Dispersion and Selective Absorption of Ultrasonic Waves in a Polyatomic Gas

B. G. Shpakovskii, Leningrad

The dispersion of sound is inseparably connected with its absorption, i.e., with the absorption of sound energy by the medium in which the sound propagates. In a whole series of their works, the founders of classical acoustics—Stokes, Helmholtz, Kirchhoff, Rayleigh—exhaustively examined the question of the propagation of sound in a deformable medium, treating this process as strictly equilibrium. Alongside this, more than 30 years ago Jeans theoretically substantiated the possibility of dispersion caused by the nonequilibrium state of the medium during the propagation of sound. Attempts to detect this phenomenon experimentally proved unsuccessful, chiefly because in the recent past there were no generators of ultrasonic frequency that were accessible to the experimenter and sufficiently stable in operation. Only the broad study of quartz oscillations in modern radio engineering made it possible, by using its piezoelectric properties, to obtain a generator of ultrasonic—or, as it is called in the English and American literature, supersonic—frequency, sufficiently reliable and stable with respect to the maintenance of the generated frequency, and emitting waves of simple and regular form.

Since the velocity of propagation of sound in an elastic medium is determined by its molecular-kinetic and thermodynamic properties, knowledge of this velocity can be applied to judging the structure and properties of the medium. The dispersion of ultrasound discovered in recent times, and the selective absorption associated with it, both entirely unforeseen by classical acoustics, have made it possible to obtain new information about the properties of the molecules of certain gases.

In the present article the question of the dispersion of sound analyzed by classical acoustics will not be touched upon at all. We shall confine ourselves to considering only that dispersion which is called the nonstationary state of the medium during the propagation of a sound wave of sufficiently high frequency. Alongside this, there will be considered—

selective absorption, observed in the region of dispersion and caused by the very same reasons, has also been considered.

This direction in modern physics, which arose comparatively recently but has already managed to become rather widespread and has an ever-growing literature, is connected chiefly with the name of Kneser.

For measuring the velocity of propagation of ultrasound in a gas at high frequencies, the method of the acoustic interferometer, first proposed by Pierce, is at present the most commonly used. The essence of this method is as follows. A plate of suitable shape, cut from a piezo-quartz crystal and correspondingly oriented with respect to its principal axes, is included in a Pierce oscillator or resonator circuit.

Fig. 1. Diagram of a piezo-quartz generator.

With a proper selection of the parameters of the electrical part of the circuit and a suitable mode of its operation, the piezo-quartz enters into its own mechanical oscillations, distinguished by a very great stability in the maintenance of frequency. Such a crystal, placed in a vessel with the gas under investigation, serves as a source of ultrasonic waves. With a sufficiently large radiating area, the emitted wave may be regarded as plane. At some distance from the crystal, a reflecting surface is placed parallel to the radiating surface of the quartz; this surface is called the reflector. If the distance between the quartz and the crystal is equal to an integral number of half-waves, then a system of standing waves is established. When the reflector is moved, it passes successively through the nodes and antinodes of the standing wave; in doing so the action of the reflected waves on the quartz changes periodically, and this reaction in turn is reflected in the operating regime of the generator in many respects,

in particular, the magnitude of the anode current varies periodically. By means of a corresponding indicator it is possible to determine the instants corresponding to the successive passage of the reflector through the nodes of the standing wave, and from this to calculate the wavelength.

With the aid of the wavelength and of the frequency of oscillation of the quartz, determined by a wavemeter or by some other method, the velocity of propagation of ultrasound is calculated. This elegant method, which uses one and the same quartz simultaneously both as a generator and as a receiver of the reflected waves, makes it possible to measure the velocity of propagation of ultrasound with a very high degree of accuracy, using small quantities of the substance under investigation.

In Fig. 1 is given Pierce’s electrical circuit, used for exciting the piezo-quartz, and in Fig. 2 is given a drawing of Kneser’s apparatus for measuring the velocity of propagation of ultrasound in carbon dioxide gas. We give the following explanations of the notation: \(Q\) denotes the piezo-quartz plates, generating waves of different frequency; \(R\) is the reflector for reflecting the ultrasound; \(M\) is the microscope for precisely reading off the position of the reflector; \(HSAH_1H_2\) is the mercury device for moving the reflector.

Fig. 2. Apparatus for measuring the velocity of propagation of ultrasound in a gas.

