ULTRASOUND

E. Hiedemann, Kiel*

Contents:

I. Review

II. Experimental Methods

A. Ultrasonic Emitters and Receivers

1. Ultrasonic emitters
2. Ultrasonic receivers

B. Methods for Investigating the Sound Field (study of the distribution of direction and intensity, the speed of sound, absorption)

1. Investigation of the field with the aid of receivers
2. Obtaining an image of the sound field by mechanical methods
3. Optical methods
4. Ultrasonic interferometer
5. Measurements of absorption

III. Propagation of Ultrasound

1. Propagation and diffraction of ultrasound in a three-dimensional medium
2. Permeability and reflecting power of interfaces
3. Oscillating crystals and plates

IV. Speed of Sound and Absorption of Sound

1. Dispersion and absorption in polyatomic gases
2. Speed of sound in liquids

V. Actions of Ultrasonic Waves

1. Formation and action of gas bubbles
2. Cavitations
3. Mechanical effects
4. Coagulation phenomena in aerosols
5. Thermal actions

Bibliography

I. REVIEW

By ultrasound are meant sound waves whose frequency lies above the limit of audibility. The earliest ultrasonic investigations arose under the influence of various technical interests. In the first ultrasonic works of König^220^ and Edelmann^97^

* E. Hiedemann, Ergebnisse d. exakten Naturwiss., Bd. 14, Berlin, 1935. Translation, with minor additions, by N. N. Malov.

the aim was pursued of obtaining ultra-acoustic waves connected, moreover, with a physiological problem (Edelmann), namely, the determination of the upper limit of audibility. The works proposed by Lebedev^42 and which prompted Martens to undertake the development of improved ultra-acoustic sources pursued purely physical aims. The catastrophe of the Titanic contributed to Richardson’s^308 discovery of the possibility of locating icebergs by means of ultrasounds, and also clarified the significance of ultrasounds for underwater signaling. During the World War, ultra-acoustic technology acquired military significance (detection of submarines); work in this direction was begun by Shilovskii and developed by Langevin^226–230. Boyle, to whom we are also indebted for the development of underwater signaling by means of ultrasound, carried out after the war a large number of ultra-acoustic investigations pursuing purely physical aims (study of the propagation of ultrasound, directivity, sound permeability, diffraction phenomena, the velocity of sound, cavitation). The investigations of Wood and Loomis^374–377 drew attention to the chemical and biological actions of ultrasound, whose systematic study continues at the present time. Pierce’s^274–277 ultra-acoustic interferometer, which made it possible to obtain high accuracy in determining the velocity of propagation of sound, led to the discovery of the dispersion of sound in polyatomic gases. Theoretical and experimental works devoted to the dispersion of sound proved the significance of ultrasound for molecular physics. The action, discovered by Debye and Sears^93, and also by Lucas and Biquard^238, ^239, of ultrasonic waves as a diffraction grating for light led to the development of an entirely new experimental technique—an optical technique—which has not yet been fully developed, but has already yielded very valuable results.

Thus, ultrasounds are applied in such diverse fields that a detailed survey of all of them is impossible within the limits of a short article (cf. reviews^132, ^393, ^407). But for all fields of application of ultrasound the experimental technique is of great interest; it will therefore be considered in greatest detail. In doing so, special attention will be devoted to the new optical methods. Of the important physical results obtained in ultra-acoustic investigations, the phenomenon of dispersion, as the most important, will be considered in the greatest detail. As for the actions and technical applications of ultrasound, only the most essential of them will be briefly discussed.

II. EXPERIMENTAL METHODS USED IN WORKING WITH ULTRASOUND

a) Ultra-acoustic transmitters and receivers

1. Ultra-acoustic transmitters. For obtaining ultrasound, piezoelectric devices are used chiefly,

emitters. But devices used in the field of the acoustics of audible sounds may also be applied.

Mechanical emitters. The excitation of sound with frequencies up to 90 kilocycles, carried out by König^220, is at present only of historical interest. The excitation of longitudinal vibrations of rods by mechanical means likewise has no practical importance in the ultrasonic region, although recently Goldmann^171, at the suggestion of Ahlberg^5, developed a method that makes it possible to obtain very considerable sound energy at frequencies of 35 kilocycles, with the constancy of the frequency maintained to an accuracy of up to 2%. The Galton whistle constructed by Edelmann (especially in the first ultrasonic octave) has retained its importance up to the present time, since it is extremely simple to handle, the energy supplied to it is quite sufficient for many purposes, and, with proper precautions, it preserves a constancy that ensures the production of a whole series of investigations. The Galton whistle is applicable chiefly in those cases where one is interested in the transition from the audible region to the ultrasonic region, i.e., for determining the upper limit of audibility and for experiments adjoining this investigation. Under especially favorable conditions Edelmann succeeded in obtaining a frequency of up to 170 kHz, but Edelmann gives 100 kHz as the mean value of the upper limit of the Galton whistle.

Greater sound energy (over a broad range of frequencies) can be obtained by means of the air-jet emitter proposed by Hartmann^138–141. Hartmann used a jet of air issuing from a nozzle at a speed exceeding the speed of sound; in accordance with Prandtl’s well-known investigations, periodic changes of pressure arise along the jet. If a suitable resonator is placed in the region of increased pressure, it is set into very intense oscillations. By using resonators of various sizes, frequencies from several hertz to many hundreds of kilocycles can thus be obtained. Owing to the great power delivered by the Hartmann emitter, it is especially suitable for investigations requiring high intensities of ultrasound in air. According to the latest data, Hartmann succeeded in obtaining at 8 kHz an acoustic power of 0.5–1 kW; at 15 kHz the power was about 150 W (both values are given for air).

Hopfield^172 showed that very weak ultrasounds (see also^86) can be obtained by the collision of two gas jets issuing from different nozzles arranged at a certain angle. When a double nozzle with a water jet is used, owing to surface tension oscillating drops are formed, and at the same time the jet emits a weak tone of high frequency. Takeuchi and Sato^354 briefly indicated that they succeeded in obtaining high-frequency sound of great power by means of sirens; this sound was intended for secret military signaling.

Thermal emitters. If, through a direct-current arc, an alternating current is simultaneously passed, then, owing to the pres—

oscillations of the current strength will arise, as well as of the pressure. Thus the arc will emit sound with a frequency equal to the frequency of the alternating current. If the amplitude of the alternating current exceeds the strength of the direct current, then the sound frequency is doubled. Palaiologos\(^{261}\), at Martens’ suggestion, constructed such a device. He used a tube generator to obtain alternating current and reached frequencies of 2000 kHz.

Altberg\(^{4}\), at Lebedev’s\(^{282}\) suggestion, had long before obtained ultrasounds with a frequency of about 300 kHz by means of a spark gap in a damped oscillatory circuit. His apparatus was later used by Neklep­aev\(^{256}\) for measurements of sound absorption, although at the same time Dickmann\(^{94}\), at Martens’ suggestion, improved this method of producing sound by making use of the Poulsen arc generator.

Electrical radiators. Of the electroacoustic apparatus used in the range of audible frequencies, only a few can be used to produce ultrasounds, since in the majority of apparatus the natural frequencies cannot be raised sufficiently, or else the electrical losses increase too much as the frequency is increased. However, Mühlwert\(^{255}\) constructed an electrodynamic radiator—a ribbon telephone—applicable up to 200 kHz; the negligible radiated power (about 0.02 W) makes this radiator suitable only for a small number of special investigations. Other electrical radiators, used in signaling by lead wires and giving frequencies lying near the audible frequencies, are described in the specialized literature\(^{225, 241, 337}\).

Klaus\(^{80}\) showed that, under the action of an alternating electric field, liquids can oscillate; he proposed using high-frequency mechanical oscillations of the surface of mercury for ultrasonic investigations. However, up to the present time, among electrical radiators only magnetostrictive and piezoelectric ones are of importance.

Magnetostrictive radiators. Black\(^{35}\), Pierce\(^{275–277}\), and Vincent\(^{364–368}\) proposed magnetostrictive radiators having practical importance; all of them use, as is known, the deformation of magnetic bodies in a magnetic field, the so-called magnetostriction (see also \(^{335, 386, 388}\)). For ultrasonic radiators, rods or tubes of ferromagnetic materials are used which, under the action of an alternating magnetic field, are set into oscillation along their length. In addition, bias magnetization is often employed; it is important, first, for eliminating the possibility of frequency doubling (as is known, the magnetostrictive effect does not change sign when the direction of the field is reversed), and, second, for creating the most favorable conditions for obtaining maximum amplitudes, i.e., for choosing the appropriate operating point on the magnetostriction curve. It is, of course, necessary to achieve resonance between the exciting alternating field and the natural frequency of the mechanical oscillations of the rod, i.e., exci-

excite the rod at its natural frequency, best of all at the fundamental frequency \(^{408}\). To improve the radiation of sound, plates of considerable cross-section are placed on the ends of the magnetostrictive rod; their thickness must be sufficient for them to be able to vibrate like a piston diaphragm. For excitation, either an oscillator or a resonator circuit may be used \(^{72}\). In Pierce’s radiator the rod is placed partly inside the coil of the oscillatory circuit of a vacuum-tube oscillator, and partly in the feedback coil. Owing to the inverse magnetostrictive effect (a change in the intensity of magnetization under elastic deformations), self-excitation of the oscillations arises.

Because of the simplicity and low cost of manufacturing magnetostrictive rods, Pierce proposed using them as frequency standards, especially in the range of several tens of kilohertz, where the use of piezoelectric resonators is not only expensive but also very difficult. According to detailed investigations by Giebe and Blechschmidt \(^{121}\), magnetostrictive rods can be used as frequency standards only in those cases where exceptionally high accuracy is not required. The point is that, owing to the change in the modulus of elasticity (and with it the natural frequency) upon magnetization, the resonance curve of a rod proves to be more diffuse than that of quartz; moreover, the temperature coefficient of the rods is rather large.

Magnetostrictive radiators can thus be used in those regions where the use of piezoelectric radiators proves too expensive, inexpedient, or altogether impossible, i.e. in the range of audible frequencies and ultrasonics up to approximately several tens of kilohertz; this range is also accessible to the Galton whistle. The advantages of magnetostrictive radiators over the Galton whistle consist in the possibility of obtaining considerably greater intensity, greater constancy of energy and frequency, in the possibility of using them in liquids, and in the ease of producing plane waves. Suitable end pieces of the rods, vibrating like piston diaphragms, create a plane wave, and diffraction phenomena can be made very small, since the size of the end piece can be chosen sufficiently large in comparison with the length of the ultrasonic wave. In addition, the Galton whistle has yet another major drawback, namely the strong influence of the steady jet of air.

Piezoelectric radiators. As is known, the piezoelectric effect is understood as the property of certain crystals to undergo mechanical deformations in an electric field, explained by the displacement of the ionic lattice of the crystal. These deformations arise in the case when the electric field has a component in the direction of a certain definite axis, called the electric axis of the crystal.

If, for example, from a quartz crystal one cuts a plane-paral-

a plate whose edges are parallel to the electrical \((x)\), optical \((z)\), and the axis perpendicular to the first two \((y)\), then, under the action of an electric field in the direction of the \(x\)-axis, there arises, for example, an extension accompanied by a simultaneous compression in the direction of the \(y\)-axis. When the direction of the field is changed, the plate will be compressed in the direction of the \(x\)-axis; in the direction of the \(y\)-axis, extension will then arise. If we create an alternating field directed along the \(x\)-axis, the quartz will enter into mechanical vibrations whose frequency coincides with the frequency of the alternating field. Since the resonance of freely vibrating quartz proves to be extremely sharp, the amplitude of the vibrations will be considerable only when the quartz is excited at its natural frequency. The use of quartz as a frequency standard is based on this principle. Owing to the sharpness of the resonance curve, in many cases it proves useful to control the frequency of the exciting alternating field, usually produced by a tube generator, by means of the same quartz, or else to stabilize the generator frequency by the latter. In the first case the quartz, according to Cady’s proposal \(^{78}\), is included in the grid circuit; in the second case—in the anode circuit (Pierce circuit). For ultrasonic investigations in gases both connections are applicable, although the Pierce circuit is used more often. For investigations in liquids, where strong damping of the quartz vibrations arises, these connections prove inapplicable; in such cases it is necessary to drive the quartz into forced vibrations.

Simple wetting of quartz with oil, as Sokolov showed, causes a strong increase in damping. A number of convincing experiments \(^{153,168}\) demonstrated a sharp smoothing of the resonance curve when quartz is placed in a liquid.

In gases, when standing waves arise, the reaction on the quartz may be so great that its vibrations cease. Therefore, when working in gases, quartz is increasingly often used as a resonator, and it is necessary to employ a tube generator stabilized by another quartz, tuned to resonance with the first.

In addition, it often proves useful \(^{159,161}\) to employ quartz provided with a variable holder which, by changing the air gap between the quartz and one of the electrodes, makes it possible to vary the frequency of the quartz within small limits; these changes are determined from a calibration curve.

For exciting ultrasonic vibrations, both vibrations along the \(x\)-axis (thickness vibrations) and vibrations along the \(y\)-axis (length vibrations) may be used. In the first case, round or rectangular plates cut perpendicular to the electrical axis (\(x\)-cut) are used, or else plates of special shape proposed by Straubel \(^{345}\). For this cut there exist three natural vibrations, one of which occurs in the direction of the electrical axis, and the other two—in the \(yz\)-plane; the latter form angles of \(71\) with the \(z\) direction.

and \(-48^\circ 25'\). These directions are the directions of the maximum and minimum modulus of elasticity in the \(yz\) plane. To create a uniform distribution of amplitudes over the entire surface of the plate, its dimensions must be chosen so that the radius in each direction is proportional to the square root of the modulus of elasticity. For the same reason, to obtain longitudinal vibrations, quartz rods are used, cut in such a way that their length does not coincide with the direction of the \(y\) axis (\(90^\circ\)—rods), but forms an angle of \(71^\circ\) with the \(z\) axis (\(71^\circ\)—rods). Such rods have a uniform distribution of amplitudes over the whole surface of the base, which oscillates like a piston membrane \(^{77,129}\). Besides the \(x\)-cut, others may also be used, provided only that the electric field has a sufficiently large component in the direction of the \(x\) axis.

