \(a/y^2\) on temperature.
Fig. 7. Dependence of \(a/y^2\) on temperature in benzene.
In accordance with this, Fig. 8 presents (on different scales) the temperature dependences of the coefficients of ordinary and volume viscosity. In this case, up to \(40^\circ\) the decrease of \(\eta\) corresponds to the decrease of \(\mu\); further, \(\eta\) decreases more slowly, and in the interval \(55\text{--}71^\circ\), in contrast to \(\mu\), \(\eta\) already increases.
Fig. 8. Dependences of the coefficients of ordinary and volume viscosities of benzene on temperature.
Besides the water and benzene considered, the dependence of \(a/y^2\) on temperature was also measured in carbon tetrachloride, in acetic acid, and in viscous liquids (glycerin, olive, castor, and linseed oils)\(^{39}\).
Of particular interest is the dependence of absorption on temperature in acetic acid, since the frequencies at which the measurements were made (\(3\text{--}11\) MHz) belong to the second region. As is seen from Fig. 9, the coefficient of volume viscosity increases rather rapidly with increasing temperature.
From Gunter’s measurements of the dependence of the absorption coefficient on temperature in glycerine and in castor, olive, and linseed oils, it follows that over the entire temperature range investigated (from 0 to 50° C), \(\alpha/\nu^2\) only slightly exceeds the Stokes value \(\alpha'/\nu^2\).
Of considerable interest is also the dependence of \(\alpha/\nu^2\) on concentration in binary mixtures. In Fig. 10 the dependence of \(\alpha/\nu^2\) on concentration in the benzene—toluene mixture is presented. It follows from the figure that \(\alpha/\nu^2\) decreases sharply with increasing toluene concentration.
Fig. 9. Dependences of the coefficients of ordinary and volume viscosities in acetic acid on temperature.
Fig. 10. Dependence of \(\alpha/\nu^2\) on concentration in the benzene—toluene mixture.
At the same time, benzene and toluene have approximately identical \(\mu\), \(\rho\), and \(\alpha\), and, consequently, \(\alpha'/\nu^2\) (of the order of \(8\cdot 10^{-17}\)). Thus, the decrease of \(\alpha/\nu^2\) with increasing toluene concentration is due to the strong decrease of \(\alpha''/\nu^2\) and, consequently, to the rapid fall with concentration of the coefficient of volume viscosity, which in benzene is almost 10 times greater than in toluene. Besides the benzene—toluene mixture, similar dependences occur in a number of other liquids. They are evidently caused by a monotonic change in relaxation time with changing concentration of the molecules of one liquid in the molecules of another. In some mixtures, such as, for example, acetone—water mixtures[^29], methyl alcohol—water, ethyl alcohol—water[^32], maxima of \(\alpha/\nu^2\) are observed at definite concentrations. In Fig. 11 the dependences of \(\alpha/\nu^2\), \(\alpha'/\nu^2\), and \(\alpha''/\nu^2\) on concentration in the methyl alcohol—water mixture are presented; in Fig. 12, for the same mixture, the dependences of the viscosity coefficients on concentration are given. The coefficient of the second viscosity has a maximum at a concentration of 40–45%, at which the coefficient of ordinary viscosity also attains a maximum value.
Consideration of the dependence of absorption on concentration we shall limit to the facts cited above. For absorption in mixtures and in electrolyte solutions, as well as in emulsions and suspensions, we refer the interested reader to the available literature.^44–50
Fig. 11. Dependence of \(a/\nu^2\) on concentration in the mixture methyl alcohol—water.
Fig. 12. Dependence of the coefficients of ordinary and volume viscosities on concentration in the mixture methyl alcohol—water.
Of definite interest is also the dependence on pressure. Biquard’s measurements^51 in toluene showed that with increasing pressure \(a/\nu^2\) decreases
\[ \left(\frac{a_{500}}{a_{\mathrm{atm}}}=0.75\right) \]
and, moreover, more rapidly than \(a'/\nu^2\)
\[ \left(\frac{a'_{500}}{a'_{\mathrm{atm}}}=0.89\right). \]
It follows from these data that the coefficient of volume viscosity decreases with increasing pressure more rapidly than the coefficient of ordinary viscosity.
5. FINE STRUCTURE OF THE RAYLEIGH LIGHT-SCATTERING LINE AND THE THEORY OF ABSORPTION
The experimental data obtained by direct measurement of the absorption coefficient are supplemented by data on absorption which can be obtained by studying the fine structure of the Rayleigh light-scattering line in liquids. The theory of this phenomenon was given principally by L. I. Mandelstam,^52 and also by L. Brillouin.^53
As is known, fluctuations of density associated with thermal motion may be regarded as the result of the superposition of a large num-
of sound waves propagating in the body in all directions with all possible frequencies. In this case Rayleigh scattering may be regarded as diffraction of light by the “spatial acoustic gratings” formed by these waves. In a certain direction, making an angle \(\vartheta\) with the direction of the incident light, diffraction will be observed caused by a grating whose constant, corresponding to the length of the sound wave \(\Lambda\), satisfies the Bragg–Wulff condition
\[ 2\Lambda \sin \frac{\vartheta}{2}=\lambda . \]
It can be shown that, in the case of diffraction by such spatial acoustic gratings, the light oscillations will be modulated with frequency \(\omega\), and thus in the diffracted light there will appear the frequencies
\[ \omega_{0}\pm\omega=\omega_{0}\left(1\pm 2\frac{a}{c}\sin\frac{\vartheta}{2}\right), \tag{5.1} \]
where \(\omega_{0}\) is the frequency of the incident light, \(\omega\) is the sound frequency, and \(c\) is the speed of light. In addition to these two frequencies (shifted components), the unshifted line with frequency \(\omega_{0}\) will also be present in the scattered light.