Fig. 3. Velocity of propagation of ultrasound in \(\mathrm{CO_2}\) relative to the velocity of propagation in argon.

As ultrasound generators Kneser used 7 different quartzes in the range from 60,000 cycles/sec to 1,500,000 cycles/sec. Carbon dioxide gas was chosen for the investigation. Experimentally, it proved considerably more advantageous to measure not the velocity of propagation of ultrasound in carbon dioxide gas, but the ratio of this velocity to the velocity of propagation of ultrasound of the very same frequency in a neutral gas—argon, which is monatomic and therefore, according to the theory set forth below, should not exhibit the phenomenon of dispersion. In Fig. 3

the results obtained by Kneser are presented graphically. Along the axis of abscissas are plotted the common logarithms of the frequency; along the axis of ordinates—the ratio of the velocity of propagation of ultrasound in carbon dioxide to the velocity of propagation in argon, in promilles. It is clear from the drawing that, although individual measurements show rather considerable discrepancies with respect to one another, nevertheless the change in the velocity of propagation exceeds the limits of the individual measurements. The increase in the velocity of propagation begins to be observed at frequencies exceeding 100,000 cycles/sec (in the drawing this point is marked 50).

On the basis of the experimental results Kneser constructed a theory of sound dispersion which satisfactorily explained both the peculiar course of the experimental curve and the numerical values of its parameters, and also connected this dispersion with the time of thermal relaxation of the vibrational degrees of freedom of the molecules—a quantity which until then had not lent itself to experimental determination. Instead of Kneser’s theory, somewhat intricate and not impeccable from both the mathematical and the physical side, there will be set forth below more simplified and schematized considerations (compiled on the basis of the works of Kneser and Henry) which, without claiming exhaustive rigor of exposition, will help to form a clear picture of the phenomenon.

The molecules of a polyatomic gas have, in the most general case, translational, rotational, and vibrational degrees of freedom. The energy possessed by the gas, its so-called internal energy, is distributed among the degrees of freedom in known proportions. Let us take carbon dioxide at room temperature. Its molecule is linear and can perform 3 vibrational motions of different frequency. The most fully expressed is the first vibrational motion, caused by a transverse or deformation vibration; in this, the two outer atomic nuclei execute vibrations relative to the central one, perpendicular to the longitudinal axis of the molecule. To each degree of freedom of translational and rotational motion there corresponds, according to the theorem on the uniform distribution of energy (at room temperatures), \(\frac{1}{2}RT\) (calculated for one gram-molecule of gas). For the vibrational degrees of freedom this condition will be fulfilled only at sufficiently high temperatures. Let us denote by \(E_a\) the energy falling to the translational and rotational degrees of freedom, and by \(E_i\) that falling to the vibrational degrees. The relation between these two quantities in the equilibrium state is determined only by the temperature of the gas:

\[ E_a : E_i = f(T) \tag{1} \]

Every change in the total energy of the gas, accompanied by a change in its temperature, leads to a redistribution of it among the corresponding degrees of freedom according to formula (1). There is no such physical process in which changes of \(E_a\)

and $E_i$ would occur simultaneously with the fulfillment of formula (1). In the emission or absorption of radiant energy, $E_i$ changes first, and then $E_a$. In adiabatic instantaneous compression of a gas, the kinetic energy of the translational and rotational motions of the molecules first increases; then, by means of a series of collisions with one another, part of the energy received passes into the vibrational energy of the molecules, while $E_i$ gradually increases until a stationary state is reached, determined by relation (1). The process by which the stationary state is established occurs, although rapidly, not instantaneously. The time practically required to reach the equilibrium state is called the relaxation time. Fig. 4 serves as a visual explanation of the process described. The lines shown in the figure give the time variation of a number of quantities: $V$—the volume of the gas, $E_i$—its internal energy,

Fig. 4. Time dependence of $E$, $E_i$, and $E_a$ under instantaneous adiabatic compression and expansion.

$E$—its total energy, where, according to the definition, $E = E_a + E_i$. The shaded part gives the course of variation of the external energy of the gas $E_a$, equal, obviously, to $E - E_i$. The segment on the time axis denoted by $\beta$ expresses the relaxation time. With great probability one may suppose that the rate of increase of the vibrational energy corresponding to the temperature $T$ is proportional to the difference between the equilibrium value $E_i$ and the nonequilibrium value $E_x$ at some moment of time $t$, namely

\[ \frac{dE_x}{dt} = \frac{1}{\beta}\left(E_i - E_x\right). \tag{2} \]

The solution of this differential equation gives the variation of $E_i$ according to an exponential law.