Plates vibrating in thickness are usually used for frequencies exceeding 300 kHz. Such plates can be made with a fundamental frequency of 50 MHz; at higher frequencies one must use overtones, or else turn to tourmaline plates, whose fundamental frequency can be brought up to 1500 MHz.

Below 300 kHz, plates vibrating longitudinally are used, provided that no considerable radiated power is required. In the case of high powers, plates vibrating in thickness are used; these prove to be very large and thick, as a result of which their price is very high \(^{376,113}\); for ultrasonic underwater signaling, a mosaic of quartz plates of equal thickness, glued to two steel plates, is used \(^{49}\).

To concentrate an ultrasonic beam emitted by a plate with a large surface, Gruetzmacher proposed using a spherical plate, the concave side of which creates a converging beam of ultrasonic rays, whose focus can easily be found experimentally; experiments have shown that concentration of the beam makes it possible to obtain an increase in the density of ultrasonic energy by many tens of times \(^{387}\). The radiation of energy in one direction can be increased if the radiator, located in a liquid, is provided on the opposite side with an electrode bounding a space that is empty or filled with gas. Here use is made of the increase of the reflection coefficient at the electrode–gas boundary, owing to which the transfer of energy into the liquid from the opposite surface of the radiator is improved.

2. Ultrasonic receivers. Mechanical receivers. The most important mechanical receiver, capable at the same time of measuring intensity quantitatively, is the acoustic radiometer. As is known, it consists of torsion balances, one arm of which carries a light reflector; the other arm is loaded with a corresponding counterweight. If the reflector is placed perpendicular to the direction of propagation of sound waves,

which are completely reflected from its surface, then the torque produced by the sound waves proves to be proportional to the sound pressure. The angle through which the suspension head must be turned in order to bring the reflector to the zero position (i.e., perpendicular to the direction of sound propagation) is also proportional to the sound pressure. The reflection coefficient of the plate depends on the difference between the acoustic densities of the plate itself and of the medium surrounding it, and also on the thickness of the plate. As the difference in acoustic densities increases, the reflection coefficient rises. If the thickness of the plate is an odd integral multiple of a quarter of the wavelength, the reflection is maximal. In liquids, where the difference between the acoustic densities of the plate and of the surrounding medium is comparatively small, care must be taken that the thickness of the plate be exactly equal to one or several quarters of a wavelength (in the plate). Detailed investigations of the operating conditions of the acoustic radiometer in a liquid belong to Boyle and his collaborators ^50,56–68,62,65,68; they also proposed the most rational forms of the radiometer. In order to make use of the advantages of a large difference in acoustic densities, Boyle employed reflectors consisting of two mica plates separated by an air gap. If the radius of the reflector is not too large in comparison with the wavelength, then distorting diffraction phenomena arise; they must either be taken into account by introducing the appropriate corrections, or else eliminated, as Altberg proposed, by placing the reflector in the plane of a reflecting stationary wall. In working with a radiometer, errors are possible owing to the influence of a steady current of air, which practically always proceeds from the sound source. A drawback of the radiometer is also its relatively low sensitivity. Therefore attempts are now being made to replace the radiometer by more perfect instruments, although in the first ultra-acoustic studies the radiometer was widely used as a measuring instrument.

Relative measurements of sound intensity in a liquid were carried out by Richards ^283–291 with the aid of a capillary, to one end of which a small funnel of approximately exponential form had been soldered. The funnel was placed in the liquid, and the displacement of the meniscus in the capillary served as a measure of the sound intensity.

Dust figures are still used to prove the formation of standing waves. Krenke ^221–223 used for this purpose thin-walled glass tubes containing a small quantity of sand. If the frequency of the ultrasonic waves was close to the tube’s natural frequency, the latter entered into natural vibrations, which were evident from the intense motion of the sand particles.

Ultrasound modulated by an acoustic frequency can be detected by simple mechanical methods. Boyle, Lehmann, and Morgan ^59,60 used a manometer closed by a mica-

washer and connected with a telephone (the ultrasound propagated in the liquid).

Unmodulated ultrasound, as Yagi and Matsuo \(^{380}\) showed, can be made audible by means of an acoustic heterodyne. These authors produced beats between a suitable ultrasound and an ultrasonic frequency close to it, generated by a heterodyne; the difference tone was made audible owing to its rectification by means of a thin glass tube.

Richardson \(^{306}\) made use of the mechanical action of ultrasonic waves on a thin oil film. He observed an interference pattern in the light reflected from the oil film; when ultrasonic waves passed through the film, the interference pattern changed—the fringes became broader.

Thermal receivers. In many works \(^{77,252,303—307}\), a heated Wollaston wire was used as the receiver, its resistance changing under the action of ultrasound. The chief advantage of this method is the negligible size of the receiver, which makes it possible to investigate the field at individual points. Changes in the temperature of a heated wire are known from the works of Giebe \(^{169}\), as well as Wetzmann and co-workers \(^{118,119,123,251,254}\). For ultracoustic waves the steady cooling effect plays a large role. Here two effects must be distinguished: nodal and oscillatory. The nodal effect, noticeable only in weakly heated wires, consists in a change in the resistance of the wire due to temperature oscillations caused by successive compressions and rarefactions arising in standing waves at the nodes of motion. In the case of strongly heated wires, cooling predominates, caused by the moving particles of air, which is maximal at the antinodes of motion (the oscillatory effect).

One may also use heating due to absorption of ultracoustic energy. Thus, Richards \(^{292}\) constructed a sound-absorbing thermopile which could register energy of the order of \(0.01\ \mathrm{W}\) per \(\mathrm{cm}^{2}\). Malov’s method \(^{244}\) is based on the same effect; he investigated the propagation of ultrasound in a liquid not by the cooling of a heated wire, but by the heating of a cold wire. In addition, Malov used a thermoelement, which gave poorer results. Johnson \(^{189}\) prepared, by cathodic sputtering of bismuth and antimony onto cellulose, a very sensitive thermoelement, which he tested at frequencies up to \(5\ \mathrm{kHz}\). From theoretical calculations he concludes that the range of frequencies in which this thermoelement can be applied extends up to \(300\ \mathrm{kHz}\). Möller and Kreft \(^{253}\) showed that a flame can be used for rectifying and listening to two superposed ultracoustic waves of approximately the same frequency (demonstration of the Doppler effect).

Electrical receivers. Boyle constructed for underwater signaling a carbon microphone, usable at fre-

up to 42 kHz. Sakerdot^323 proposed a condenser microphone operating in the frequency range up to 90 kHz.

At higher frequencies only magnetostrictive and piezoelectric receivers are applicable. In the first case the inverse magnetostrictive effect is used, i.e. a change in the intensity of magnetization under elastic deformations. If ultrasonic waves fall on a membrane fastened to the end of a magnetostrictive rod, then the latter is set into mechanical vibration; the resulting changes in the intensity of magnetization produce induction currents in a coil surrounding the rod; these currents can be detected with a suitable amplifier. Pierce, in his patent,^276 indicates a whole series of applications and methods of using this effect. In practice, magnetostrictive receivers have found application in new designs of echo sounders.^225 In addition, the inverse magnetostrictive effect can be used in an acoustic interferometer. In work with interferometers containing quartz, the inverse piezoelectric effect of quartz is used. In many studies piezo-quartz was used only as a receiver, so that the whole instrument contained two tuned piezo-quartz plates.^2, 3, 49, 129, 144, 191, 381, 382 Piezoelectric receivers were first used by Langevin and later by Hayes,^49 and recently by Marro^248 for underwater telephony, in which an ultrasonic wave was used as the carrier wave.

B. Methods of investigating the sound field (study of the distribution of direction and intensity, velocity of sound, absorption).

1. Investigation of the field by means of receivers.

For investigating the sound field at individual points at low frequencies, mechanical receivers (radiometers, etc.) may be used. At high frequencies the dimensions of these receivers become too large in comparison with the wavelength, as a result of which electrical receivers must be employed.^244 A substantial shortcoming of all these methods is the distortion of the field by the receiver, which is made in the form of a thermoelement or a resistance thermometer.

In standing waves their distribution can be investigated by moving the receiver.^77, 244, 252, 305 In the case of traveling waves, as Altberg showed, one can make use of a diffraction grating.^4, 94, 261 An exceedingly elegant method for measuring the wavelength in traveling waves was proposed by Richardson.^306 He uses two wires, the distance between which is continuously varied. If it amounts to half the wavelength, then, owing to the adiabatic temperature oscillations in the sound wave, the two wires, falling into regions with opposite phases, behave differently; if the distance is equal to a whole wave, then both wires fall into regions where the oscillations occur in identical phases. In this way it is possible to determine

distance between two points with identical or opposite phases, i.e. the wavelength. Buks and Müller^77 found that the accuracy of measuring wavelength (in standing waves) with wire receivers approaches the accuracy attainable with an interferometer; Richardson, however, considers his method still more sensitive. The application of piezoelectric receivers to the measurement of wavelength will be discussed in connection with absorption measurements.

1. Obtaining an image of the sound field by mechanical methods. The old methods of Kundt’s and Chladni’s dust figures are also applicable in the field of ultrasound. In investigating the velocity of sound in solid media, standing waves can be made visible with the aid of dust figures; this method has not lost its significance up to the present time.

In gases, dust figures make it possible to find regions of maximum intensity^129, 133, 345; in studying propagation in tubes filled with gas, the condensation and coagulation effects are used for the same purpose. Thus, Buks and Müller determined the distribution of standing waves from the condensation of an alcohol mist in the velocity antinodes. Brandt and Freund^72 and Pearson^262, independently of one another, simultaneously discovered the coagulation and subsequent settling of tobacco-smoke particles in a standing sound wave. The settling particles produced a distribution analogous to Kundt’s dust figures. Pearson used this method to measure the dispersion of sound. It seems to the author that Pearson’s method is hardly applicable to very precise measurements, since an influence of the settling smoke particles on the velocity of propagation is possible. At the author’s suggestion, his collaborators, Brandt and Freund, investigated the possibility of using, for precise measurements, Dvořák’s method, recently improved by one of these authors. In Dvořák’s experiment there is used the phenomenon that, in a standing wave arising in a tube, owing to the finite amplitude of the sound wave, there appears a periodic spatial distribution (with a period of half a wavelength) of the pressure (time-averaged). According to Dvořák, when an indicator reacting to the time-averaged pressure distribution—namely a layer of liquid—is placed in the tube, the standing waves can readily be detected from small elevations of the liquid surface in the velocity antinodes. Brandt and Freund^74 proceeded from the assumption that the curved elevation formed on the surface of the liquid would act as a cylindrical lens. Therefore, with suitable illumination of the tube, a system of bright parallel bands is obtained at the corresponding places, the sharpness of which is illustrated by Fig. 1. If, in doing this, a liquid with negligibly small vapor pressure is used, it appears possible to improve this method for precise measurements.

To detect standing waves, Boyle^55 poured coke dust into a liquid. It settled slowly, collecting on the nodal surfaces. Collecting it on a horizontal plate, he obtained

pattern of the distribution of nodal lines, and, in the case of propagating waves, the directional characteristic of the radiation ^195. Boyle and his collaborators used this method to carry out a whole series of investigations of the propagation of ultra-acoustic waves and of their velocity. No detailed account of this method is given here, since at the present time Giedeman and his collaborators ^7, ^8, ^157–^168 have developed optical methods of considerably greater sensitivity. Boyle proposed still another method for observing nodal surfaces: at sufficiently high energies, under suitable conditions, gas bubbles are liberated from the liquid; the bubbles collect at the nodal surfaces and, under favorable conditions, can be used to measure the distance between these surfaces ^47.

Fig. 1. Standing wave in air, after Brandt and Freund.

At very high frequencies (5 MHz) this phenomenon was investigated by the author and Seifen; it turned out that at such a high frequency the method is no longer applicable as a measuring method, but it can be used, for example, to determine the maximum amplitude of oscillations at the surface of quartz.

3. Optical methods. Theoretical part. The discovery of the action of ultra-acoustic waves as an optical grating was made independently and almost simultaneously by Debye and Sears ^93, and by Lucas and Biquard ^238, ^239; this discovery led to the development of a whole series of new, extraordinarily fruitful methods.

The impetus for such investigations came from the results of Brillouin’s theory of light scattering in solids ^75. Just as Debye in his theory of specific heats took into account the influence of the thermal motion of neighboring atoms, considering the thermal motion of each atom as the result of the superposition of sound waves, Brillouin in his theory of light scattering introduced the concept of thermal elastic waves. Brillouin himself also pointed out that it would be very interesting to investigate the scattering of light by artificially produced sound waves. These considerations, as well as the increase in the intensity of the Lauegrams given by quartz under piezoelectric excitation, discovered by Fox and Carr (see also the new work of Clauer ^198), gave impetus to the investigations of Lucas and Biquard.

According to Brillouin’s theory, the scattered ray is regarded as the result of optical reflection from sound waves of a suitable

directions, i.e., from moving reflecting surfaces. Therefore a Doppler effect must arise, owing to which the primary ray is split into two components, the frequencies of which prove to be greater or less than the frequency of the primary ray by the amount of the frequency of the sound wave. In the free thermal motion of molecules, for example in gases, the scattering of light can be accompanied only by a broadening of the spectral lines. Therefore the investigation of the fine spectral structure of Rayleigh scattering should decide the question of whether the thermal molecular motion in liquids is analogous to motion in gases, or whether it more closely resembles thermal motion in solids. After Gross first discovered the splitting of spectral lines, at Debye’s suggestion an investigation was carried out by Meyer and Ramm281, who found that in toluene the scattered light gives not a doublet but a triplet, the middle line of the triplet having a frequency equal to the frequency of the primary beam. During his stay in America Debye, together with Sears, investigated this scattering of light from the standpoint of Brillouin’s theory on artificially produced sound waves in a liquid.

Fig. 2. Diagram of the optical apparatus, according to Debye and Sears.