Since the ratio of the speed of sound to the speed of light is very small, it is clear that the deviations of the shifted components from the fundamental frequency must be very small. These deviations (fine structure) were first observed by E. F. Gross\(^{54}\).
As is easy to see, expression (5.1) makes it possible to determine the speed of sound from spectral data. But, in addition, the study of the fine structure can also be used for qualitative conclusions concerning absorption.
Indeed, the lines of the doublet will be resolved only when the decrease in the amplitude of the acoustic wave over the course of its length is small, i.e., if
\[ \alpha\Lambda \ll 1. \tag{5.2} \]
It may therefore be stated with certainty that the value of the absorption coefficient in acetone, toluene, \(\mathrm{CCl}_{4}\), benzene, glycerine, castor oil, and other liquids in which the fine structure of the Rayleigh line has been observed satisfies condition (5.2).
It is interesting to note that extrapolation of expression (2.1) for the absorption coefficient to frequencies \(10^{10}\)–\(10^{11}\) Hz, which play an essential role in Rayleigh scattering, leads to the conclusion that, for the majority of liquids in which fine structure has been observed, \(\alpha\Lambda > 1\). This is seen, for example, from Table IV, where, in addition to \(\alpha\Lambda\) calculated from (2.1), the value of \(\alpha\Lambda\) calculated from Stokes’ expression (1.3) is given.
Table IV
| No. | Liquid | From (21) $\alpha/\gamma^2 \cdot 10^{17}$ | From (21) $\alpha\lambda$ | According to Stokes $\alpha/\gamma^2 \cdot 10^{17}$ | According to Stokes $\alpha\lambda$ |
|---|---|---|---|---|---|
| 1 | Acetone | 27 | 0.34 | 7 | 0.08 |
| 2 | Toluene | 80 | 1.05 | 8 | 0.10 |
| 3 | CCl$_4$ | 560 | 5.2 | 20 | 0.19 |
| 4 | Benzene | 850 | 11 | 8.6 | 0.11 |
| 5 | Glycerin | 2600 | 52 | 2500 | 47 |
| 6 | Castor oil | 9500 | 140 | 8100 | 125 |
It follows from this table that fine structure in acetone can be observed even in the case when $\alpha/\gamma^2$ does not depend on the frequency. The situation is different for toluene, CCl$_4$, and benzene. For them, in the frequency region $10^8—10^{10}\ \text{c/s}$, $\alpha/\gamma^2$ must decrease as the frequency increases; this will occur if the component $\alpha''/\gamma^2$ decreases with frequency.
Then, at the frequency $10^{10}\ \text{c/s}$, the absorption will be determined mainly by the quantity $\alpha'/\gamma^2$, and condition (5.2), under which the fine structure can be observed, will be satisfied.
These facts also served Mandelstam and Leontovich as the premise for creating a theory of absorption that takes into account relaxation phenomena associated with bulk viscosity.
Already after the publication of Mandelstam and Leontovich’s theory, Raman and Venkateswaran$^{55}$ observed the fine structure of the Rayleigh scattering line in glycerin. As is seen from Table IV, the assumption that for glycerin the Stokes portion of the absorption $\alpha'/\gamma^2$ does not decrease with frequency leads to the value $\alpha\lambda \simeq 50$. Such a result would contradict the data of Raman and Venkateswaran. In order for condition (5.2) to be fulfilled, one must assume that relaxation associated with the ordinary viscosity takes place, and that the relaxation time must be greater than $10^{-11}\ \text{s}$.
The same may be said also with regard to castor oil, in which the fine structure of the Rayleigh scattering line was likewise observed$^{56}$.
Let us dwell somewhat more fully on the connection between the observed fine structure of the Rayleigh scattering line and the absorption of hypersonic waves. As is known, in the derivations of L. I. Mandelstam and L. Brillouin the absorption of the acoustic wave was not taken into account. Only later did M. A. Leontovich$^{57}$ give a theory allowing for this factor. From this theory it followed that the shifted components have a finite width, and that the intensity distribution near the center
each line has a dispersive character
\[ I=\frac{I_0}{a^2+(q-\beta)^2}. \tag{5.3} \]
In the scattering of light waves at a given angle \(\vartheta\), all acoustic waves whose wave numbers lie in the interval \(\Delta\beta=\alpha\) take part. Correspondingly, the frequency interval lies in the range \(\Delta\omega=\alpha a\). Hence it follows that one or another dependence of absorption on temperature, viscosity, and other factors must substantially affect the width and intensity of the lines.
As is known, according to Landau and Placzek\(^{58,59}\), the ratio of the intensity of the central line \(I_c\) to the total intensity of the doublet components \(I_a\) is given in the form
\[ \frac{I_c}{I_a}=\frac{C_p-C_v}{C_v}. \tag{5.4} \]
If, however, one takes into account that the triplet lines have a finite width associated with absorption of hypersonic waves, then for the ratio of the maximum intensities one can obtain\(^{60}\)
\[ \left(\frac{I_c}{I_a}\right)_{\max} = \frac{C_p-C_v}{C_v}\cdot\frac{a\alpha}{\beta^2\chi}, \tag{5.5} \]
where \(\chi\) is the coefficient of thermal diffusivity.