The pressure of the gas, determined according to molecular-kinetic conceptions by the translational energy of the molecules contained in a unit volume, is proportional to the value $E_a$ (represented by the shaded area in Fig. 4), and at the moment of adiabatic compression will have its greatest value, gradually reaching, after the relaxation time has elapsed, the corresponding equilibrium

values. In an instantaneous adiabatic expansion the pressure first falls below the equilibrium value and then gradually rises to it. Fig. 5 gives a graphical representation of the process described.

When an elastic sinusoidal wave with a period considerably exceeding the relaxation time propagates in a gas, the adiabatic compressions and expansions will occur not instantaneously, but gradually in time and so slowly that practically at every moment the equilibrium relation between the gas pressure and the corresponding specific volume or density will be preserved. In the case where the period of the wave is comparable with the relaxation time, there will be a phase shift between the pressure and the compression, since the pressure maximum will be determined not only by the maximum of compression, but also by the rate of change of the compression; this circumstance

Fig. 5. Time dependence of pressure under adiabatic compression and expansion.

will cause the pressure wave to lead in phase with respect to the compression wave, the magnitude of which depends on the relation between the period of the wave and the relaxation time.

In this case the equations for the propagation of a plane sinusoidal wave of pressure and compression will take the following form:

\[ \pi = \pi_0 e^{\,i\left[\omega\left(t-\frac{x}{v}\right)+\psi\right]}, \qquad S = S_0 e^{\,i\omega\left(t-\frac{x}{v}\right)} . \tag{3} \]

\(\pi\) denotes the change in pressure caused by the wave, \(S\) is the compression, determined by the formula \(\rho = \rho_0(1+s)\), where \(\rho_0\) is the initial density of the gas and \(\rho\) is the density during passage of the wave; \(\psi\) denotes the phase shift; the remaining notation is conventional. The differential equation of Euler’s motion, as applied to a plane wave, has the form:

\[ \rho \frac{\partial^2 \xi}{\partial t^2} = -\frac{\partial \pi}{\partial x}, \tag{4} \]

where \(\xi\) denotes the displacement of a particle of the medium.

According to the definition

\[ S=-\frac{\partial \xi}{\partial x}. \tag{5} \]

With the aid of (4) and (5) we obtain the wave equation:

\[ \frac{1}{\rho}\frac{\partial^{2}\pi}{\partial x^{2}}=\frac{\partial^{2}S}{\partial t^{2}}. \tag{6} \]

Substituting into the last equation, in place of the second derivatives, their values determined from (3), we obtain the following relation:

\[ v^{2}=\frac{1}{\rho}\frac{\pi}{S}, \tag{7} \]

or

\[ v^{2}=\frac{1}{\rho}\frac{\pi_{0}}{S_{0}}e^{i\psi}, \tag{8} \]

or

\[ v^{2}=v_{0}^{2}e^{i\psi}. \tag{9} \]

The modulus of this complex expression determines the square of the propagation velocity of the elastic wave, while \(\psi\) mathematically expresses the phase shift between pressure and compression. For \(\psi=0\), i.e., for a phase shift equal to zero, we have the classical case of propagation of a plane wave.

To derive the dispersion formula for the propagation velocity of ultrasound, we shall apply the following considerations. Suppose that a plane sinusoidal wave passes through a given point of a gaseous medium, causing, owing to the adiabatic nature of the process, periodic sinusoidal changes of temperature, expressed by the following formula:

\[ T=T_{0}+T_{1}e^{i\omega t}. \tag{10} \]

Here \(T_{0}\) is the mean temperature of the gas, \(T\) is the temperature changed by the passing wave, \(\omega\) is the angular frequency, and \(T_{1}\) is the amplitude of the change in temperature.