Debye and Sears, as well as Lucas and Biquard, used the apparatus shown in Fig. 2. In a cell filled with liquid, quartz was placed which produced a plane wave propagating along the length of the cell. Light passing from a narrow illuminated slit, as in the usual grating experiment, was converted by a lens into a beam which passed through the cell parallel to the wave front of the ultrasonic waves; a lens placed behind the cell gave an image of the beam on a screen. According to Brillouin’s theory, the diffraction pattern of the slit was to appear in the form of a doublet, whereas according to the experiments of Meyer and Ramm one should expect the occurrence of a triplet. However, it turned out that the diffraction pattern was very similar to the pattern given by an ordinary diffraction grating, and its intensity, as well as the number of spectra obtained on both sides of the zero-order spectrum, were very large. Debye and Sears at first supposed that the appearance of spectra of higher orders could be explained by the influence of overtones of the ultrasonic waves, but Lucas and Biquard showed that this supposition was incorrect. Later Debye gave a theory of the scattering of light by sound waves, supplementing Brillouin’s theory; from this theory it followed that, in calculating the scat-

in the first approximation spectra of the first order are obtained; in the second approximation spectra of the second order can be obtained, and so on. From his theory it further followed that spectra scattered in the direction of the sound beam must have a frequency equal to the frequency of the primary beam plus the sound frequency multiplied by the order of the spectrum; whereas for scattering in the direction opposite to the propagation of the sound beam, an analogous decrease in frequency must occur. The wavelength of the ultrasonic wave was taken as the constant of the ultrasonic grating; such a grating could then be calculated in the usual way. The theory was able to explain the considerable intensity of the observed interference pattern only in general terms, and requires further extension.

Brillouin \(^{76}\) somewhat later improved his theory and was also able to explain the occurrence of spectra of higher orders. He was able to indicate a method for calculating the distribution of intensity in spectra of different orders; however, these calculations involve such considerable mathematical difficulties that they have not yet been carried through to completion. From his theory there naturally follows a change in frequency due to the Doppler effect. For a light beam whose dimensions are small in comparison with the wavelength of the sound wave, according to his theory, only a slight spreading of the beam should be produced, which agrees with the results of the experiments of Lucas and Biquard.

A considerably more visual and fruitful treatment of this question was given by Lucas and Biquard \(^{238}\), who proceeded not from an exact calculation of the scattering field, but from simple ideas of geometrical optics. The course of their reasoning is approximately as follows. If a sound wave propagates in the direction \(z\) (the wavelength being equal to \(\Lambda\)), then at some instant of time a periodic distribution of the refractive index arises, determined by the relation

\[ n = n_0 + \Delta n \cdot \cos \frac{2\pi z}{\Lambda}. \]

Using the known laws of the optics of inhomogeneous media, one can calculate the curvature and trajectory of a light ray entering a medium penetrated by ultrasonic waves parallel to their front. In doing so it is assumed that the distribution of the refractive index does not change during the passage of the beam through the liquid, which is quite legitimate, since the speed of sound is considerably less than the speed of light. Figure 3 shows the trajectories of the rays. It is seen from the figure that the light rays leave the sound field as if they had entered it as narrowly bounded beams. The lines of convergence of the rays lie at a distance of one sound wavelength from one another; they form a grating propagating with the speed of sound \(v\Lambda\) in the direction of the \(z\)-axis. For the distance \(y\) of the first line of convergence from the plane in which the light beam enters the sound field, one obtains

...one obtains an expression determining the dependence of this distance on the wavelength and the sound intensity, or, what is the same thing, on \(\Delta n\):

\[ v=\frac{\Lambda}{2\pi}\cdot K_{0}\cdot \sqrt{n_{0}}\cdot \frac{1}{\sqrt{\Delta n}}. \]

For a given sound wavelength, the quantity \(v\sqrt{\Delta n}\) remains constant; hence it is clear that the position of the convergence line can serve as a measure of the sound intensity.

Giedemann and his collaborators\(^{157-16}\), in particular Bachem\(^{7,8}\), showed that at the places where the rays converge one can observe very

Fig. 3. Path of light rays in a medium permeated by sound waves, according to Lucas and Biquard.

bright bands both in the case of standing waves and in the case of traveling waves. For traveling waves studied under stroboscopic illumination, this is completely understandable from the point of view of Lucas and Biquard. The appearance of a similar pattern in the case of standing waves, however, requires explanation. Since the compression maxima over the course of a half-period change their magnitude from zero to a maximum and again to zero, the convergence lines during a half-period must move from infinity to some minimum distance and then again go off to infinity. If, as a first approximation, one assumes that in standing waves as well the change in the refractive index is distributed sinusoidally in time, then from the Lucas–Biquard equations one obtains, for the time variation, an inverse proportionality to the square root of the sine. Therefore the convergence lines remain for a long time near the minimum distance, but in other regions must be very short-lived; consequently, the time-averaged pattern must give sharp bands. For measurement purposes it is very important to know the width of the bands, which depends on the position of the region of convergence. In one particular case Bachem\(^{8}\) was able to calculate the distribution of the regions of convergence for different instants of time. In good agreement with experiment, he [[unclear: word continues on next page]]

...led to the fact that for 50% of the time the eye sees a band whose width does not exceed the value \(\frac{\Lambda}{15}\), while during the remaining time, when the region of convergence moves rapidly, the bands are sharply weakened. Therefore, in stroboscopic observations the width of the visible band depends on the duration of illumination.

In standing waves a system of bands is observed, separated from one another by a distance \(\frac{\Lambda}{2}\); in traveling waves the distance is equal to the wavelength. To prove the change in the frequency of light required by the theory of Brillouin and Debye, Debye, Zach, and Coulon\(^{92}\) proposed the following method. They screened out all spectra except those of zero and first order, and then caused these spectra to interfere once more. If the resulting frequency changes were determined by the sound frequency, then in the secondary interference the intensity of the light should have varied periodically with the sound frequency; consequently, through a given point of the field of view there should have passed \(\nu\) interference fringes each second. If, however, observations were made with stroboscopic illumination at this same sound frequency, then the interference fringes should have remained motionless. An analogous effect should also arise for spectra of other orders, if one works with a suitable frequency of stroboscopic illumination. Thus the change of frequency in traveling waves could be proved with complete certainty. From observations with standing waves the authors concluded that here too one can prove a frequency change due to the Doppler effect. In both cases Debye, Zach, and Coulon used a scheme coinciding with the previously published scheme of Bachem and Hiedemann\(^{7,162,163}\), since according to Abbe’s theory the microscopic image of such a grating and the secondary interference coincide with one another. Bachem\(^{8}\) succeeded in showing in an elementary way that, when the grating is displaced with the speed of sound, the calculated Doppler effect has the same magnitude. In standing waves, however, according to his calculations, there should be no frequency change due to the Doppler effect. In Bachem’s opinion, the frequency changes arising in this case are caused by the following reason: since in a standing wave during a half-period the acoustic grating appears and then disappears again, the intensity of the diffraction spectra during a half-period increases from zero to a maximum and again falls to zero. Therefore the light in all spectra is modulated with frequency \(2\nu\), i.e., in addition to the fundamental frequency of the light, in all spectra there should also arise frequencies equal to

\[ \nu_{\text{light}} \pm 2m\nu_{\text{sound}}, \]

where \(m = 0, 1, 2, 3 \ldots\)

Berg\(^{12}\) discovered a remarkable periodicity in the distribution of intensity among spectra of different orders, illustrated by Fig. 4; this periodicity depends on the length of the segment traversed by the light in the sound field, on the intensity of the sound, and on the ratio

of the light wavelength to the sound wavelength. The author drew attention to the similarity between the photographs of Berg and those of Цубер[^384], who investigated the distribution of intensity in diffraction spectra obtained by means of gratings with the same constants, but with a varying transparent part. At the author’s suggestion, Bachem[^8] considered this case and arrived at results qualitatively coinciding with Berg’s results. Exact agreement can be obtained only after Brillouin’s theory has been developed for this case. The theory of the diffraction of light by an ultrasonic grating was also developed by Raman and his collaborators[^399–^401]. The same question is the subject of Rytov’s work[^406].

Fig. 4. Dependence of the intensity of diffraction phenomena on the intensity of ultrasound, according to Berg.

Experimental methods. Already in the first communications on diffraction phenomena obtained by means of an ultrasonic grating, measurements were made of the speed of sound in various liquids. From the measured angle of diffraction, using the well-known grating formula, it is easy to calculate the grating constant, equal to the wavelength. If a parallel beam enters a cuvette, it is easy to show that the refraction arising at the exit from the cuvette has no influence on the magnitude of the diffraction angle. If, moreover, the frequency is determined with the aid of a wave meter, then the speed of sound is readily calculated. Debye pointed out that this method should be very accurate, and proposed applying it to the measurement of the speed of sound in liquefied gases, and also to the investigation of the dependence of the compressibility of electrolytic solutions on concentration. Lucas and Biquard likewise determined from diffraction phenomena the speed of sound in a liquid and in quartz itself.

In the frequency region easily accessible to experiment, the sound wavelength is of the order of \(0.1\) mm; the diffraction angle is very small, and its precise determination is difficult. It is therefore understandable that the absolute accuracy of measuring the speed of sound by means of diffraction phenomena proves to be not very satisfactory. The obtained values of the speed differ from values measured by other methods by approximately \(1\%\). But relative measurements can be made with very high accuracy; moreover, one can photograph the diffraction patterns (when exciting different quartz overtones or changing the concentration of a solution) one above another on one and the same plate[^20,^22]. Cаваu[^351] succeeded in this way in attaining a relative accuracy of the order of \(10^{-4}\). The possibility of increasing the accuracy in measurements of high-order spectra is limited by the circumstance that the number of the resulting

spectra depends on the intensity of the sound. To obtain spectra of very high orders it is necessary to produce sounds of great intensity, causing considerable heating, as a result of which maintaining a constant temperature is very difficult. The attainment of high absolute accuracy is apparently possible only in a range of frequencies which at present are very difficult to excite. Moreover, the absorption of sound, increasing as the square of the frequency, may cause known errors.

Using diffraction phenomena, it is easy to detect the overtone of a quartz oscillator vibrating in a liquid1. Bergmann succeeded in showing that, if quartz excites several of its own natural frequencies simultaneously, they all appear in the diffraction patterns. He therefore proposed calibrating wavemeters by measurements of diffraction patterns2. It should be pointed out, however, that the smoothing of the resonance curve of quartz vibrating in a liquid is so great3 that large errors are possible. It is more rational to use, for this purpose, a quartz oscillator vibrating in air. In this case one can attain accuracies accessible to ordinary electrical methods, but this method presents a number of difficulties.

As Bär and Meyer showed4, by means of diffraction phenomena one can make the distribution of the sound field visible. For illumination they used not a slit, but a screen with a large number of small circular apertures. If the rays emerging from some aperture undergo diffraction while passing through a liquid penetrated by a sound beam, then a diffraction image of this aperture arises; moreover, the images of higher orders are situated on both sides of the image of zero order in the direction of sound propagation. Thus, the position of the diffraction pattern characterizes the direction of sound propagation. Since the number of diffraction images depends on the sound intensity, the distribution of intensity can be inferred from this same pattern. Bär and Meyer, using this method, demonstrated the reflection, refraction, diffraction, and absorption of ultrasonic waves. Fig. 5 shows the reflection and refraction of a sound beam at the boundary between two liquids.

Bär and Meyer, by causing sound waves to diffract on a wire grating and optically measuring the diffraction angles, measured the velocity of propagation of sound; the purpose of this investigation was to check measurements of the sound velocity with the aid of the diffraction of light at the same sound frequency. They found agreement between the two results within the limits of the rather considerable experimental errors. Bär5 used the same method to measure the velocity of sound relative to water from the angle of refraction of it in various media. With the aid of the Bär–Meyer method it can also be proved that maxima of permeability of a wedge-shaped plate exist6, whence the wavelength in the mate-

of the plate. Further, by this same method it was possible to study the distribution of amplitudes in front of quartz oscillating in a liquid[^155], and it became clear that, instead of a row of holes, it is more convenient to use a slit, so that the sound field can be seen as a whole, without gaps. It is also quite possible to use the Berg—Meyer method to demonstrate the ordinary phenomena of geometrical optics in a sound beam. Thus, photographs were taken[^156] which testify to the influence of concave and convex mirrors on the beam, as well as photographs of a number of other phenomena which were later investigated from another point of view[^27].

Fig. 5. Refraction and reflection of ultrasonic rays at the boundary between xylol (above) and water (below), according to Berg.

In front of the reflector, diffraction phenomena may arise, owing to the mutual superposition of sound waves; the resulting pattern corresponds to crossed gratings. In some portions of Fig. 5 such patterns are clearly visible. In their first paper Berg and Meyer presented an excellent photograph of diffraction from crossed gratings. Later, similar diffraction was studied by a number of other authors[^156,^326].

The possibility of producing several intersecting sound rays, which should create a three-dimensional grating in the liquid, and the possibility of using such a grating for studying diffraction phenomena were pointed out, independently of one another, by Schaefer and Bergmann[^326] and by Hiedemann and Asbach[^156]. The latter used a single sound ray which underwent several reflections from reflectors arranged at different angles; the diffraction pattern thus obtained resembled a Debye—Scherrer diagram. Schaefer and Bergmann, systematically

Fig. 6. Schaefer–Bergmann interference patterns.
1—Lauégram of a liquid penetrated by three mutually perpendicular sound beams.
2—Diffraction patterns of an oscillating glass cube.

those who studied diffraction from a spatial grating used a more perfect apparatus, in which the grating was produced by means of three piezoelectric quartzes of the same frequency; the grating obtained was strictly periodic. In Fig. 6₁ is shown a Laue diagram obtained in this way. In order to eliminate the distortion of the picture caused by strong heating of the liquid, Schaefer and Bergmann in a later paper ³²⁷ used glass cubes of various sizes, to which two or three, as far as possible identical, piezoelectric quartzes were glued. The interference patterns obtained in the vibrating cubes are also shown in Fig. 6. Schaefer and Bergmann ³²⁸—³³⁰ also used this method for the study of anisotropic media. We shall return to this question below.

Optical methods of observing the sound field. Tavill ³⁵⁶—³⁵⁸ used Toepler’s method (Schlierenmethode) to photograph a sound wave emitted by quartz. The sharpness of the image was very poor; no improvements of this method were made. Considerably better photographs had previously been obtained in the study of shock waves ¹⁰⁸; these photographs, while having no special value from the point of view of measuring technique, are excellent illustrations of certain propagation processes. It may be thought, however, that with further development of this method the possibility is not excluded of obtaining considerably better results, which could also be used for quantitative measurements. Recently, Pohlman ³⁵⁹ published photographs considerably superior in quality to Tavill’s photographs.