Investigating the fine structure of the Rayleigh-scattering line in alcohols, Bai\(^{61}\) found that this ratio increases in passing from simpler to more complex alcohols. This contradicts formula (5.4), from calculations according to which it follows that the intensity ratio for more complex alcohols is smaller than for less complex ones. At the same time, the intensity ratio calculated from (5.5) increases in passing from simpler alcohols to more complex ones. This increase can be explained by a considerable increase in the absorption coefficient in passing from simple to more complex alcohols. The latter is confirmed by direct measurements by ultrasonic methods. In the case under consideration we were dealing with low-viscosity liquids, for which the relaxation times are apparently sufficiently small \((<10^{-10}\ \mathrm{sec})\). In viscous liquids, in all probability, one should assume that \(\tau_1\) and \(\tau_2>10^{-10}\ \mathrm{sec}\). In glycerine the fine structure was observed by Venkateswaran\(^{56}\), and it turned out that with decreasing temperature (increasing viscosity) the intensity ratio decreases. At first sight this contradicts relation (5.5), if one assumes that with increasing viscosity the absorption coefficient also increases. Below we shall show that in fact, in the frequency region \(\sim 10^{10}\ \mathrm{cps}\), in glycerine \(\alpha\) does not increase, but decreases with increasing \(\mu\), and thus relation (5.5) agrees with experiment in this case as well.
Data on dispersion, whose value can be determined, are of very great importance for the discussion of the relaxation theory.
I. G. MIKHAILOV AND S. B. GUREVICH
...to determine, by comparing the velocities determined at hypersonic frequencies by the scattering method with the velocities determined at frequencies \(\sim 10^6 — 10^7\) Hz by ordinary ultrasonic methods. Venkateswaran determined the velocity at hypersonic frequencies for a whole series of liquids\({}^{62}\). Comparing his results with the results of measurements of the sound velocity at ultrasonic frequencies, he came to the conclusion that for most liquids, including such as cyclohexane, chloroform, \(\mathrm{CCl}_4\), and benzene, dispersion is absent in the frequency range \(10^6 — 10^{10}\) Hz. For some liquids this result contradicts relaxation theory. Indeed, in order for fine structure to be observed in such liquids as benzene, it is necessary that the relaxation time be greater than \(10^{-10}\) sec. And this means that the dispersion region in these liquids must lie between the frequencies \(10^6 — 10^7\) Hz and \(10^{10} — 10^{11}\) Hz. As for the magnitude of the dispersion, an idea of it may be obtained from Table V. The dispersion values given in it are calculated from \(\eta_0\) and \(\tau_2\), the limiting values of \(\tau_2\) (and consequently also of the dispersion) being determined, on the one hand, by the condition that at hypersonic frequencies \(\alpha\Lambda < 1\), and on the other hand, by those limiting frequencies at which
Table V
| Substance | \(a^2/\mu \cdot 10^{17}\) | \(\eta_0\), poise | Lower limit: \(\tau_2\) | Lower limit: \((a_\infty^2-a_0^2)_\eta\) | Lower limit: increase of \(a_\infty^2\) over \(a_0^2\), % | Upper limit: \(\tau_2\) | Upper limit: \((a_\infty^2-a_0^2)_\eta\) | Upper limit: increase of \(a_\infty^2\) over \(a_0^2\), % |
|---|---|---|---|---|---|---|---|---|
| Cyclohexane | 195 | 0.11 | \(10^{-9}\) | \(1.4\cdot 10^8\) | 0.4 | \(3\cdot 10^{-11}\) | \(0.5\cdot 10^{10}\) | 15 |
| Chloroform | 475 | 0.23 | \(0.6\cdot 10^{-8}\) | \(0.5\cdot 10^8\) | 0.3 | \(5\cdot 10^{-11}\) | \(10^{10}\) | 30 |
| Carbon tetrachloride | 500 | 0.27 | \(3\cdot 10^{-9}\) | \(5.5\cdot 10^7\) | 0.3 | \(0.6\cdot 10^{-10}\) | \(0.3\cdot 10^{10}\) | 12 |
| Benzene | 850 | 0.64 | \(0.8\cdot 10^{-9}\) | \(0.9\cdot 10^9\) | 2.7 | \(0.8\cdot 10^{-10}\) | \(0.9\cdot 10^{10}\) | 22 |
there is as yet no observed decrease of \(\alpha/\nu^2\) with increasing frequency. From Table V it follows that the magnitude of the dispersion in benzene must be considerable even for the greatest possible relaxation time. The fact that dispersion in benzene was not detected should evidently be attributed to the low accuracy of determining the sound velocity from the fine structure of the Rayleigh-scattering line. From Table VI it follows that the discrepancy between the ultrasonic and hypersonic velocities for glycerin and castor oil is very considerable. From this table it also follows that for glycerin, as the temperature decreases, the magnitude of the dispersion increases. However, this increase occurs very slowly. This fact, as we shall see furth—
…, is of substantial importance in determining the dependence of \(\alpha\) on \(\mu\). The table also gives the relaxation times and the values of \(\alpha/\nu^{2}\) that may be expected according to the relaxation theory for \(\nu = 1.53\cdot 10^{10}\); this frequency follows from the measurements of Raman and Venkateswaran.