Depending on the temperature, changes will occur in the internal energy of the gas with the same angular frequency and with a certain amplitude according to the following law:

\[ E_{i}=E_{0}+E_{x}e^{i\omega t}. \tag{11} \]

Let us denote the total molecular heat capacity by \(C_{v}\), the heat capacity due to translational and rotational degrees of freedom by \(C_{a}\), and that due to vibrational degrees of freedom by \(C_{i}\). According to the definition we have:

\[ C_{v}=C_{a}+C_{i}. \tag{12} \]

The amplitude \(E_{x}\) may be represented in the form:

\[ E_{x}=C_{i}T_{1}, \tag{13} \]

and formula (11) will be rewritten as follows:

\[ E_{i}=E_{0}+C_{i}T_{1}e^{i\omega t}. \tag{14} \]

Substituting in (2), instead of \(E_i\), its value from (14) and solving the differential equation with respect to \(E_x\), we obtain:

\[ E_x = E_0 + \frac{C_iT_1}{1+i\omega\beta} e^{i\omega t}. \tag{15} \]

Comparing the expression obtained with (14), we arrive at the conclusion that, in a nonstationary process, the internal heat capacity of a gas must be regarded as a complex quantity, namely:

\[ C_i'=\frac{C_i}{1+i\omega\beta}. \tag{16} \]

Consequently, the total heat capacity of the gas will also be a complex quantity:

\[ C_v' = C_a + \frac{C_i}{1+i\omega\beta}. \tag{17} \]

The equation of propagation of a plane wave has the form:

\[ \frac{\partial^2 \xi}{\partial t^2} = v^2 \frac{\partial^2 \xi}{\partial x^2}, \tag{18} \]

where \(v^2\)—the square of the velocity of propagation of the elastic wave—is expressed by the following formula in terms of the pressure \((p)\), density \((\rho)\), gas constant \((R)\), and molecular heat capacity at constant volume \((C_v)\):

\[ v^2=\frac{p}{\rho}\left(1+\frac{R}{C_v}\right). \tag{19} \]

Substituting for \(C_v\) its value, determined by means of (17), we obtain the required dispersion formula in the general form:

\[ v^2=\frac{p}{\rho}\left(1+\frac{R}{C_a+\dfrac{C_i}{1+i\omega\beta}}\right). \tag{20} \]

Representing this complex expression in the usual trigonometric form

\[ v^2=v_0^2 e^{i\psi}, \tag{21} \]

we conclude, on the basis of the considerations set forth above, that the existence of a nonzero argument indicates a phase shift between the pressure and the condensation in the propagating wave, while the modulus expresses the square of the speed of sound. Instead of the modulus we take its real part (which in practice does not introduce a significant error), and finally obtain the following dispersion formula:

\[ v^2=\frac{p}{\rho}\left(1+R\frac{C_v+\omega^2\beta^2 C_a}{C_v^2+\omega^2\beta^2 C_a^2}\right). \tag{22} \]

In Fig. 6 a curve is shown giving the form of \(v^2\) as a function of \(\lg \omega\).

From the figure it is seen that, at small frequencies from \(\omega=0\) to a certain \(\omega_1\), the velocity of propagation of the elastic wave remains constant, equal to \(v_0\). Then, in the interval from \(\omega_1\) to \(\omega_2\), the function

\(v^2\) increases, having an inflection point in its largest part at \(\omega_w\), and then remains constant at all frequencies greater than \(\omega_2\) and equal to \(v_\infty\). The region of dispersion occupies the interval from \(\omega_1\) to \(\omega_2\), equal to \(\Delta \lg \omega\). From the formula we obtain the following limiting values:

\[ v_0^2=\frac{p}{\rho}\left(1+\frac{R}{C_v}\right), \qquad v_\infty^2=\frac{p}{\rho}\left(1+\frac{R}{C_a}\right). \tag{23} \]

The steepest part and the inflection point lie at the frequency

\[ \nu_w=\frac{1}{2\pi}\cdot\frac{1}{\beta}\cdot\frac{C_v}{C_a}. \tag{24} \]

The region of dispersion is determined chiefly by the quantity \(\frac{1}{\beta}\), which plays in the theory of acoustic dispersion a role similar to that of the resonance frequency in the theory of optical dispersion. In deriving this formula, certain simplifying assumptions were made,

Fig. 6. Frequency dependence of the square of the velocity of propagation of elastic waves.

in particular it was assumed that the relaxation time for the rotational degrees of freedom is considerably shorter than the relaxation time for the vibrational degrees of freedom. If this is taken into account, then, repeating the very same considerations, we obtain on the dispersion curve, beyond the dispersion region considered, a new rise at periods comparable with the time required for the establishment of equilibrium for the rotational degrees of freedom, corresponding to a new region of dispersion. Thus the complete dispersion curve must have a step-like form.