The methods described below ⁷, ⁸, ¹⁰²ᵃ, ¹⁵²—¹⁶⁸ give a considerably brighter and sharper picture than the other methods considered. They have already been used for studying propagation processes and can be applied for measurements with very high accuracy ¹⁸, ¹⁰²ᵃ, ¹⁶¹. The methods consist in using lines of convergence (in the sense of Lucas and Biquard) of light rays.

They differ, however, from the ordinary Toepler method in that regular changes of density are used as a kind of system giving an image. With a sinusoidal distribution of density changes, a system is obtained which acts as a series of cylindrical lenses arranged close to one another, so that a parallel beam of light, whose cross-section considerably exceeds the length of the acoustic wave, is collected by them in various lines.

The optical arrangement of this method differs from the Debye-Sears and Lucas-Biquard arrangement only in that behind the cuvette there is mounted a microscope focused on the lines of convergence of the rays in the cuvette, or on the lines of convergence lying outside the cuvette. The possibility of observation at a considerable distance from the cuvette was demonstrated by Winkelmann’s investigations, carried out with an ordinary grating ³⁷³. The distance between neighboring lines is equal to half the wavelength only in the case where the light beam entering the sound field is strictly parallel. This requirement must be exactly

to be observed in all investigations of the ultrasonic field and, in particular, when measuring the distance between the lines.

Reliable investigations of the sound field by the method of Berg and Meyer are likewise possible only with a beam incident on the cuvette almost in parallel, since otherwise the paths of the rays in the sound field will be incorrect.

Measurement of the image of an ultrasonic grating was used by Bachem and Giedemann instead of measuring diffraction spectra, since from measurement of the grating constant one can obtain substantially more accurate results than from measurement of the diffraction angle. These authors \(^{161}\) achieved an accuracy in measuring the length of the sound wave of the order of \(10^{-4}\), and in the most favorable cases even \(10^{-5}\); these figures refer to relative measurements. As for absolute measurements, owing to the elementary method of determining the frequency the accuracy was only about \(1\%\). But under suitable experimental conditions one may expect that the indicated method will make it possible to attain an absolute accuracy of the order of \(10^{-4}\), and in individual cases the possibility of increasing the accuracy by another factor of 10 is not excluded. Thus this method proves to be the most accurate one in the frequency region presently in use.

Since the distance between the fringes is exactly equal to half the wavelength only when the light beam is strictly parallel—a condition that can be attained only with great difficulty—a measuring method was developed in which the parallelism of the light beam did not play an essential role. In this method the measuring cuvette was moved on the carriage of a comparator while the positions of the microscope and the light source remained unchanged. In this way the setting of the crosshair on individual fringes always took place under the same optical conditions. Even with imperfect parallelism of the light beam, the distance between fringes obtained in this manner was exactly equal to half (or a whole) wavelength. It was then possible to use a large measuring interval, which helped to increase the accuracy of the measurement. The parallelism of the glass walls of the measuring cuvette plays here a less essential role than in photographing diffraction spectra. But high measurement accuracy requires exceptional constancy of temperature. To eliminate possible local heating near the quartz, and also to shield the measuring space from the electric field, the emitting quartz is separated from the measuring space and a whole series of precautions is employed. The temperature gradient that forms near the quartz may entail an increase in the distance between the fringes \(^{159}\) as a result of additional bending of the trajectories of the light rays as they pass through a thermally nonuniform medium. Figure 7 gives an example illustrating the distribution of fringes near the quartz. Furthermore, the parallelism of the quartz and the ref-

of the reflector, and it must be borne in mind that the sharpness of the resulting pattern cannot serve as a criterion of parallelism. Indeed, sharp bands parallel to the reflector may be images of the amplitude maxima of mutually superposed waves^160.

The ultrasonic grating can likewise be photographed in transparent solids^157, 158, 165–167; in this case, too, a high accuracy of absolute measurements can be achieved.

4. The ultra-acoustic interferometer. The acoustic interferometer, proposed by Pierce^274, is based on the following simple principle: the radiating surface is placed parallel to a plane reflector. When the distance between the reflector and the emitter is varied, the reaction of the reflected

Fig. 7. Temperature gradient near quartz; frequency 3300 kilocycles, according to Seifen.

sound wave changes periodically with a spatial period equal to half the wavelength. The maxima of the reaction are so sharp that they permit very high measurement accuracy. Without special difficulty the accuracy can be brought to several per mille; under especially favorable conditions it can be increased to tenths of a per mille. Hubbard^175–185, Herzberger^147–150, and Pielemeier^264–273 took part in the development of the theory of the interferometer. Hubbard proposed an equivalent electrical circuit of the interferometer, which is a development of the equivalent circuit of piezo-quartz proposed by Dye. From his theory it follows that, when the position of the reflector is changed, the following electrical quantities vary periodically: first, the anode current of the generator; second, the frequency; then, the current flowing through the quartz, and the voltage applied to the quartz. Analogous results can also be obtained for magnetostrictive emitters investigated with the aid of the interferometer. It turns out that, for a strict periodicity of the reaction of the reflected sound wave, exact parallelism of the reflector and the emitter is necessary, as well as the existence of a plane wave. If the diameter of the radiating surface is not sufficiently large in comparison with the wavelength, then diffraction phenomena must be taken into account; as a result of their action, the distances between the maxima of the reaction in the immediate vicinity of the quartz differ from half the wavelength^285, 286. The corrections required in these cases were calculated by Graby^126 and Grossmann^130–131.

According to Burgers ^43 it is also necessary to take into account the influence of absorption of the sound wave over the distance between neighboring nodal lines; however, for the most part this influence is so slight that it may be neglected. Near the quartz the intensity of the radiation is greatest, as a result of which, under suitable conditions, considerable errors may arise if the medium has large absorption. The possible effects of sound diffraction, as well as of the temperature gradient, may be manifested most sharply near the quartz. Therefore, in measurements, the distance between the quartz and the reflector should not be chosen too small.

For measuring displacements of the reflector or the quartz there are two different methods. Either the reflector is displaced by means of a micrometric screw ^274, or it is raised and lowered by means of a suitable mercury float, changing the height of the mercury level ^204. The latter method has the advantage that it does not depend on possible errors of the micrometric screw.

For determining the maxima of the reaction in most works dealing with gases, measurement of the anode current is used. Ziolke ^385, at Knezer’s suggestion, developed an objective recording device making it possible simultaneously to record changes in the anode current and the corresponding displacements of the reflector.

The most accurate method used in the study of liquids is the method of Freyer, Hubbard, and Andrews ^116, ^117, based on the idea proposed by Hubbard and Loomis ^182–185. The maxima of the reaction are determined from changes in frequency, the control of which is easily carried out by means of the beat method, giving very high accuracy.

Wiss ^379, at Greinacher’s suggestion, developed an ultrasonic interferometer in which the maxima of diffraction phenomena arising on an ultrasonic grating served as the indicator. According to the author, the accuracy of the measurements reached 0.2‰. The difference in the velocity values found by Wiss deserves attention, arising when the same apparatus was used once as an interferometer, and another time as a wavelength meter from diffraction spectra. Chester-Swanson ^347, ^349 described a special interferometer for high pressures. Andrews ^6 described a reflectometer used at frequencies up to 4 MHz over a very broad temperature range: from −63° to 150°C. Klein and Hershberger ^199–201 showed in what way, with a certain improvement of the interferometer, it can be used for measuring the velocity of sound in solids, and also with small quantities of liquid. The use of the interferometer for measuring absorption will be discussed in the corresponding section of the article.

Before the development of optical methods of measurement, the acoustic interferometer was the most accurate instrument for studying ultra-

sound. In the frequency range up to 1500 kHz it is still the most suitable.

1. Absorption measurements. The absorption of ultrasound in the case of plane waves is determined from the decrease in the intensity of sound with distance from the source; in the case of spherical waves it is determined from the decrease proceeding faster than inversely proportional to the square of the distance. Thus, for measurements it is necessary to have a constant source of sound and a sound receiver permitting quantitative measurement of the intensity. For example, one may use a sound radiometer \(^{1,29,30}\), but it should be borne in mind that very substantial errors may be caused by a constant source of error in radiometric measurements—the existence of a steady air current.

Richardson \(^{305}\) determined the absorption of sound by means of a heated wire; the latter also reacts to a steady flow (cooling effect). In Richardson’s absorption measurements \(^{289}\), in which he used a thermoelement surrounded by a strongly sound-absorbing medium, the influence of a steady flow was also noticeable. By placing a barrier between the sound source and the measuring space, one can eliminate the steady flow coming from the source; Abello \(^{1,2}\) separated the measuring space from the sound source by a thin celluloid film.

If piezoquartz is used as the receiver, the source of error indicated above disappears. Abello \(^{2,3}\) used a plane wave produced by quartz; the receiver was likewise quartz, tuned to the same frequency. Grossmann \(^{129}\) shielded the surface of the emitting quartz until the emitted wave ceased to be spherical; he measured the absorption by observing deviations from the \(1/r^2\) law.

A very interesting method for determining the velocity of sound and absorption was recently described by Igli \(^{381,382}\). He used an apparatus externally resembling an interferometer, but instead of a reflector he employed a receiving quartz, positioned relative to the emitting quartz (of course tuned to the same frequency) in such a way that one of its edges was farther from the emitting quartz by \(\Delta/4\) than the other. In this way he eliminated acoustic resonance, since the influence of each twice-reflected sound ray was compensated by another ray, likewise reflected twice but having the opposite phase of oscillation. The energy absorbed by the receiving quartz was converted into electrical energy and fed to an amplifier. At the same time, a certain constant and controlled amount of electrical energy from the generator circuit feeding the emitting quartz was supplied to the amplifier. By means of a tube voltmeter the (amplified) vector sum of the two alternating voltages corresponding to electrical and acoustic reception was measured. When the distance between receiver and emitter was changed, the phase difference between the electrical and acoustic components changed, and these changes varied periodically with a period corresponding to the length

wave. From this it was possible to calculate the speed of sound. Owing to the absorption of sound, the acoustic component decreased with increasing distance, as a result of which the value of the maximum observed when the phases of the two voltages coincided, or of the minimum arising when the phases were opposite, changed. From these data it was possible to calculate the absorption. Apparently, the elimination of acoustic resonance was quite complete, since no periodicity corresponding to half a wavelength could be detected on the recorded curves. The accuracy of the method was estimated by the author as \(1^{0}/_{00}\) for the speed of sound, and \(1\%\) for the absorption. Since the accuracy of absorption measurements is usually considerably lower, further study of the present method is highly desirable. The absorption, varying as the quartz is moved away, can be calculated from observations made with an interferometer, by reducing the reverse effect on the quartz. After such measurements had first been carried out by Pielemeier \(^{268}\), the interferometric measurement technique was greatly improved by Hubbard \(^{175,176,179,181}\) and Hiedemann \(^{150}\).

Optical methods have also been used or proposed for measuring absorption in liquids. However, up to the present time there is not a single method that has been developed to such an extent that its application would have practical significance. Biquard \(^{32,33}\) made measurements in which light from a zero-order spectrum fell on a photocell. He assumed that, as the intensity of the sound increases, the amount of diffracted light also increases, so that at small intensities the photocell current should decrease proportionally to the increase in sound intensity. To this it may be objected that the periodicity, discovered by Berg \(^{12}\), in the distribution of intensity among spectra of different orders makes the above assumption not entirely probable. Since the exact law of the distribution of light intensity among spectra of different orders is still not known, the question of the validity of the assumption made by Biquard can be decided only after a sufficient amount of experimental material has been accumulated. The values of the absorption coefficients found by Biquard are very remarkable. At \(7.558\ \mathrm{kHz}\) they were approximately 100 times greater than the values that could be expected from the classical theory. Bachem and Hiedemann \(^{161}\), in measurements of the speed of sound at a frequency of \(5.200\ \mathrm{kHz}\), found no deviations of the speed from the values obtained at lower frequencies (the accuracy of the measurements was of the order of \(1^{0}/_{00}\)); this circumstance makes the above-mentioned large values of the absorption coefficients very puzzling. Zak \(^{324}\) proposed measuring absorption in solids by Biquard’s method. As a measure of the sound intensity he took the ratio of the intensities of the zero- and first-order spectra observed in front of the absorbing solid and behind it. However, the still unstudied distribution of intensities among spectra of different orders makes this method also somewhat doubtful. However, Zak found that a good crystal of common salt posses—

gives less absorption than a crystal having inhomogeneities. This result agrees with Sokolov’s data\(^{340}\), who used this property to study the quality of various specimens. For the investigation of steel blanks Sokolov used observation of the diffraction pattern obtained in a liquid through which ultrasonic waves, produced by piezo-quartz and first passed through the specimen under investigation, were propagated. In the presence of flaws in the specimen, irregular reflections and other effects arose, considerably increasing the absorption of energy, as a result of which the intensity of the diffraction pattern fell sharply. Theoretically, the method of measuring absorption from the position of the line of convergence of light rays is beyond reproach. It was investigated in 1933 by Bachem, who tried to use it for measuring absorption; however, the experimental difficulties encountered have not been overcome to this day. Somewhat later, independently of Bachem, Baumgart\(^{17}\) proposed an analogous method. The chief difficulties of this method consist in the fact that a sharp image of the lines is obtained at a certain finite depth; it should further be pointed out that, according to Winkelmann’s data\(^{273}\), images of the grating can be observed in different planes. It is possible that the optical method proposed by Lucas\(^{237}\) will prove more reliable. In Lucas’s opinion, the angle \(\alpha\) at which a light beam (whose dimensions are small in comparison with the length of the sound wave) entering a medium penetrated by ultrasonic waves, parallel to the wave front, leaves this medium is a function of the change in refractive index \(\Delta n\), the value of the refractive index \(n\), the length of the sound wave, and the thickness of the medium \(y\). Over a rather wide range there exists proportionality between \(\Delta n\) and \(\operatorname{tg}\alpha\). From this one can calculate the absorptive and reflective capacity of the medium.