Table VI
| Substance | \(t^\circ\) | \(\mu_0\) | \(\alpha_0\), ultrasound | Author | \(\alpha_\infty\), hypersound | Author | \((\alpha_\infty^2-\alpha_0^2)10^{-10}\) | \(\tau\cdot 10^{10}\) | \(\alpha/\nu^2\cdot 10^{17}\), ultrasound | \(\alpha/\nu^2\cdot 10^{17}\), hypersound | \(\alpha\Lambda\) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Glycerin | 26 | 6 | 1960 | Bagavantam and Rao \(^{63}\) | 2500 | Raman and Venkateswaran \(^{55}\) | 2.6 | 2.7 | 1700 | 2.5 | 0.1 |
| same | 42 | 1.5 | 1936 | Same | 2412 | Same | 2.0 | 1.1 | 650 | 0.5 | 0.24 |
| same | 110 | 1840 | Same | 898 | same | ||||||
| Castor oil | 28 | 6 | 1478 | Gunter | 1626 | Venkateswaran | 0.5 | 1.7 | 5200 | 0.2 | 0.005 |
Venkateswaran \(^{55}\), and the values of \(\alpha\Lambda\). We see that \(\alpha\Lambda\) is less than unity, i.e., condition (5.2) is satisfied. It is interesting to note that with increasing temperature (for example, in glycerin), owing to the decrease of the relaxation time and the shift of the dispersion region toward higher frequencies, the quantity \(\alpha/\nu^2\) increases, and together with it \(\alpha\Lambda\) also increases. According to Leontovich’s theory \(^{57}\), such growth must be accompanied by a broadening of the shifted lines. According to Venkateswaran’s data \(^{56}\), such a broadening of the lines does in fact take place*).
6. ABSORPTION IN VERY VISCOUS LIQUIDS
Despite the relatively large number of experimental works on the absorption of ultrasound, until recently no measurements at all had been made of absorption in very viscous liquids. Meanwhile, such investigations are of considerable interest and may prove very useful for discussion of the theory.
It follows from the theory that in the pre-relaxation region (the first region, Fig. 3) the absorption coefficient is proportional to the sum of the viscosity coefficients. As regards the dependence of the absorption coefficient on viscosity in the region of very large viscosities, the relaxation theory does not determine it uniquely. Therefore an experimental investigation of the dependence of the absorption coefficient on viscosity may provide very interesting and valuable material.
*) Venkateswaran himself erroneously believed that his data contradicted Leontovich’s theory.
Of no lesser importance in discussing the theory is the investigation of the frequency dependence in the region \(\omega\tau \gg 1\) for all \(\tau\). However, this region has experimentally not been studied at all. This is explained by the fact that, for ordinary low-viscosity liquids, the condition \(\omega\tau \gg 1\) is fulfilled at frequencies \(>10^{10}\) cycles, whereas the limiting ultrasonic frequency at which it has been possible to measure absorption is \(10^{9}\) cycles[^21].
However, the condition \(\omega\tau \gg 1\) can also be fulfilled at ordinary ultrasonic frequencies if liquids of very high viscosity, and consequently with a long relaxation time, are investigated.
Such investigations were carried out by the authors of the present article on molten rosin[^64][^65][^66].
Figure 13 presents the results of measurements of the absorption coefficient in rosin as a function of temperature (viscosity). They may be formulated as follows.
Fig. 13. Dependence of the absorption coefficient on temperature (molten rosin).
1) For all the frequencies investigated, the absorption coefficient increases with increasing viscosity in the region of small viscosities, reaches a maximum in the region of viscosities of the order of \(10^{4}\) poises, and decreases with increasing viscosity at larger viscosities.
2) In the region of small viscosities, the absorption coefficient is approximately proportional to the square of the frequency. This proportionality is not observed for all values of \(\mu\), which may be explained, for example, by relaxation of the volume viscosity at small values of the coefficient of ordinary viscosity. As for the frequency dependence of \(\alpha\) in the region of very large viscosities (above \(10^{4}\) poises), here the proportionality of \(\alpha\) to the square of the frequency is not observed at all. At the same time, the independence of \(\alpha\) from frequency, which is assumed by the simplest relaxation-theory scheme (for \(\omega\tau \gg 1\)), is also not observed. The proportionality of the absorption-coefficient magnitude to the square root of the frequency agrees best with the experimental data.
Regarding the observed decrease of the absorption coefficient with increasing viscosity, the following may be noted.
If one uses the simplest scheme of relaxation theory, then, as follows from § 3, in the region \(\omega^2\tau_1^2 \gg 1\) and \(\omega^2\tau_2^2 \gg 1\) expression (3.5) is valid; if (3.7) and (3.8) are substituted into it, it is transformed into
\[ \alpha=\frac{\rho}{2a^3}\left[ \frac{(a_\infty^2-a_0^2)_\mu^2}{\frac{4}{3}\mu_0} + \frac{(a_\infty^2-a_0^2)_\eta^2}{\eta_0} \right]. \tag{6.1} \]
It follows from (6.1) that in the region under consideration (\(\omega\tau_1 \gg 1\) and \(\omega\tau_2 \gg 1\)) the absorption coefficient may decrease with increasing viscosity in the case where the quantities \((a_\infty^2-a_0^2)_\mu^2\) and \((a_\infty^2-a_0^2)_\eta^2\) increase more slowly than the viscosity increases. Let us note here that in most cases the term containing the bulk viscosity may be neglected, since from expression (3.5) it follows that the ratio of the effects of ordinary and bulk viscosity depends mainly on the ratio of the relaxation times of ordinary and bulk viscosity; moreover, the greater \(\tau\), the smaller the effect exerted on the magnitude of the absorption. In most cases the relaxation time of bulk viscosity is considerably greater than the relaxation time of ordinary viscosity, which makes it possible to neglect bulk viscosity not only when \(\omega\tau \sim 1\), but also in the region \(\omega\tau \gg 1\).