Let us try to represent those molecular processes which give rise to the phenomena of dispersion. Excitation of a molecule, i.e. bringing it into a vibrational state, is accompanied by the formation of a quantum of vibrational energy at the expense of the non-quantized energy of the translational motion of the molecules. Loss of the vibrational quantum returns the molecule to its former unexcited state. This loss may occur either by the reverse transition of the quantum of vibrational energy into the non-quantized energy of motion, or by the excitation of vibrations in another molecule, if suitable conditions are present during a collision.

It is also possible to suppose that a given quantum of vibrational energy passes through several molecules before it passes into

unquantized energy of motion. An excited molecule, i.e. one in a vibrational state, may undergo a large number of impacts in collisions with other molecules without losing its excited state. The statistically defined time that elapses from the moment of formation of a quantum of vibrational energy until its reverse transition into unquantized energy is called the mean lifetime of a vibrational quantum. At a given temperature, a quite definite fraction of the molecules of a polyatomic gas is in an excited state; moreover, this equilibrium state is dynamic, i.e. the number of newly formed vibrational quanta is equal to the number passing into unquantized energy. With an instantaneous increase of the temperature to a higher value, the former equilibrium state is disturbed. The new state, corresponding to a larger percentage of excited molecules, will be attained, according to the definition given above, after a relaxation time \(\beta\). It is not difficult to see that

Fig. 7. Comparison of the theoretical dispersion curve with experimentally obtained data.

the phenomenologically defined quantity \(\beta\) coincides in essence with the concept, just introduced, of the mean lifetime of a quantum of vibrational energy.

Let us apply the theory set forth above to the interpretation of Kniezer’s experimental data on the velocity of propagation of ultrasound in carbon dioxide gas. According to measurements by numerous investigators, the molecular heat capacity of \(\mathrm{CO}_2\) at \(0^\circ\mathrm{C}\) is equal to \(6.65\ \dfrac{\mathrm{cal}}{\mathrm{mole}\,\mathrm{degree}}\); here the share of the translational and rotational degrees of freedom accounts for the value required by the classical theory of equipartition of energy and equal to \(C_a - 5\cdot\dfrac{R}{2}\), i.e. \(4.96\ \dfrac{\mathrm{cal}}{\mathrm{mole}\,\mathrm{degree}}\), while the share of the vibrational degrees of freedom accounts for

\[ C_i = 1.69\ \dfrac{\mathrm{cal}}{\mathrm{mole}\,\mathrm{degree}}. \]

Substituting these values for \(C_a\) and \(C_i\) into the theoretical dispersion formula and choosing a suitable value for the parameter \(\beta\), we obtain the curve shown in Fig. 7, which agrees fairly well with the experimentally determined points. The one-step character of this curve indicates the falling out only of the vibrational degrees of freedom in the measured frequency range and, conse-

...consequently, the falling out of the rotational degrees of freedom should be expected at frequencies lying above 1,000,000 cycles/sec.

From the position of the rising part of the curve one determines the value, fundamental for the theory of dispersion, of the quantity \(\beta\), equal to \(1.2\cdot 10^{-6}\) sec. Since at \(0^\circ\) about 4% of the carbon-dioxide molecules are in the excited state, i.e. each molecule is excited, on the average, for 0.04 sec. in 1 sec., the number of collisions per second that bring a molecule into the excited state will be statistically equal to 40,000.

Since, according to the classical kinetic theory of gases, the number of collisions experienced by a molecule in 1 sec. is equal to \(10^{10}\), it follows that out of 100,000 collisions only once can translational energy pass into vibrational energy. The rarity of such collisions is explained by the considerable magnitude of the quantum of vibrational energy, equal to \(1.3\cdot 10^{-13}\) ergs, in comparison with the mean kinetic energy of the molecule. In addition to purely energetic considerations, the mutual arrangement of the molecules at the moment of collision also plays a large role. For excitation there is required a collision of molecules possessing kinetic energy several times greater than the magnitude of the vibrational quantum, the degree of this excess depending on the character of the collision. In particular, for one of the most favorable cases, shown in Fig. 8, only \(1/3\), \(1/4\) of the translational energy can be transformed into vibrational energy.

Fig. 8.

If one calculates what number of molecules possesses translational energy exceeding the mean value by 3–4 times, then one obtains that only 1 in 100,000 has energy sufficient to excite vibrations, i.e. that only 1 in 100,000 collisions will prove sufficiently effective, in full agreement with the conclusions based on measurements of the velocity of propagation of ultrasound.