If a light ray is subjected to the simultaneous action of two crossed sound beams, Lissajous figures arise, which, when the frequencies are equal, have the form of ellipses. From the ratio of the principal axes of the ellipse one can determine the ratio of the intensities of the sound beams, from which the absorptive or reflective capacity of a given medium can be calculated. Practical applications of this method have not yet been published. Nevertheless, one may think that one of the optical methods will prove to be the most accurate method for measuring the absorption of sound.

III. Propagation of Ultrasound

1. Propagation and diffraction of ultrasound in a three-dimensional medium.

In the ultracoustic region the wavelengths are so small in comparison with the dimensions of sound sources, various obstacles, etc., that here, much better than in the region of audible frequencies, directed beams are obtained, undistorted by diffraction.

As is known, the impetus for Langevin’s work was the consideration that, with the aid of ultrasonic rays, it would be possible to create directed signals, and that, by means of their reflection, it would be possible to determine the position of a reflecting object. Boyle \(^{45,71}\), in his fundamental works, repeatedly pointed out that with the aid of ultrasonic waves it is extremely convenient to investigate various details of directional phenomena, as well as diffraction phenomena in the immediate vicinity of a sound source or some obstacle. He pointed out that by means of ultrasonic waves one can study the details of diffraction phenomena, for example near a diaphragm, in a region constituting a fraction of a wavelength or a small number of wavelengths, which, as is known, is completely impossible in analogous cases of optical phenomena. For diaphragms or sources whose dimensions are very large or very small in comparison with the sound wavelength, the corresponding phenomena have been studied fairly well; of special interest are those cases when the dimensions of these objects are commensurable with the wavelength. Boyle and Reid \(^{61}\) carried out systematic investigations of the intensity distribution for such radiators. For a region separated from the source by a distance considerably exceeding the wavelength, they established formulas analogous to the formulas for optical diffraction at a circular aperture. Experimental investigation of the sound field gave results in good agreement with the theoretically calculated intensity values. For a region separated from the source by a small number of wavelengths, Boyle could make use only of Lommel’s calculations. It turned out, however, that the experiment did not agree with the theory. Namely, Boyle and Reid \(^{61}\) established that in the immediate vicinity of the source the intensity distribution depends on the individual features of the sound radiator (in these experiments a quartz mosaic glued to steel plates was used).

Using the dust-figure method, Boyle \(^{49}\) investigated the directed action of sound radiators and obtained directivity characteristics that agreed well with theoretical calculations. As is known, for points considerably removed from the source, it follows from the formulas of diffraction theory that a ray diffracted by a diaphragm forms with the axis passing through the center of the diaphragm an angle determined by the expression:

\[ \sin \theta = 1.22 \frac{\lambda}{D}, \]

where \(D\) is the diameter of the aperture. The needs of technical acoustics (for example, the theory of the loudspeaker) led to the detailed development of the corresponding theory even before Boyle’s works \(^{193–194}\). Thus, for example, Strutt \(^{346}\) and Stenzel \(^{343,344}\) calculated the directional characteristics for various membranes and apertures. Such characteristics can easily be detected in air with the aid of dust figures \(^{127}\). In Fig. 8 are shown dust figures, po-

emitted from the slit, whose width is 2.65 wavelengths (at a frequency of 100 kHz). The figure shows good qualitative agreement of the boundaries of the dust figure with the directivity characteristic. The dust figures are formed under the influence of radiation pressure, and also of a steady air current known as the quartz wind. If only the radiation pressure acted, then in front of the slit aperture there would be formed a dust-free surface whose boundaries would be determined by curves of equal amplitude of the sound pressure.

Fig. 8. Dust figure in front of a slit of width 2.65 wavelengths; frequency—100 kilocycles. Below, for comparison, are shown the envelope of the surface free of sand and the directivity characteristic according to Grossmann and Hiedemann.

Backhaus \(^{9,10}\) succeeded in giving a solution (for the case of a circular piston membrane) for points whose distance is not too large in comparison with the dimensions of the membrane. Grossmann \(^{130,131}\) and Rüdy \(^{319,321}\) made use of Backhaus’s solution for certain special cases. In Fig. 9 is shown the principal section, calculated by Grossmann, of the wave field produced by a piston membrane of radius equal to 1.75 wavelengths. The heavy lines represent surfaces of equal phase of pressure, i.e. wave surfaces. The thin lines connect points with the same pressure amplitude \(P\). The numbers indicated in the drawing characterize the ratio \(P/P_0\), where \(P_0\) defines the pressure that would arise in a tube whose cross-section is equal to the cross-section of the membrane if the membrane oscillated not in free half-space, but in this tube. It should be noted that, as is seen from Fig. 9, the curvature of the wave surfaces surrounding the piston membrane is not everywhere positive, but may at certain points also be negative.

Grossmann calculated this case, since measurements with an ultrasonic interferometer showed that near the oscillating quartz the maxima of the reaction do not possess the periodicity corresponding to half the wavelength. Discussing the computed wave field, he showed that these deviations are due to diffraction phenomena, depicted in Fig. 9. He gave an approximate formula that makes it possible to calculate the minimum distance between the source and the reflector which

Fig. 9. Wave field of a circular piston membrane of radius \(1.75\lambda\), according to Grossmann and Giedemann.

Fig. 10. Ultrasonic wave emerging from a slit of width \(2\) wavelengths, according to Grossmann and Giedemann.

Fig. 11. Sound field in front of a rectangular piston membrane. Quartz width \(4.85\) wavelengths, frequency \(3300\) kilocycles, according to Giedemann and Asbach.

must be chosen when working with an interferometer. The formula also made it possible to estimate the errors associated with this phenomenon. Further, it should be noted that the characteristic minima and secondary maxima arise already at small distances from the membrane. Recently \(^{133}\) an attempt was made to photograph these curved wave fronts. In this case an ultrasonic wave emerging from a certain aperture was photographed (Fig. 10). The experimental conditions made it necessary to use a rectangular aperture formed by two closely spaced wedge-shaped obstacles. In this process distorting reflections occurred in front of the slit, but nevertheless all the wave fronts visible beyond the aperture clearly prove the existence of curvatures, pre-

stated by Grossmann. In later, as yet unpublished investigations by Hiedemann and Asbach, an attempt was made to photograph the propagation of sound in front of a quartz plate oscillating like a piston membrane. The great influence of the mounting of the quartz on the distribution of amplitudes (in thickness oscillations), established both optically and in Malov’s work \(^{244}\) by means of a resistance thermometer, gave grounds for using a plate cut in a very unusual way. With the aid of this plate the photographs shown in Figs. 11 and 12 were obtained. In Fig. 11

Fig. 12. Sound field in front of the membrane shown in Fig. 11, at high sound intensity, according to Hiedemann and Asbach.

the wave field was photographed at a very small excitation (far from the resonance frequency of the quartz); here the curvatures are quite clearly visible. In Fig. 12 the wave field of the very same quartz is shown, but at a considerably larger amplitude of oscillation. In this case as well, the minima and subsidiary maxima are quite clearly visible; in addition, the wave fronts in this photograph prove to be strongly deformed; the latter phenomenon has not yet received a definitive explanation.

There is reason to suppose that the density changes arising in the quartz wind are connected with changes in the temperature of the quartz; apparently these changes, like the quartz wind, propagate in the direction of the maxima. If this is so, then the exceptionally good agreement between the calculated directional characteristics and the experimentally obtained dust figures becomes comprehensible. If the supposition—also confirmed by other observations—that under the action of a steady stream small changes of density arise is correct, then they can be detected optically, and thus the direction of propagation of this steady stream can be studied in detail; up to the present time there have been no exhaustive investigations of this phenomenon, although a number of works have been devoted to it \(^{355, 369, 370}\).

The photographs of the sound field presented here make it possible to think that this optical method may also prove suitable for co-

...of quantitative study of the process of sound propagation. Boyle had already attempted to solve various questions of propagation by using an ultrasonic field, the dimensions of which in many cases are convenient for experiment; however, the sharpness of the field images obtained by him was satisfactory only for a few special cases. It should be noted that quantitative experiments in an ultrasonic field (but not by means of the optical method) were carried out by Goldmann \(^{124}\), who investigated the reaction of a funnel-shaped obstacle.

2. Permeability and reflecting power of interfaces. In the simplest case of a sound wave incident on the interface between two media, the problem was solved already by Rayleigh \(^{283}\), who gave both a general formula and its expression for particular cases. Boyle and Rawlinson \(^{62}\) continued Rayleigh’s calculations. Boyle and his co-workers succeeded in proving the validity of Rayleigh’s formula for the ultra-acoustic region. For perpendicular incidence on an interface, the ratio of the reflected energy to the incident energy is determined, according to Rayleigh, by the expression:

\[ R=\left(\frac{\rho_1 V_1-\rho_2 V_2}{\rho_1 V_1+\rho_2 V_2}\right)^2, \]

where \(\rho\) and \(V\) represent the densities and the velocities of sound in the corresponding media. Thus the reflecting power \(R\) depends only on the difference of the acoustic densities of the two media. When the acoustic densities differ greatly (for example, water—air), the permeability \((D=1-R)\) is very small; when the differences are small (water—aluminum), it is relatively large. Boyle and Taylor \(^{68}\) investigated the reflecting power of several materials of interest for underwater signaling (ice, steel, rocks) and obtained good agreement with Rayleigh’s formula. Boyle and Rawlinson developed the theory for the case of oblique incidence of a ray on an interface and calculated a number of special cases.

Of great importance is the case of three spatially separated media, where the middle medium has a thickness comparable with the wavelength. In this case the most interesting situation is when the first and third media are identical, so that the second medium represents a wall separating them. According to Rayleigh, for the case of perpendicular incidence on a wall we have:

\[ R= \frac{ \left(\dfrac{V_1 \rho_1}{V_2 \rho_2}-\dfrac{V_2 \rho_2}{V_1 \rho_1}\right)^2 }{ 4\,\operatorname{ctg}^2 \dfrac{2\pi l}{\lambda_T} + \left(\dfrac{\rho_1 V_1}{\rho_2 V_2}+\dfrac{\rho_2 V_2}{\rho_1 V_1}\right)^2 } \]

where \(l\) is the thickness of the wall, and \(\lambda_T\) is the wavelength of the ultrasound in the material of the wall. From this formula it is evident that, for all thicknesses equal to an integral multiple of half the wavelength, the permeability is maximal and the reflecting power minimal. The opposite phenomenon occurs for thicknesses equal to an odd integral multiple of a quarter wavelength. Boyle and Rawlinson calculated this equation for the case water—duralumin—water. Neglecting energy absorption, they obtained the diagram shown in Fig. 13,

characterizing the dependence of \(R\) and \(D\) on the wall thickness, with the ratio of the wall thickness to the wavelength in the wall plotted along the abscissa. The solid curve represents the course of \(D\), while the dotted curve characterizes the changes in \(R\). The transmission maxima are very sharp, whereas the minima are very diffuse. Rayleigh’s law is of great importance for ultrasonic investigations. In all experimental work in which it is required to transmit through a wall the maximum amount of sound energy, and also in those cases where maximum reflection is required, as, for example, when working with an acoustic radiometer, Rayleigh’s law must be taken into account. Furthermore, the sharpness of the transmission maxima may be used to determine the wavelength in the material of the wall. Boyle and his collaborators \(^{48, 50, 56, 57, 65}\) carried out a careful investigation of the conditions of applicability of this law. In some investigations radiometer vanes of different thicknesses were used; in others—with an unchanged radiometer—walls of different thicknesses were used.

Fig. 13. Dependence of the transmission and reflecting power of a separating wall on the ratio of the wall thickness to the wavelength, according to Boyle—Rawlinson.

They found an exceedingly good agreement of theory with experiment. Of course, complete agreement was not to be expected, since Rayleigh’s formula was derived on the assumption that the dimensions of the wall (along the boundary) are very large in comparison with the wavelength. If this is not the case, then, in addition to everything else, diffraction phenomena arise, especially pronounced in those cases where the dimensions of the radiometer vane amount to a small number of sound wavelengths. Boyle and Lehmann \(^{58}\) investigated this case both theoretically and experimentally. Further, it should be taken into account that the sound waves arising in the wall propagate not only as longitudinal waves; for this reason there arises, first, a decrease of the maximum reflection, and, second, the velocity of propagation of these waves proves to be different from the velocity of propagation of longitudinal waves. Boyle and Sproule \(^{65}\) found that the velocity of sound calculated from the maximum transmission agrees well with the velocity of sound determined by other methods if, in the latter, the modulus of volume compression is introduced instead of the modulus of linear extension. Instead of changing the wall thickness, one may also vary the frequency of the oscillations. Hiedemann and Asbach \(^{154}\) developed an optical method for determining the thickness corresponding to maximum transmission. They used frequencies of \(10\,000\ \mathrm{kHz}\); in this case it proved convenient to use a wedge as the wall, since the number of diffraction spectra obtained, produced by the ultrasonic grating behind the wedge, made it possible to determine the position of the transmission maxima. Behr and Valty \(^{15}\)

they used this method to study glass plates. By the same method they investigated the maxima of permeability of plane glass plates, the normals to which made various angles with the direction of propagation of the sound. In these cases the intensity of the transverse waves in the glass increased, which made it possible to determine Poisson’s ratio. Boyle and Rawlinson generalized Rayleigh’s law to the case of oblique incidence of a ray on a wall, neglecting the complications arising from the formation of transverse waves.

Malov and Rzhevkin ^245, and also Sokolov ^339, studied transmission; in their experiments agreement with Rayleigh’s law was not obtained. They came to the conclusion that sound conductivity is to a considerable degree determined by the vibrations of the wall as a whole. Since sound conductivity depends both on the actual passage of sound and on the vibrations of the wall, it would be very desirable to clarify in what cases one effect prevails over the other.