The qualitative measurements of Raman and Venkateswaran \(^{55}\) for certain liquids indicate that, with an increase in the coefficient of viscosity, the dispersion increases. From the quantitative data, available only for glycerin, it follows that the magnitude of the dispersion increases considerably more slowly than the viscosity increases, so that the ratio
\[ \frac{(a_\infty^2-a_0^2)_\mu^2}{\mu}, \]
despite the increase in dispersion, decreases (Table VII).
Table VII
| \(t^\circ\mathrm{C}\) | \(\mu\), poise | \(a_0\), ultrasound | \(a_\infty\), hypersound | \(a_\infty^2-a_0^2\) | \(\dfrac{(a_\infty^2-a_0^2)_\mu^2}{\mu}\) |
|---|---|---|---|---|---|
| 25 | 6 | \(1.95\cdot10^5\) | \(2.5\cdot10^5\) | \(2.81\cdot10^{10}\) | \(1.35\cdot10^{20}\) |
| 42 | 1.5 | \(1.923\cdot10^5\) | \(2.412\cdot10^5\) | \(2.11\cdot10^{10}\) | \(2.97\cdot10^{20}\) |
The data given in the table indicate that in glycerin, in the region \(\omega\tau \gg 1\), absorption should decrease with increasing viscosity. From this point of view, the fact cited in § 5 of the decrease in the ratio \(\left(\dfrac{I_c}{I_d}\right)_{\max}\) in (5.5) with increasing viscosity becomes understandable.
There are also other indications that, apparently, the square of the quantity \((a_\infty^2-a_0^2)\) grows more slowly than the viscosity coefficient. One may refer, for example, to the fact that, as the temperature increases, the sound velocity in most liquids falls considerably more slowly than the viscosity decreases. Thus, in benzene in the interval from \(20\) to \(70^\circ\mathrm{C}\), the viscosity falls from \(0.0065\) to \(0.0035\) poise, whereas the velocity decreases only from \(1.32\cdot10^5\) to \(1.10\cdot10^5\) cm/sec\({}^{67}\). Even if one assumes that at higher frequencies the velocity almost does not decrease, then, for a dispersion value (at \(20^\circ\mathrm{C}\)) of \(10\%\), the quantity \((a_\infty^2-a_0^2)^2\) falls from \(0.13\cdot10^{20}\) to \(0.08\cdot10^{20}\), and thus
\[ \frac{(a_\infty^2-a_0^2)^2}{\mu} \]
increases from \(20\cdot10^{20}\) to \(25\cdot10^{20}\).
Therefore, for most liquids in the region under consideration (large viscosities and frequencies), one may assume a decrease of \(\alpha\) with increasing \(\mu\).
Thus, comparison of the simplest scheme of the relaxation theory with the experimental data indicates qualitative agreement between theory and experiment.
Let us turn again to the experimental data presented in Fig. 13.
When the absorption coefficient, as a function of viscosity (and consequently also of \(\tau\)), reaches a maximum, the relaxation time \(\tau\) becomes equal to \(\frac{1}{\omega}\) (\(\omega\tau=1\)).
Knowing \(\tau\), and also the magnitude and position of the maximum, one can easily calculate some important constants of the liquids. These data for rosin are given in Table VIII.
Table VIII
| \(\nu\) MHz | \(\mu\) poise | \(\alpha\) | \(\tau\) | \(G\) | \(a\) | \(a_\infty^2-a_0^2\) | Increase of velocity in % |
|---|---|---|---|---|---|---|---|
| \(0.66\) | \(10^3\) | \(1.5—2\) | \(2.4\cdot10^{-7}\) | \(4\cdot10^9\) | \(1.5—1.6\cdot10^5\) | \(3\cdot10^9\) | \(8\) |
For other, higher frequencies these calculations were not made, since for them the position of the maximum is uncertain. Nevertheless, from Fig. 13 it follows that, in passing from \(\nu=0.66\) MHz to \(\nu=1.52\) MHz, the maximum shifts toward lower viscosities. This fact is easily explained by the relaxation theory. Indeed, since the absorption maximum corresponds to \(\omega\tau\sim1\), then, as \(\omega\) increases, the condition \(\omega\tau\sim1\) will be fulfilled when the value of \(\tau\) decreases, which will occur when the viscosity decreases.
The relaxation theory considered by us is approximate (since the Maxwell relaxation scheme for ordinary viscosity is approximate*), and from it one can demand only qualitative agreement with experiment. However, even in its simplest form this theory makes it possible to explain correctly a number of observed phenomena.
The discrepancy between theory and experiment, found in studying the dependence of \(\alpha\) on \(\nu\) in liquids with high viscosity, can probably be eliminated with further development of the theory.
7. THEORY OF ABSORPTION AND RELAXATION PROCESSES IN LIQUIDS
The relaxation theory of absorption considered above was not connected with any definite mechanism of absorption.
The available experimental data in some cases make it possible to draw preliminary conclusions about the nature of the processes in a liquid that cause the absorption of ultrasonic waves.
Even before the publication of the theory of Mandelstam and Leontovich, Leontovich put forward the supposition that processes similar to Kneser processes in gases may take place in liquids. Debye \(^{68}\), and also Frenkel and Obraztsov \(^{7}\), suggested that absorption may be caused by a change in short-range order. Below we shall consider to what extent one or another process may affect absorption.