Fig. 9. Dependence of pressure on volume under rapid changes.

For the interpretation of this circumstance it is convenient to introduce the concept of a decrease in the effective diameter of the molecule.

Along with dispersion, in the same frequency range there is also observed a considerable absorption of wave energy, the so-called selective absorption. The reason for this phenomenon is easy to understand from consideration of Fig. 5 already given above.

Let us represent the process considered, namely the course of the functions \(p\) and \(v\), under instantaneous adiabatic compression, in the coordinates \(pv\), in order to establish the relation between them (Fig. 9). The resulting closed curve characterizes an irreversible process, accompani-

…associated with the absorption of energy. To clarify this circumstance, let us present the following experiment. A heat-impermeable cylinder \(A\) (Fig. 10) is filled with a polyatomic gas (let this be, for definiteness, \(CO_2\)), in which dispersion is observed. The piston \(B\), which hermetically closes the cylinder, moves in it without friction and is under a constant external pressure. We neglect the inertia of the gas located in the cylinder and outside it. With sinusoidal motion of the piston about the equilibrium position, alternating compressions and rarefactions of the gas in cylinder \(A\) will occur. If the piston oscillates slowly, with a period considerably greater than the time of thermal relaxation, denoted above by \(\beta\), the energy expended on compressing gas \(A\) will be wholly returned by the gas back during expansion, since the pressure exerted on the piston by the gas, at a given position, will be the same both when the piston moves in one direction and when it moves in the opposite direction. In exactly the same way, the work of the gas during expansion will be compensated back to it during compression and, consequently, the piston set in motion will perform undamped oscillations at the expense of the energy of the initial impulse.

Fig. 10.

Let us suppose that the motion of the piston occurs more rapidly, namely with a period comparable with the time of thermal relaxation \(\beta\). When the piston rapidly passes through the equilibrium position—during compression of the gas—the energy transferred by it to the gas will not have time to be distributed in equilibrium over all degrees of freedom, and the piston will have to overcome a pressure greater than that which corresponds, in the equilibrium state, to the given temperature and volume. The time during which the piston, with a considerably lower velocity (according to the nature of sinusoidal motion), approaches its extreme position, stops, and begins to move back, may prove sufficient for the establishment of the equilibrium state. During the further motion of the piston with ever increasing velocity, the pressure of the gas falls below that value which it had during the motion of the piston compressing the gas, and, consequently, the support provided by the gas to the piston during the motion rarefying the gas will be less than the resistance which the piston encountered from the gas during the compressing motion. Likewise, during the further motion of the piston, rarefying the gas, after passing through the equilibrium position the pressure of the gas in the cylinder decreases more than does its value corresponding to the corresponding values of temperature and density in the equilibrium state and, consequently, when overcoming the external pressure the piston is supported by the gas less than in the case where the gas pressure corresponded to the equilibrium state.

After the piston reaches the other extreme position, equilibrium is also established during the reverse motion of the piston; the force causing this motion, and equal to the difference between the external and internal pressures, is less than the force that opposed the motion of the piston from the equilibrium position to the extreme position.

In this case, i.e., when the period is comparable with the relaxation time \(\beta\), the motion of the piston will be damped, and continuous addition of energy is required to maintain it.

Finally, for completeness, let us consider a third case, when the motion of the piston occurs with a period considerably smaller than the relaxation time. Without repeating in detail the whole sequence of the preceding arguments, it is sufficient to indicate that in this case the pressure exerted by the gas on the piston will be the same at a given position of the piston both when it moves in one direction and when it moves in the directly opposite direction; consequently, in this case, just as in the first, no absorption of energy by the gas will occur and the oscillations of the piston will be undamped.

Fig. 11. Thermodynamic diagrams of adiabats for slow (1) and fast (3) compression of a gas.