3. Oscillating crystals and plates. As has already been indicated, Schaefer and Bergmann ^326–329 in their investigations obtained Laue diagrams by means of visible light; they excited vibrations of glass cubes, and later also of various crystals, by gluing piezoelectric quartz plates to them. In Figs. 14₁—14₃ are shown the diffraction patterns of a circular aperture when light is propagated along the \(z\)-, \(y\)-, and \(x\)-axes. In Figs. 14₁₀—14₁₂ are shown diffraction patterns for barite. Schaefer and Bergmann, already in their first paper, pointed out that the interference patterns are closely connected with the symmetry of the crystal; this connection was explained by the investigations of these authors, as well as by the theory of Fues and Ludloff ^330.

Schaefer and Bergmann experimentally established the following: the interference patterns are exceedingly rich in interference spots. They possess low selectivity: there is no need to work in a certain fairly broad spectral region, as when taking X-ray Laue diagrams; in practice it is possible to work with homogeneous light. The observed phenomena do not depend on small rotations of the crystal relative to the light ray. The character of the interference patterns also does not depend on the external shape of the crystal, but only on the direction of illumination. With small vibrations of the exciting sound frequency the character of the interference pattern is preserved, but individual interference spots disappear, while new ones arise in their place. With a generator whose frequency is insufficiently constant, peculiar “scintillations” of the interference spots are observed. Further, Schaefer and Bergmann investigated a quartz cube in polarized light. When illuminated in the direction of the \(y\)-axis, the diffraction pattern split into two parts depending on whether the electric vector of the light wave was parallel or perpendicular to the optical axis of the crystal being illuminated. When illuminated in the direction of the optical axis they found no difference between the pattern obtained in

To E. Hiedemann’s article.

Fig. 14. Interference patterns of vibrating crystals according to Schaefer and Bergmann: 1—3—quartz, experimental patterns; 4—6—sections of the quartz lattice (experimental); 7—9—diffraction phenomena on a section of the quartz lattice (experimental); 10—12—barite, experimental curves; 13—15—quartz, theoretical curves; 16—18—barite, theoretical curves.

polarized or natural light. By means of the dark-field method they were able to photograph a section of a certain scattering surface of the crystal (see Figs. \(14_4\)—\(14_6\)). They succeeded in carrying out a harmonic analysis of such a section, for which they used this photograph as a grating for monochromatic light. The corresponding photographs, shown in Figs. \(14_7\)—\(14_9\), are repetitions of the figures obtained in Figs. \(14_1\)—\(14_3\), though somewhat imperfect and slightly distorted.

Fuss and Ludloff developed the theory of Schaefer–Bergmann interference figures, the basic ideas of which are set forth below.

The large number of observed interference spots makes it impossible to suppose that in these photographs we are dealing with a Laue pattern of a combined space grating (Translationsgitter). Indeed, in that case a large number of interference spots would have to correspond to an equally large number of interference orders.

If the faces of the crystal are quadrangular, then from the nature of the equations of elastic vibrations, in the first approximation, a harmonic distribution of density is obtained, i.e. only first-order interference. Further, the arrangement of the interference spots does not agree with that which might be expected for a Laue pattern. Photographing a section of the scattering surface does not give a simple two-dimensional combined grating (Translationsgitter), as it ought to if it were a section of a three-dimensional combined grating. On the contrary, it can be explained by the existence of several mutually superposed, intersecting gratings with different constants. These considerations lead to the following basic idea: the explanation of the phenomenon must be given from the point of view of elastic natural vibrations resonating with the exciting frequencies. The number of natural vibrations whose existence must be admitted turns out to be very large. For example, in the case of quartz, the resonance curve, although sharp, nevertheless has a finite width, the lower limit of which is the quantity

\[ \frac{\Delta \nu}{\nu} = 10^{-4}. \]

But in this spectral region there fit approximately one hundred different natural frequencies. The scintillation phenomena also speak in favor of this basic assumption. The experimentally established independence of the arrangement of the interference spots from the shape of the crystal indicates that the general course of these curves is little connected with the boundary conditions. Therefore, for a formal description of the phenomenon, one may restrict oneself merely to considering plane waves. Each natural vibration of a crystal of one structure or another may be regarded as a sum of elastic plane waves satisfying a double Fourier integral. The boundary conditions will affect only the amplitudes, i.e. the intensities. The next step in the reasoning consists in establishing the conditions determining the possibility of the occurrence of these plane waves in the crystal. Further, from

of all possible waves it is necessary to select the waves compatible with the given exciting frequency. Then it is necessary to determine what role each of these waves plays in the formation of the interference pattern. In this connection it may be said at once that only the longitudinal components of the oscillations should be taken into account. Such a wave creates a layered harmonic distribution of the scattering power, the distance between the layers being equal to \(d=\frac{2\pi}{k}\). According to Bragg’s equation:

\[ 2d\sin\theta=\lambda_i \tag{1} \]

such layers can reflect light of wavelength \(\lambda_i\) only at the angle \(\theta\). Thus a certain interference point is determined. From equation (1) and the relation between the wavelength of the elastic wave \(d\) and the light wave \(\lambda_i\), one obtains for the diffraction angle a value of \(10^{-2}\), corresponding to the experiment. Consequently, only those plane waves take part in the formation of the interference pattern whose propagation vector \(\mathbf{k}\) is approximately perpendicular to the primary beam; consequently, for the interference pattern only those oscillations are essential which, in the longitudinal direction (with respect to the primary beam), have not a single node, or not more than two nodes, which explains the slight selectivity.

An important condition leading to the possibility of determining the elastic constants is obtained by considering the question of which of the plane waves that can exist in the crystal are compatible with the exciting frequency. From the elastic potential of the medium one can obtain a system of vibrational equations for the displacement vector \(\mathbf{s}\). In the usual notation we have:

\[ \rho \ddot{s}_{\alpha'}=\sum_{\beta\alpha'\beta'} \frac{\partial}{\partial x_\beta} \left( c_{\alpha\beta\alpha'\beta'} \frac{\partial s_{\alpha'}}{\partial x_{\beta'}} \right). \tag{2} \]

To obtain a plane propagating wave

\[ \mathbf{s}=\mathbf{a}\cdot e^{i(\mathbf{k}\mathbf{r}-\omega t)} \tag{3} \]

it is necessary that the following condition be satisfied for the amplitudes and the propagation vector:

\[ \sum_{\alpha'}\left(\sum_{\beta\beta'} c_{\alpha\beta\alpha'\beta'} k_\beta k_{\beta'}-\rho\omega^2\delta_{\alpha\alpha'}\right)a_{\alpha'}=0. \tag{4} \]

Since this system of equations must be satisfied for \(a_\alpha\), the symmetric determinant of its coefficients must vanish, i.e., it must be:

\[ D(k_1 k_2 k_3,\ \omega)=0. \tag{5} \]

This equation determines a certain spatial surface of the sixth order, the surface \(F\). Introducing the direction cosines \(\alpha_1,\alpha_2,\alpha_3\), we obtain:

\[ D(k,\ \omega,\ \alpha_1,\ \alpha_2,\ \alpha_3)=0. \tag{6} \]

For given \(\omega,\alpha_1,\alpha_2,\alpha_3\), this equation is cubic with respect to \(k^2\). The condition determined by equations (5) and (6) is related to the numerical values of the elastic constants. Thus, for given elastic constants one can determine this condition, and conversely. In order to be able to calculate the interference pattern, it is necessary also to know the relation between the indicated condition and the interference figures. Fig. 15 shows one of such elastic waves causing the phenomenon of diffraction. The propagation vector \(\mathbf{k}\) and the primary ray lie in the plane of the drawing. The propagation vector must terminate on the surface \(F\). The interference ray, arising owing to Bragg reflection, goes at an angle \(2\theta\); upon emerging from the crystal, owing to refraction, the angle increases by \(n_0\) times. The point at which the interference ray falls on the photographic plate, located at a distance \(A\), proves to be displaced relative to the primary ray by a segment \(r\), determined by the expression:

Fig. 15. Formation of interference figures, according to Fues and Ludloff.

\[ r=A\cdot n_0\cdot \sin 2\theta, \]

which, owing to the smallness of \(\theta\), may be written in the form: \(r=An_0\cdot 2\theta\), or

\[ r=A\cdot n_0\lambda_i\cdot \frac{k}{2\pi} = A\cdot \lambda_a\cdot \frac{k}{2\pi}. \tag{7} \]

Since the vector \(\mathbf{k}\) is almost perpendicular to the primary ray, it is easy to see from the figure that the interference pattern gives geo-

metrically similar image of the section of the surface \(F\), perpendicular to the incident ray, magnified by a factor of \(\dfrac{A\lambda a}{2\pi}\). Thus, from the interference pattern one can calculate the section of the surface \(F\) perpendicular to the direction of incidence of the light ray, whence the numerical values of the elastic constants of the crystal can be determined.

Fues and Ludloff gave examples of calculations for isotropic bodies (a glass cube), trigonal crystals (quartz), and rhombic crystals (barite). In Figs. \(14_{13}\)—\(14_{18}\) the curves calculated according to their theory are shown. The excellent agreement with the diffraction patterns obtained experimentally is striking. The system of elastic constants (adiabatic) for barite determined in this way was compared with determinations of these quantities (isothermal) made by Voigt, it being assumed that the adiabatic constants should be equal to the corresponding isothermal ones. The agreement proved excellent. The enormous advantage offered by this method in comparison with the methods used earlier will be clear if one takes into account that in this method only a few photographs are required, whereas Voigt had to make 15 thousand separate measurements.

Another advantage of this method is the absence of elastic aftereffect and the possibility of determining the entire system of elastic constants on a single specimen of a crystal, which previously had been impossible. The accuracy of the method already now surpasses the accuracy of the former static methods. The relative accuracy in determining the elastic constants of one and the same system is exceptionally high. The absolute accuracy is close to \(1\%\).

The idea that, owing to the smallness of the diffraction angle on the wave grating formed by ultrasound, the speed of sound can be determined from measurements of the grating constants with a considerably greater absolute accuracy than from determination of the diffraction angle itself—this idea, which gave the impetus to the development of an accurate method for measuring the speed of sound in liquids \(^{161}\) by photographing the grating, also formed the basis of attempts to apply this method to solids. True, this method is limited by the circumstance that the grating constants can be measured only in one or two directions. Nevertheless even this is valuable, since measurement of the grating, performed with high absolute accuracy, makes it possible to increase the accuracy of determining the system of elastic constants measured by the Schaefer–Bergmann method with high relative accuracy.

Avtor, together with Gesche and Asbach \(^{157,158}\), succeeded in photographing the standing wave grating in piezoquartz itself; using very simple means, they succeeded in attaining a rather high accuracy of measurement.

In this connection it turned out that the distance between the bands in the oscillat-

in quartz being studied coincides very accurately with the ratio of the thickness of the quartz to the number of the harmonic. At present it is being checked on a larger body of material whether this agreement always occurs; if this is so, it would be sufficient to determine the distance between the bands (which is done by very simple methods) in order to establish the number of the harmonic with complete certainty. From the thickness of the plate and the number of the harmonic one can determine the wavelength, again using very simple means, and the accuracy obtained will be very great.

In measurements with quartz oscillating through its thickness it was further established that all the higher harmonics are in the proper relation to one another, but not to the fundamental frequency of the quartz. Bergmann\({}^{23}\), almost simultaneously, came to the same conclusion while studying diffraction phenomena.

Subsequently it proved possible to make visible a wave in a crystal, oscillating through its thickness, perpendicular to the direction of the field\({}^{165}\). Later\({}^{152, 166, 167}\), attempts were made to measure the ultrasonic grating in a piece of glass excited by piezo-quartz by the Schaefer–Bergmann method. The accuracy already attained at present in measuring the grating constant exceeded one per mille. At the same time it was also possible to measure a wave perpendicular to the direction of excitation. Of very great significance is the quite unexpected observation\({}^{166, 167}\): in investigations in polarized light it was found that, when a glass cube is illuminated with light whose plane of polarization is parallel to the front of the sound wave, and when observation is made with crossed polarizer and analyzer, a system of bands appears, the distance between which is considerably smaller than when observed in natural light or at another position of the plane of polarization. Attempts to detect this system of bands in natural light proved fruitless.

This circumstance permits the assumption\({}^{152}\) that these bands should not be regarded as an image of the compression wave in the sense of Lucas and Biquard, but as an image of the places of maximum stresses in the crystal, visible thanks to birefringence, in complete agreement with the classical investigations of Biot and Tyndall. It may further be supposed that this wave has no longitudinal component, i.e., should be regarded as a pure transverse wave. If one assumes that both observed waves are plane waves and takes into account (in the sense of the Fuchs–Ludloff theory) that the boundary conditions may practically be disregarded, then the velocities of propagation of both waves must be determined by equations applicable to an elastic medium unbounded in all directions.

Strict agreement is not to be expected, since these velocities are limiting values, which the observed velocities approach asymptotically as the ratio of the linear dimensions of the elastic medium to the wavelength increases. In addition to the Fuchs–Ludloff theory, such a method of consideration is confirmed by measure-

observations by Derfler ^95, who observed on quartz plates the transition of a flexural wave into a transverse wave. He found that a flexural wave arises only when the thickness of the plate is small in comparison with the wavelength. In addition, he established that for high harmonics, i.e., for small values of \(\lambda/d\), the velocity asymptotically approaches the velocity of transverse waves in an unbounded medium. In the case of longitudinal waves a similar assumption is also valid, as is proved by the behavior of the higher harmonics of quartz vibrating through its thickness ^23,158. In some works it was found that the nonharmonic deviations of the overtones are already very small at relatively low orders of the overtones ^122; hence one may conclude that the asymptotic approximation to the limiting case is in practice attained very rapidly.

Let us suppose that the distance between the fringes observed in natural light is exactly equal to one half of the wavelength of the longitudinal elastic wave propagating in an unbounded medium; and that the distance between the fringes observed in polarized light is equal to one half of the wavelength of the transverse wave. Then, from the measured distances and the known frequency, the elastic constants of the glass body can be determined. For one kind of glass which, according to Schott’s data, had a modulus of elasticity \(E = 7.471\) kg per square millimeter, there was found \(E = 7.531\) kg/mm\(^2\), a torsion modulus of \(3.119\) kg/mm\(^2\), and Poisson’s constant \(0.2072\) (at a frequency of \(4 \cdot 894\) kHz). If one takes into account that the modulus of elasticity was determined under different conditions—isothermally and adiabatically—then the agreement must be regarded as so good that the interpretation of the two waves as longitudinal and transverse may be considered proved. On other glass plates it was likewise possible to observe transverse waves in experiments with polarized light. The assumption that the fringes observed with crossed polarizer and analyzer should be associated with transverse waves makes clear the importance of such investigations. Recently, using the same apparatus, Giedemann and Hesch were able to observe diffraction phenomena in which the grating constant was determined by transverse waves; it is interesting to note that only first-order diffraction spectra were obtained. The dependence of the observability of transverse waves on the position of the plane of polarization corresponds to the data of one of Koenig’s old works ^390.