The influence of Kneser effects on absorption was discussed by Herzfeld \(^{69}\). He obtained the following expression:
\[ \alpha_i \nu^2 = \frac{2\pi^2}{a^2}\varepsilon\tau . \tag{7.1} \]
In the case of not too high frequencies, when \(\alpha_i/\nu^2\) retains a constant value,
\[ \varepsilon_0 = \frac{C_i \Delta}{C_p(C_p-\Delta)} , \tag{7.2} \]
where the quantity \(C_i\)—the heat capacity at constant volume due only to internal degrees of freedom—can be obtained from spectroscopic investigations, \(C_p\) from direct measurements, \(\Delta=C_p-C_v\) from data on compressibilities and on \(C_p\). Table IX gives data for water (\(C_i\), \(C_p\), and \(\Delta\) in cal/mol·deg, \(\varepsilon\) is a dimensionless quantity). In order now to calculate \(\alpha_i/\nu^2\),
* Ya. I. Frenkel and Yu. N. Obraztsov considered, in addition to purely viscous deformation, also the so-called relaxation elastic deformation investigated by Kobeko, Kuvshinskii, and Gurevich. However, the expression they obtained at very high viscosities (large \(\tau\)) leads to the same dependence on frequency as the simplest scheme of the relaxation theory.
one must also know \(\tau\). Using measurements in the ultrasonic region, one may assert that for water \(\tau\) should not be greater than \(\sim 10^{-9}\). Thus, the quantity \(\alpha/\nu^{2}\), due to Kneser effects, cannot exceed
\[ \alpha/\nu^{2}=\frac{2\pi^{2}}{1.5\cdot 10^{5}}\cdot 5\cdot 10^{-5}\cdot 10^{-9}\simeq 0.6\cdot 10^{-17}, \]
i.e., is approximately 30 times smaller than \(\alpha''/\nu^{2}\). It is more probable that \(\tau\) is still smaller than \(10^{-9}\), and thus the influence
Table IX
| \(C_i\) | \(C_p\) | \(\Delta\) | \(\omega\) |
|---|---|---|---|
| 0.152 | 18 | 0.11 | 0.00005 |
of Kneser effects in the case of water is negligibly small. The attempt to ascribe to Kneser effects the absorption caused by bulk viscosity is also contradicted by the fact that at \(4^\circ\)C, when for water \(\Delta=0\), \(\alpha''/\nu^{2}\) is not only not equal to zero, but is even greater than at \(20^\circ\)C.
Let us now turn to the case of benzene. According to the data of Fox and Rock \(^{42}\), \(\varepsilon\) for benzene at \(20^\circ\) is equal to 0.146. In order not to contradict the experimental data in the ultrasonic region, one must assume that \(\tau<10^{-9}\). It is not difficult to verify that the calculated \(\alpha/\nu^{2}\) satisfies the experimental value if one sets \(\tau=4\cdot 10^{-10}\). Such a value of \(\tau\) is quite probable, since from observations of the fine structure it follows that \(\tau\) must be no less than \(1.5\cdot 10^{-10}\). Thus, it is very probable that the absorption in benzene can be explained by relaxation processes associated with the exchange of energy between internal and external degrees of freedom. At the same time, of course, one still cannot assert that only these processes are responsible for the absorption.
As has already been indicated, an essential role in absorption associated with bulk viscosity may, in the opinion of Debye, as well as of Frenkel and Obraztsov, be played by a change in the “short-range order.”
A. I. Anselm \(^{70}\) carried out a calculation of this kind of absorption, making use for this purpose of Eyring’s expression for the free energy and Lennard-Jones’s expression for the “free volume” of a molecule in a liquid. The “free volume” depends on the parameter \(\xi\), which may be characterized as a “packing factor.”
Assuming that the sole cause of the absorption caused by the second viscosity is the change of the “packing factor” during propagation of the wave, A. I. Anselm calculated all the quantities entering into Mandelstam and Leontovich’s expression (2.5), and computed the value of \(\alpha''/\nu^{2}\).
These calculations are, of course, of a very approximate character. Nevertheless, the calculations carried out for monatomic mercury gave for $\tau$ a value of $\sim 10^{-12}$ sec., which also follows from other calculations (for example, from Frenkel’s formula $^{71}$). Moreover, the calculations showed that absorption caused by a change of short-range order is relatively small and should manifest itself noticeably only at very high frequencies.
It is possible that one of the most important causes of absorption is the orientation of molecules or groups of molecules in the ultrasonic field.
The existence of double refraction in an ultrasonic field $^{72-74}$ indicates that orientation of molecules actually takes place. As is known, orientation accompanied by friction, and consequently by dissipation of energy, plays a role in the Kerr effect, and is also one of the most important causes producing dispersion of the dielectric constant $^{75}$. Undoubtedly, in all these cases one and the same process (orientation of molecules) is involved, so that for the same substances in all these effects the relaxation times should be of the same order. In this connection, a parallel investigation of the absorption of ultrasonic waves and of effects in which orientation is manifested is of interest.
It is possible, although this question has not yet been investigated in detail, that association is also responsible for absorption in liquids.
In a number of special cases, for example absorption in strong and weak electrolytes, the relaxation processes may be ascribed a definite specific character (see, for example, the works of Shaposhnikov and Leontovich $^{76}$ and of Leontovich $^{77}$).
The experimental material available at the present time is still insufficient to indicate definitely which particular effects play the most important role in the absorption process.
ADDENDUM
After this article had been completed, the authors became acquainted with several works on the absorption of ultrasonic waves in liquids that had appeared very recently.
We give here a brief survey of these works.