In Fig. 11, the first and third cases are represented in the form of thermodynamic diagrams. The lines shown express adiabats; moreover, the adiabat corresponding to the third case is steeper, since in the third case, owing to the omission of the vibrational degrees of freedom from the overall balance for \(C_v\), there will be an increase of

\[ \gamma=\frac{C_p}{C_v}. \]

In Fig. 5 the process corresponding to the second of the cases described is shown graphically. The area enclosed by the closed curve represents the energy that is absorbed by the gas, i.e., irreversibly transformed into heat. It is obvious that this area, and hence also the absorption of wave energy, will be greater the closer the wave period is to the relaxation time. A detailed mathematical analysis (not presented here) leads to the conclusion that in this case the greatest phase shift in a sinusoidal wave between density and pressure will occur. From formula (20) we obtain the following expression for the phase shift, denoted by \(\psi\):

\[ \tg \psi = \frac{\left(v_{\infty}^{2}-v_{0}^{2}\right)\gamma_{w}} {v_{0}^{2}w^{2}+v_{\infty}^{2}\gamma^{2}} . \]

From analysis of this formula it follows that \(\psi\) becomes zero in the limiting cases when \(\gamma=0\), \(\gamma=\infty\), and reaches a maximum at

\[ \gamma_{\max}=\frac{v_{0}}{v_{\infty}}\gamma_{w}. \]

Fig. 12 gives a graphical representation of this function. According to the theory set forth, the absorption of ultrasound in the region of maximum absorption should considerably exceed the values calculated on the basis of considerations of classical acoustics. In passing to a comparison of these theoretical conclusions with experimental data, it should first of all be noted that the experimental material available, both on the measurement of sound absorption in general and, in particular, in the region of ultrasonic frequencies, is not very extensive.

Fig. 12. Frequency dependence of the phase shift between pressure and condensation in a plane elastic wave.

The great experimental difficulties with which these measurements are associated compel one in some cases to be satisfied with knowledge only of the order of magnitude of the measured quantity; in other cases, an error of 50 and even 100% must be regarded as satisfactory.

Measurements of the absorption of ultrasound in CO\(_2\) were carried out by Abello, Pilemer, and Grossmann. The values obtained are shown in Fig. 13. The curve shown is theoretical, constructed on the basis of measurements of the dispersion of ultrasound by Kneser. The double arrow in the figure gives an indication of the limits of the measurement error (20%).

Fig. 13. Comparison of the theoretical curve of selective absorption in CO\(_2\) with experimental data.

Taking into account the above considerations concerning the degree of accuracy of the measurements, it must be acknowledged that the experimental material in general confirms the theory of selective absorption.

The comparison of the theory with experimental data relating to other gases—for example, N\(_2\)O, SO\(_2\)—in air, oxygen—is less clear, owing to the extremely small number of measurements. Nevertheless, it may be concluded that the theory set forth

the theory gives a considerably closer approximation to reality than classical acoustics, both as regards the magnitude of the quantities calculated with its aid and as regards their general variation as a function of frequency.

The considerations set out above made it possible to substantiate and observe such phenomena as the dispersion and selective absorption of elastic (sound and ultrasonic) waves, which had not previously been known to classical theoretical and experimental acoustics. The time of establishment of equilibrium for the oscillatory degrees of freedom, calculated on the basis of the observed dispersion, proved to be 100,000 times greater than the values obtained by means of the classical kinetic theory. These data compelled a considerable change in the customary ideas. The number of works devoted to the question touched upon in this article is continually increasing, which testifies to the undoubted interest that this new direction in contemporary physics has acquired. The fundamental idea running through all these works consists in using the nonequilibrium state to determine the time of occurrence of rapid processes. This method can be applied to the determination of such constants as, for example, the ionization constant, the dissociation constant, etc.

References

1. H. O. Kneser, Ann. d. Phys. 5, 11, 761, 1931.
2. —, —, 11, 777, 1931.
3. —, —, 16, 337, 1933.
4. H. O. Kneser u. I. Zühlke, Zs. f. Phys. 77, 619, 1932.
5. P. S. H. Henry, Pros. Cambr Phyl Soc. 28, 249, 1932.
6. I. H. Jeans, Phil. Mag. 6, 2, 638, 1901.
7. H. A. Lorentz, Arch. Néerland. 16, 1, 1881.
8. K. F. Herzfeld u. F. O. Rice, Phys. Rev. 31, 691, 1928.
9. D. G. Bourgin, Phil Mag 7, 821, 1929.
10. A. Einstein. Berl Ber. 380, 1920.
11. A. I. Rutgers. Ann d Phys. 16, 350, 1933.
12. O. Heil, Zs. f Phys. 74, 31, 1932.
13. W. Heilemeyer, Phys Rev. 34, 1184, 1929.
14. E. Grossman, Ann d. Phys. 5, 13, 681, 1932.
15. Eucken u Becker, Zs physikal. Ch. 13, 1933.
16. Frank u. Euken, Zs. physikal. Ch 13, 1933.