The method of measuring the grating in solids is apparently the most accurate and can be used for determining the elastic constants of transparent isotropic bodies. In the case of anisotropic bodies, however, one must use the Schaefer–Bergmann method, whose absolute accuracy can easily be increased by simultaneous use of the first method.

We shall not dwell here on the consideration of the propagation of ultrasonic waves in rods ^1,5,313,318,320 and in liquids filling tubes ^106,372, referring those interested to Grossmann’s article ^132, and also to recent works ^11,18,334.

IV. Speed of Sound and Absorption of Sound

1. Dispersion and absorption in polyatomic gases. Classical theory of the speed of sound and of its absorption in gases. Speed of sound.

As is known, the speed of sound can be calculated with the aid of Euler’s equation. Considering the propagation of sound as an adiabatic process, we obtain Laplace’s formula:

\[ V=\sqrt{\frac{p\gamma}{\rho}}, \]

where \(p\) is the pressure of the gas, \(V\) the speed of sound, \(\rho\) the density, and \(\gamma\) the ratio of the specific heats.

This equation does not imply the existence of a dependence of the speed of propagation on frequency—dispersion of sound. However, if one considers frequencies so high that the length of the sound wave becomes comparable with the mean free path of the molecules, then the assumptions used in deriving Euler’s equation are no longer admissible. Moreover, in this case the process cannot be regarded as adiabatic. Herzfeld and Rice \(^{151}\), and also Condon \(^{85}\), showed by elementary arguments that at such high frequencies the process proceeds isothermally; see also the work of Van der Dungen \(^{96}\). A detailed calculation of the frequency dependence of the thermal conductivity was given by Rocard \(^{309}\), who showed that, in the case of air, already at frequencies of \(5\cdot 10^{8}\) Hz the process changes from adiabatic to isothermal, i.e. Laplace’s equation must be replaced by Newton’s equation. Although this frequency range has practically not yet been reached, we have mentioned it because the state of affairs contradicts the usual notions—which are not entirely accurate—that the propagation of sound occurs adiabatically owing to very rapid transitions from compressions to rarefactions. Proceeding from this idea, one might think that the transition to an isothermal process should take place in the region of very low, rather than very high, frequencies. This phenomenon can be understood only by an exact investigation of the frequency dependence of the thermal conductivity. It is true that the time during which heat removal occurs increases as the frequency decreases, but the amount of heat removed per unit time is proportional to the magnitude

\[ \frac{\partial x}{\partial T}, \]

i.e., for a given amplitude, inversely proportional to the square of the wavelength. Therefore the heat removed during a period decreases with decreasing frequency, since it decreases with increasing wavelength to a higher degree than the period increases.

If the influence of internal friction, thermal conductivity, and heat radiation on the speed of sound is taken into account, then an exact calculation yields a frequency dependence of the speed of sound which, however, is so small that it cannot be detected with the present accuracy of measurement.

Until 1925 there were no experimental data on the frequency dependence of the velocity of propagation of sound. In that year Pierce \(^{274}\), working with an interferometer, measuring the velocity of sound in carbon dioxide at three different frequencies, found an increase of \(0.6\%\) at the highest frequency. An entirely undoubted increase of the velocity of sound in carbon dioxide, reaching \(4\%\), was measured much later by Kneser \(^{202,204,205}\) in the ultrasonic region. Subsequent experimental and theoretical works were devoted to the development of the theory of dispersion of sound in polyatomic gases; below we shall consider these works, taking into account the influence of intramolecular processes.

Absorption of sound. According to the old theory, absorption in gases must be attributed to three different factors, namely—the viscosity of the gas, thermal conductivity, and heat radiation. Since the absorption of sound due to radiation is very small in comparison with the absorption caused by thermal conductivity (about \(\frac{1}{200}\)), it is possible to confine oneself only to the first two causes. The decrease in the intensity of a plane wave, caused by absorption, may be represented by the equation: \(I_x = I_0 e^{-\frac{\alpha}{\lambda}x}\), where \(\frac{1}{\alpha}\) determines the number of wavelengths after traversing which the intensity falls to \(\frac{1}{e}\) of its initial value. According to the classical works of Stokes and Kirchhoff, the absorption coefficient \(\alpha\) is determined by the equation:

\[ \alpha=\frac{4\pi^2\nu}{V^2\cdot\rho}\cdot\left(\frac{4}{3}\eta+\frac{\gamma-1}{c_p}\cdot K\right), \]

here the first term characterizes the influence of internal friction, and the second—thermal conductivity; the sound frequency is denoted by \(\nu\), the gas density by \(\rho\), the coefficient of internal friction by \(\eta\), and the coefficient of thermal conductivity by \(K\).

Numerical calculation of \(\alpha\) from the known values of \(\rho,\eta\), etc. for the case of air and water shows that (at equal frequency) absorption in air is almost 1000 times greater than in water. Therefore in air ultrasonic waves can propagate only over very insignificant distances, whereas in water ultrasonic signaling covers enormous spaces. Multiplying by \(\lambda\), we obtain an expression independent of frequency:

\[ A=\alpha\cdot\lambda=\frac{4\pi^2}{V\cdot\rho}\cdot\left(\frac{4}{3}\eta+\frac{\gamma-1}{c_p}\cdot K\right). \]

Experimental investigations of the absorption of sound have shown that the experimentally established value \(A\) in many cases is significantly—

considerably exceeds the theoretical value and, moreover, has a frequency dependence. Only in monatomic gases do the two values agree fairly well with one another.

The theory of dispersion and absorption of sound in polyatomic gases. The explanation of deviations from the classical theory in polyatomic gases was given by taking intramolecular processes into account. In a polyatomic gas, besides the external degrees of freedom (translational motion), one must reckon with the presence of internal degrees of freedom (rotational and vibrational motions). In the equilibrium state, the number of molecules using, at a given moment, a particular internal degree of freedom (in other words, the number of molecules in which a particular vibrational or rotational motion is excited) depends only on the temperature. When a sound wave passes through a gas, adiabatic pressure oscillations are produced, and consequently also temperature oscillations. These temperature oscillations also give rise to oscillations in the concentration of excited molecules; it is assumed here that the time required for the establishment of concentration equilibrium is not too large in comparison with the period of the sound wave. If the concentration equilibrium is established in a finite interval of time, then at low frequencies the changes in concentration follow the temperature oscillations; at frequencies whose period is commensurable with the time of establishment of equilibrium, the amplitude of the concentration oscillations will decrease. Finally, at high frequencies, whose period is considerably smaller than the time required for the establishment of equilibrium, the concentration equilibrium will not change at all. But this means that the propagation of sound at very high frequencies will take place as though the corresponding degree of freedom did not exist at all. To calculate the velocity of propagation of sound at these frequencies, one must use an increased ratio of specific heats, i.e., the velocity of sound must increase.

Proceeding from these considerations, Herzfeld and Rice[^151] gave the first approximate theory of the dispersion of sound, taking internal friction and thermal conductivity into account. The construction of the corresponding theory, in which the influence of internal friction and thermal conductivity was not taken into consideration, is due to Bourgin[^38–^44], Kneser[^203], Rutgers[^322,^208], Richards[^249,^295], Grossmann[^129], and Henry[^145,^146], the last of whom gave the simplest derivation of the dispersion formula. Kneser’s work, although not the first, played the greatest role, since his arguments were distinguished by great clarity and he had at his disposal extensive experimental material that served to test his theory. Kneser made use of the equation for the reaction between excited and unexcited molecules; at the same time he also succeeded in using calculations previously carried out by Einstein[^98] for the case of sound propagation in a dissociated gas.* The assumption that in dissoci-

* Prunkhagen and Gens found no dispersion in partially dissociated \(N_2O_4\) at a frequency of 15 kHz. In the ultrasonic region, dispersion

Consequently, we obtain:

\[ C_\omega - C_\infty = (C_0 - C_\infty)\cdot \frac{\Delta n_1}{\Delta n_1\big|_{\omega=0}} = (C_0 - C_\infty)\cdot \frac{k_0+k_1}{j\omega+k_1+k_0}. \tag{11} \]

Whence

\[ C_\omega - C_\infty = \frac{C_0-C_\infty}{1+\dfrac{j\omega}{k_1+k_0}}. \]

Fig. 16. Dispersion and absorption of sound in polyatomic gases according to Kneser.

Using equation (4), we have:

\[ V= \frac{p}{\rho} \left( 1+ \frac{R}{ C_\infty+ \dfrac{C_0-C_\infty}{1+\dfrac{j\omega}{k_1+k_0}} } \right). \tag{12} \]

Owing to the smallness of the imaginary part, it may be assumed that the real part of \(V^2\) is equal to the absolute value of \(V^2\). Introducing

\[ \frac{1}{k_0+k_1}=\beta, \]

we find, after simple transformations:

\[ V^2= \frac{p}{\rho} \left( 1+ R\frac{C_0-C_\infty\omega^2\beta^2}{C_0^2+C_\infty^2\omega^2\beta^2} \right). \]

For very small frequencies \((\omega\to 0)\) we find

\[ V_0^2=\frac{p}{\rho}\left(1+\frac{R}{C_0}\right), \]

and for very large frequencies \((\omega\to\infty)\) one obtains

\[ V_\infty^2=\frac{p}{\rho}\left(1+\frac{R}{C_\infty}\right). \]

Constructing the dependence of \(V^2\) on the logarithm of the frequency, we obtain the upper curve of Fig. 16, the dispersion curve.

The dispersion curve has an inflection point at the frequency \(\omega_w=\dfrac{1}{\beta}\dfrac{C_0}{C_\infty}\).

The slope of the curve at this point is easily determined from the expression:

\[ \left.\frac{dV^2}{d\ln\omega}\right|_{\omega_w} = \frac{1}{2}\Delta V^2, \quad \text{where } \Delta V^2=V_\infty^2-V_0^2 . \]

It is a measure of the maximum dispersion arising in the region \(\Delta\ln\omega=2\), i.e. approximately within three octaves.

The existence of a complex value of the speed of sound means the presence of a phase difference between the pressure wave and the compression wave; \(\operatorname{tg}\varphi\) is equal to the ratio of the imaginary and real parts of \(V^2\).

The calculation gives:

\[ \operatorname{tg}\varphi = \frac{(V_\infty^2-V_0^2)\cdot\omega_w\cdot\omega} {V_0^2\omega_w^2+V_\infty^2\omega^2} \]

The maximum value of the tangent turns out to be:

\[ \operatorname{tg}\varphi_m = \frac{V_\infty^2-V_0^2}{2V_0V_\infty}. \]

It is obtained at the frequency:

\[ \omega_m = \frac{V_0}{V_\infty}\omega_w = \frac{1}{\beta} \sqrt{ \frac{(C_0+R)C_0}{(C_\infty+R)C_\infty} } \]

The width of the \(\operatorname{tg}\varphi\) curve at half height covers a frequency range of about 3.8 octaves.

The phase shift indicates energy losses due to absorption.

A visual interpretation of this phenomenon was proposed by Kneser in his thought experiment \(^{207}\). Generally speaking, for any sound wave it may be asserted that absorption can always be represented in the form of a phase shift between the pressure and compression waves; see, for example, Rockar’s work \(^{309}\).

Substituting into the equation for the intensity of a plane wave

\[ j_x=j_0\cdot e^{2j\omega\left(t-\frac{x}{V}\right)} \]

complex sound velocity \(V^2 = V_e^2 e^{i\varphi}\), we obtain:

\[ j_x = j_0 e^{2j\omega\left(t-\frac{x}{V\cdot e^{j\varphi/2}}\right)} \]

or, after transformations:

\[ j_x = j_0 e^{2j\omega\left(t-\frac{x}{V}\cos\frac{\varphi}{2}\right)} \cdot e^{-2\omega \frac{x}{V}\sin\frac{\varphi}{2}} \]

The second term, depending on distance, determines the attenuation of the wave. The absorption coefficient, referred to the wavelength, is determined by the expression:

\[ \alpha = 4\pi \sin\frac{\varphi}{2}. \]

But since

\[ \operatorname{tg}\varphi = \frac{2\sin\frac{\varphi}{2}}{1-\operatorname{tg}^2\frac{\varphi}{2}}, \]

then for all values \(\operatorname{tg}^2\frac{\varphi}{2} \ll 1\) one may take:

\[ \alpha = 2\pi \operatorname{tg}\varphi. \]

Since in all known investigations the angle \(\varphi\) is always less than \(6^\circ\), in calculating the absolute value of \(V^2\) it is quite legitimate to neglect the imaginary part in comparison with the real part, as was done above.

The curve \(\operatorname{tg}\varphi\) characterizes the course of the absorption coefficient. It is shown in Fig. 16 below, where \(\alpha = 2\pi \operatorname{tg}\varphi\) is assumed.

A simplified formula for absorption can be derived \(^{206}\), if it is taken into account that in most cases \(k_0 \ll k_1\), i.e. \(F e^{-h\nu/kT} \ll 1\). Neglecting \(k_0\) in comparison with \(k_1\) and \(\chi\) in comparison with unity, we have:

\[ \operatorname{tg}\varphi = \frac{R\omega k_1 (C_0 - C_\infty)} {k_1^2(C_0^2 + RC_0) + \omega^2(C_\infty^2 + RC_\infty)}. \]

Since \(C_0\) does not differ too greatly from \(C_\infty\), one may introduce a further simplification:

\[ \operatorname{tg}\varphi = \frac{R}{C_0(R+C_0)}(C_0-C_\infty)\cdot \frac{k_1\omega}{k_1^2+\omega^2}. \]

The maximum is obtained at $\omega = k_1$, and the maximum value of the absorption coefficient proves to be equal to:

$$ \alpha_{\max} = \pi \frac{R(C_0 - C_\infty)}{C_0(R + C_0)} . $$

This simplified formula has the advantage that in it $\alpha_{\max}$ does not depend on $\omega$ and $k_1$ and can be precalculated for each vibrational degree of freedom, if we have the corresponding spectroscopic data, calculating from them the difference of the specific heats $(C_0 - C_\infty)$ for the degree of freedom under consideration by the Planck–Einstein formula.