Pellam and Squire $^{22}$ measured the absorption and velocity of ultrasonic waves in liquid helium as a function of temperature in the interval from $1.57^\circ\mathrm{K}$ to $4.5^\circ\mathrm{K}$. Absorption was measured at a frequency of 15 MHz, and the velocity at frequencies of 15 MHz and 1.3 MHz. It was established that the velocity at the indicated frequencies has one and the same value. In measuring the absorption coefficient the following results were obtained: 1) In the upper temperature interval He I, the measured absorption coefficient agrees with the absorption coefficient calculated by Stokes’ formula; 2) at the $\lambda$-point
absorption increases sharply, apparently to infinity; 3) below the $\lambda$-point (the first sound of HeII) the absorption coefficient increases as the temperature is lowered.
Pinkerton[^78] investigated the absorption of ultrasonic waves in water in the frequency range from $7.37$ Mc/s to $66.1$ Mc/s and in the temperature interval from $0$ to $60^\circ$C. He established that, over the entire frequency range investigated, $(\alpha/\nu^2)_{\mathrm{exp}}$ does not depend on frequency. With increasing temperature $(\alpha/\nu^2)_{\mathrm{exp}}$ decreases, changing from $56.9\cdot 10^{-17}$ at $0^\circ$C to $10.24\cdot 10^{-17}$ at $60^\circ$C. The ratio
$$ \frac{\alpha_{\mathrm{exp}}}{\alpha_{\mathrm{Stokes}}} $$
remains approximately constant at all temperatures and equal to $\sim 3$, increasing appreciably only in the temperature interval from $0$ to $10^\circ$C. Pinkerton, like several other authors, used the new, so-called pulse method. The accuracy of measuring the absorption coefficient by this method is $0.5$–$1.0\%$.
Hall[^79] gave a theory of the absorption of ultrasonic waves in water, considering the relaxation of compressibility.
Finally, let us mention here the discussion[^80] on the question of the absorption of ultrasonic waves in liquids, held on November 13, 1947, by the Acoustics Group of the Physical Society at Imperial College (England). In individual contributions to this discussion a number of controversial and erroneous opinions were expressed. Thus, for example, Richardson stated that Tisza was the first to call attention to the importance of taking the second viscosity into account in explaining the experimental results on absorption. Let us recall here once again that Tisza’s work was published in 1942, whereas the fundamental work of L. I. Mandelstam and M. A. Leontovich was published in 1937. Moreover, Tisza added nothing new to the theory of L. I. Mandelstam and M. A. Leontovich.
In his contribution Pinkerton proposed dividing all liquids in which absorption measurements have been made into 4 classes: 1) liquids in which $\alpha/\nu^2$, found experimentally, is greater than $\alpha/\nu^2$ calculated by Stokes by a factor of 1500–10; 2) liquids in which $(\alpha/\nu^2)_{\mathrm{exp}}$ is greater than $(\alpha/\nu^2)_{\mathrm{Stokes}}$ by a factor of 10–3; 3) liquids in which the ratio of $(\alpha/\nu^2)_{\mathrm{exp}}$ to $(\alpha/\nu^2)_{\mathrm{Stokes}}$ has a value from 3 to 1; and, finally, 4) very viscous liquids, in which $(\alpha/\nu^2)_{\mathrm{exp}}$ is approximately equal to $(\alpha/\nu^2)_{\mathrm{Stokes}}$.
The proposed classification is artificial, since it does not take into account certain theoretically important and characteristic features of the absorption of ultrasonic waves in liquids. Indeed, such liquids as acetic acid and ethyl acetate, on the one hand, and benzene and carbon tetrachloride, on the other, fall into the same group, whereas absorption in them has an entirely different character. In the frequency range investigated ($10^6$–$10^8$ Mc/s), $(\alpha/\nu^2)_{\mathrm{exp}}$ in acetic acid and ethyl acetate depends on
frequency, while in benzene and \( \mathrm{CCl}_4 \) it does not depend on it. On the other hand, such liquids as benzene and toluene, in which absorption has an analogous character, turn out to be in different groups.
It seems to us that, for understanding the processes occurring in the absorption of ultrasonic waves in liquids, the classification given in § 4 is more convenient, since it is based on the relaxation theory, which in the main correctly explains the experimental results.
Cited Literature
- G. Stokes, Camb. Trans. Phil. Soc., 8, 287 (1845).
- Rayleigh, The Theory of Sound, vol. II, Gostekhizdat (1944).
- G. Kirchhof, Pogg. Ann., 177, 1934 (1868).
- R. Lucas, C. R. 203, 611 (1936).
- R. Biquard, Ann. d. Physique, 9, 193 (1936).
- L. I. Mandelstam and M. A. Leontovich, ZhETF, 7, 498 (1937).
- Ya. I. Frenkel and Yu. N. Obraztsov, ZhETF, 9, 1081 (1939).
- M. A. Isakovich, DAN, 23, 782 (1939).
- G. Stokes, Math. Phys. Pap., 1, 68 (1889).
- L. Brillouin, Leçons sur la viscosité des liquides et des gaz. Paris (1907).
- L. I. Mandelstam and M. A. Leontovich, DAN, 3, 111 (1936).
- M. A. Leontovich, Izv. AN (phys. ser.), 5, 633 (1938); ZhETF, 6, 561 (1936).
- N. Kneser, Ann. d. Phys., 37, 177 (1938).
- N. Kneser, Phys. Zeits., 39, 860 (1938).
- N. Kneser, Ann. d. Phys., 16, 337 (1933).
- Z. Tisza, Phys. Rev., 61, 531 (1942).
- Ya. I. Frenkel, Kinetic Theory of Liquids. Publishing House of the Academy of Sciences of the USSR, 1945.
- C. Maxwell, Phil. Trans., 157, 49 (1867).
- R. Bär, Helv. Phys. Acta., 10, 332 (1936).
- Rickmann, Phys. Zeits., 40, 582 (1939).
- G. R. Ringo, I. W. Fitzgerald and B. G. Hurdle, Phys. Rev., 72, 87 (1947).
- I. R. Pellam and G. F. Squire, Phys. Rev., 72, 1242 (1947).
- P. A. Bazhulin, ZhETF, 8, 437 (1938).
- J. Clayes, J. Errera and N. Sack, Trans. Faraday Soc., 33, 136 (1939).
- G. Grobe, Phys. Zeits., 39, 333 (1938).
- G. J. Willard, J. Ac. Soc. Am., 12, 438 (1941).
- E. F. Fox and G. D. Rock, J. Ac. Soc. Am., 12, 505 (1941).
- E. T. Hsu, J. Ac. Soc. Am., 17, 127 (1945).
- P. A. Bazhulin and Yu. M. Merson, DAN, 24, 107 (1939).
- S. Parthasarathy, Curr. Sci., 6, 501 (1937).
- A. Lindberg, Phys. Zeits., 71, 457 (1940).
- I. G. Mikhailov and S. B. Gurevich, DAN, 52, 630 (1946).
- I. R. Pellam and I. Galt, J. Chem. Phys., 14, 608 (1946).
- R. A. Rapuano, Phys. Rev., 72, 78 (1947).
- J. Quinn, J. Ac. Soc. Am., 18, 185 (1946).
- E. Baumgardt, C. R. 204, 416 (1937).
- E. I. Gregg, Phys. Rev., 58, 208 (1940); Rev. Sci. Instr., 12, 149 (1941).
- E. T. Beyer and M. Smith, J. Ac. Soc. Am., 18 (No. 2) (1946).
- J. I. Hunter, J. Ac. Soc. Am., 13, 36 (1941).
- P. A. Bazhulin, DAN, 31, 114 (1941).
- L. Zachoval, J. de Physique, 10, 350 (1939).
- E. F. Fox and G. D. Rock, Phys. Rev., 70, 68 (1946).
- E. Baumgardt, C. R., 202, 203 (1936).
- P. A. Bazhulin, JETP, 9, 1151 (1939).
- P. A. Bazhulin, DAN, 19, 153 (1938).
- F. H. Willis, J. Ac. Soc. Am., 19, 242 (1947).
- W. Bus, Ann. d. Physik, 33, 193 (1938).
- Rüfer, Ann. d. Physik, 41, 301 (1942).
- S. M. Rytov, V. V. Vladimirsky, A. Galanin, JETP, 8, 614 (1938).
- V. V. Vladimirsky and A. Galanin, JETP, 9, 233 (1939).
- P. Biquard, C. R., 206, 797 (1938).
- L. I. Mandelstam, ZhRFKhO, 58, 381 (1928).
- L. Brillouin, Ann. d. Physique, 17, 88 (1922).
- E. Gross, Nature, 126, 201, 400, 603 (1930); 129, 722 (1932).
- C. V. Raman and C. S. Wenkateswaran, Nature, 142, 791 (1938); 143, 728 (1939).
- C. S. Wenkateswaran, Proc. Ind. Ac. Sci. (A), 15, 360 (1942).
- M. Leontovich, Zeits. f. Phys., 72, 247 (1931).
- L. Landau and G. Placzek, Sow. Phys., 5, 172 (1934).
- E. F. Gross, JETP, 16, 129 (1946).
- V. L. Ginzburg, Izv. AN (ser. fiz.), 9, 174 (1945).
- R. S. Bai, Proc. Ind. Acad. Sci. (A), 15, 349 (1942).
- C. S. Wenkateswaran, Proc. Ind. Acad. Sci. (A), 15, 371 (1942).
- Bhagavantam and Rao, Proc. Ind. Acad. Sci. (A), 9, 312 (1939).
- I. G. Mikhailov and S. B. Gurevich, DAN, 58, 221 (1947).
- I. G. Mikhailov, Bulletin of Leningrad University, No. 3, 5 (1947).
- S. B. Gurevich, Dissertation, LSU (1947).
- P. Porodnov and V. Nozdrev, JETP, 9, 625 (1939).
- P. Debye, Zeits. f. Electrochem., 45, 174 (1939).
- R. F. Herzfeld, J. Ac. Soc. Am., 13, 33 (1941).
- A. I. Anselm, JETP, 15, 751 (1945).
- Ya. I. Frenkel, Izv. AN (ser. fiz.), 3, 287 (1937).
- R. Lucas, J. d. Physique, 206, 827 (1938).
- N. Petralia, Euc. Am., 348 (1940).
- V. Tsvetkov, A. Mindlin, G. Makarov, JETP, 10, 891 (1946).
V. Tsvetkov and V. Eskin, DAN, 59, 1089 (1948). - Debye and Sack, Theory of the Electrical Properties of Molecules, ONTI (1936).
- Shaposhnikov and Leontovich, ZhFKh, 13, 781 (1938).
- M. A. Leontovich, JETP, 18, 40 (1938).
- Pinkerton, Nature, 160, 128 (1947).
- Hall, Phys. Rev., 71, 318 (1947).
- Hall, Nature, 160, 913 (1947).