The position of the region of dispersion and absorption is determined by the quantity $\beta$. But $\beta$ represents the time for establishing equilibrium of the concentration of excited and unexcited molecules, i.e. in the case of a single vibrational heat capacity—the time for establishing this heat capacity; the quantity $\beta$ is also called the relaxation time. From equations (6) and (9) we find the relation:

$$ -\frac{\partial n_1}{\partial t} = \frac{n_1}{\beta} - k_0 N, $$

whose solution is the expression:

$$ n_1 = A e^{-\frac{t}{\beta}} + k_0 N \beta . $$

Thus $\beta$ represents the time during which some disturbance of equilibrium decreases to $1/e$ of its initial value. The quantities $k_1$ and $k_0$, from which $\beta$ is formed, represent, by definition, the number of transitions of a quantum into translational or rotational energy per unit time. The quantity $1/k_1$ therefore represents the lifetime of the vibrational quantum. It should be borne in mind that the lifetime of the vibrational quantum, generally speaking, is greater than the lifetime of the excited molecule, since the vibrational quantum can pass from one molecule to another; the quantum in this case continues to exist, while the excitation of the first molecule disappears.

From the equations

$$ \beta = \frac{1}{k_1 + k_0} $$

and

$$ x = \frac{k_0}{k_1} = F e^{-\frac{h\nu}{kT}} $$

and

From the computed \(\beta\) and the known vibrational heat capacities one can compute \(k_1\) and \(k_0\). If it is assumed that the transition from translational energy to vibrational energy and back occurs in a collision of the first kind, then the quotient obtained by dividing \(k_1\) or \(k_0\) by the number of collisions of a molecule per unit time determines the probability of destruction or creation of a vibrational quantum in a collision. But since the probability calculated for one collision, of course, cannot change, an increase in the number of collisions, i.e. an increase in pressure, must cause a decrease in the relaxation time; that is, it must be

\[ p \sim \frac{1}{\beta}. \]

This dependence of \(\beta\) on pressure is very important, since by means of a small number of constant frequencies (i.e. a small number of quartz plates) it proves possible to make a large number of measurements of \(\beta\). Thus a change in pressure acts so that the region in which the sound velocity increases when the pressure is changed is displaced parallel to the \(\omega\)-axis, and this displacement is proportional to the pressure. According to Wallmann’s measurements, this conclusion is confirmed within the limits of observational error.

To compute \(\beta\), the expression used is \(\beta = \dfrac{1}{\omega_w} \cdot \dfrac{C_0}{C_\infty}\); consequently, it is necessary to determine \(\omega_w\). If one takes one of the dispersion curves, measured at a sufficient number of points, then one can use the fact that the slope of the dispersion curve at the point \(\omega_w\) is equal to \(\dfrac{\Delta V^2}{2}\). If at a distance \(\dfrac{\Delta V^2}{2}\) between the limiting values of the velocity one draws a straight line parallel to the \(\omega\)-axis, then the point of its intersection with the dispersion curve will determine \(\omega_w\). But in many cases it is more expedient to compute \(\omega_w\) for each measured value of \(\omega\) and \(V\). In doing so, from the dispersion formula one can, as Wallmann showed, obtain the following expression:

\[ \omega_w^2 = \omega^2 \frac{V_\infty^2 - V^2}{V^2 - V_0^2}. \]

This expression has the advantage that \(V_0\) and \(V_\infty\) can be computed taking into account deviations from the ideal-gas state by the method of Eucken and Mücke \(^{99}\). The absorption coefficient can be computed from the difference between its experimental value \(\alpha_{\mathrm{exp}}\) and the value calculated from the classical theory. In deriving the dispersion formula we neglected the loss of energy by excited molecules through radiation. Knezer \(^{207}\) investigated this case in detail. He found that spontaneous emission or absorption of quanta plays an insignificant role in dispersion, but can have a large influence on absorption. However, we still do not possess sufficient

experimental material in order to clarify this question completely.

The theory set forth above considers the simplest case, assuming that there is one single energy level and one single finite time for the establishment of equilibrium. Of course, the simultaneous existence is possible of several quantum states of one vibrational degree of freedom, or of different degrees of freedom, with a finite time for the establishment of equilibrium, as has also been indicated in a number of works. The corresponding theoretical considerations are due to Richards^295 and Rose^314. As the frequency increases, different parts of the vibrational heat capacity drop out gradually, so that the speed of sound changes in jumps and various regions of absorption arise; it is assumed, of course, that the times for the establishment of equilibrium of the different quantum states are substantially unequal. However, a large number of dispersion steps and a number of absorption regions have not yet been observed experimentally.

This theory has already been tested on a large body of experimental material, and it has become clear that some ambiguities revealed in this test will require rather a certain deepening of the theory than fundamental changes in it.

Deviations from the theory appeared chiefly in works studying the temperature dependence of sound dispersion. Richards and Reid^296 introduced, for characterizing the temperature dependence, the concept of the “activation energy of collisions.” Richardson^306 pointed out that dispersion may be caused by selective (resonance) absorption and by an anomalous course of the viscosity at a very high frequency of oscillations. Kneser^206 concluded from Knudsen’s^216 investigations, devoted to the propagation of sound in oxygen and air, that the temperature dependence of the time for the establishment of equilibrium coincides with the increase in the quantity \(x\); in this case the relative kinetic energy of the colliding molecules may pass entirely into the energy of vibrational motions. To clarify the temperature dependence, still more experimental material is undoubtedly necessary; however, the data already available permit one to think that the theory set forth will have to be somewhat refined. There is reason to think that the formal considerations of Richards and Reid will hardly lead to satisfactory results; on the contrary, it is apparently necessary to carry out a more detailed investigation of the processes of molecular collision. Extremely interesting work in this direction has been begun by Eucken and Becker^102, and also by Huntington^187.

Pearson^262 found an anomalous course of the dispersion curve which, however, may be due to experimental errors not taken into account by the author. Relston and Richardson^279 also obtained an anomalous dispersion curve; the reason for the anomaly has so far remained unexplained. Despite the large number of works by various authors, the experimental material now available needs considerable supplementation. This is explained, first of all,

short duration of the existence of the theory of sound dispersion, and, secondly, because in many works sufficient attention was not paid to the purity of the gas, since the strong influence of impurities has become clear only recently. Below the most reliable results of investigations of sound propagation in gases are given in the briefest form.

Carbon dioxide. In carbon dioxide dispersion was first discovered by Pierce, whose work gave rise to a large number of further investigations of this gas. For the most part what was investigated was the establishment of vibrations of the deformation type (Deformationsschwingung, ↑O↓C↑O). Wallmann \(^{212,371}\) found at \(21^\circ\mathrm{C}\) and 600 mm Hg \(\beta = 4.6 \cdot 10^{-6}\) sec. This value agrees well with the value \(\beta = 5.7 \cdot 10^{-6}\) sec found by Eucken and Becker at \(18^\circ\mathrm{C}\), if recalculated to equal temperatures and pressures. The number of collisions necessary for the transition of a vibrational quantum into translational energy, amounting at \(18^\circ\mathrm{C}\) to 51,000, increases between \(-32^\circ\) and \(145^\circ\) almost fourfold. Further, Wallmann’s experiments proved that excitation of vibrations of the deformation type is possible in a double collision. Finally, he proved that symmetric vibrations of the valence type (Valenzschwingung, \(O \leftarrow C \rightarrow O\)) must have a relaxation time greater than or equal to the relaxation time of vibrations of the deformation type. Richards and Reid \(^{299}\) found that part of the specific heat of carbon dioxide drops out already at 3 kHz; from the measurements of other authors they concluded that this part drops out already at room temperature and at a considerably lower frequency. This conclusion agrees with the conclusion of Eucken and Moque \(^{99a}\), made on the basis of their own investigations, that the heat capacity corresponding to vibrations of the valence type drops out already in the region of sound frequencies.

This result could have been predicted from geometrical considerations, since the probability of excitation of such vibrations must be smaller than that of vibrations of the deformation type.

In the audible frequency range the speed of sound in carbon dioxide at various pressures (from 1 to 85 atm) and temperatures (from 15 to \(50^\circ\mathrm{C}\)) was studied by Shpakovskii, who found an increase of the heat capacity \(C_v\) with increasing pressure \(^{398}\).

Pumper \(^{404}\) investigated the propagation speed of ultrasound (frequency 44 kHz) in air and carbon dioxide at reduced pressure (down to 0.02 atm) and found an insignificant increase of the speed: \(0.65\%\) for air and \(2.2\%\) for carbon dioxide.

In measurements of absorption in carbon dioxide \(^{87}\) Grossmann \(^{128,129}\) found an increase of \(\alpha_m\) by \(20\%\), which was explained by Kneser as the result of radiation losses \(^{207}\). In a recently published work by Richardson and Relston the absorption also proved to be greater than the theoretical value.

CS₂. According to measurements by Richards and Reid \(^{299}\), symmetric vibrations of the valence and deformation type drop out at frequencies,

exceeding 450 kHz. Within the errors of the experiment, the conditions for excitation of these oscillations are the same.

N₂O. Kneser and Zöhlke²¹³ found that the dispersion region for oscillations of the deformation type lies near 100 kHz. The time for the establishment of equilibrium is \(\beta = 1 \cdot 10^{-6}\) sec. On average, one quantum is excited in every 50,000 collisions; for approximately 5,000 collisions it does not pass into the energy of translational motion. The absorption was measured by Abello, and his results agree well with the results that could be expected on the basis of the dispersion measurements.

Cl₂. Dispersion in Cl₂ was investigated by Eucken and Becker¹⁰⁰–¹⁰³. The establishment time at \(18^\circ\)C is \(4.2 \cdot 10^{-6}\) sec. The number of collisions necessary for taking away a vibrational quantum from an oscillating molecule is, at room temperature, 34,000. When the temperature is changed from \(-32\) to \(145^\circ\)C, it increases by approximately a factor of 7.

H₂. Since at room temperature the vibrational heat capacity is not yet excited, the dispersion can be due only to the loss of rotational heat capacity. Richards and Reid³⁰⁰ supposed that they had discovered the loss of rotational heat capacity in the frequency region between 94 and 451 kHz. However, careful investigations of pure hydrogen carried out by Kneser and Wallmann²¹², and also by Becker and Jäckel, showed that in the frequency region exceeding 1481 kHz dispersion is absent. Roy and Rose³¹⁵–³¹⁷ almost simultaneously calculated, from quantum-mechanical considerations, the equivalent radius \(r\) corresponding to excitation of rotational energy. From their calculations it follows that the dispersion region may occur at frequencies of about \(10^7\) Hz.

Experimental investigations³¹⁶ at 388 and 1465 kHz, at pressures from 424 to 772 mm Hg, showed that the dispersion region lies above 1500 kHz. Both in hydrogen and in other gases, it has so far not been possible to observe the loss of rotational heat capacity; apparently, it can occur only at considerably higher frequencies than those at which the measurements were made. This is explained by the high probability of excitation of rotational oscillations, following from geometrical considerations, as well as by the smallness of the rotational quantum.

CO. Sherratt and Griffiths³³⁶ investigated the speed of sound in CO at 7.9 and 27.4 kHz, at temperatures from 1000 to \(1800^\circ\)C. The true heat capacities, calculated from the apparent heat capacities and the theory of dispersion, agreed well with the values calculated from data obtained in the investigation of band spectra. It was assumed here that the dispersion formula is valid also in the case when not a single quantum state is excited. The relaxation time was found to be almost constant and equal to \(1 \cdot 10^{-5}\) sec.

N₂. Wallmann³⁷¹ found that in pure N₂ at 700 mm pressure, at \(21^\circ\)C, in the frequency region 65.7–716.8 kHz, dispersion is abs--

exists. This result agrees with the studies of Eucken, Mücke, and Becker[^99]. These authors determined the vibrational heat capacities of \(N_2\) and \(O_2\) by the Lummer–Pringsheim method and obtained agreement with the Planck–Einstein formula. However, determinations of the heat capacities from the speed of sound in the audible region gave somewhat low values. From this they concluded that the loss of vibrational heat capacity in \(N_2\) and \(O_2\) must already occur in the region of audible frequencies.

In agreement with these data are the absorption measurements made by Kneser and Knudsen[^201] in the audible region by the echo method. They found, for very pure oxygen, that the time for the establishment of equilibrium must be greater than or equal to \(10^{-3}\) sec. Measurements of absorption in oxygen in the region of ultrasonic frequencies gave values more than \(100\%\) exceeding the classical value \(\alpha\). But since, far from the region of dispersion, no additional absorption should arise, one may think that the measured large values should be attributed to insufficient purity of the oxygen.

Dispersion and absorption in gas mixtures. In gas mixtures an additional absorption arises, caused by mutual diffusion of gas molecules. The sound wave carries the lightest molecules from places of compression into regions where rarefaction has been created. The sound energy expended on this undergoes an irreversible transformation, causing an increase in absorption. A detailed theoretical treatment of this question was made by Rocard[^309]. The additional absorption \(\alpha_D\) (referred to one centimeter) is strictly proportional to \(\omega^2\) and, in the case of air, has the same order of magnitude as the absorption caused by thermal conductivity, i.e., about \(0.1\) of the absorption caused by viscosity.

This loss of energy, caused by diffusion, cannot explain the considerable increase of absorption in gas mixtures. Its cause lies in the “molecular” absorption, as Kneser calls it. Measurements in gas mixtures are of great interest: first, because air is a mixture of gases, from a purely technical point of view; and second, because measurements of the times for the establishment of equilibrium for various combinations of gases may provide a number of important data for molecular-kinetic theory.

\(C_2H_4\) and impurities. The first systematic measurements of dispersion in gas mixtures were carried out by Richards and Reid[^293,^300]. They investigated the speed of sound in ethylene at 94 and 451 kHz, \(15\)—\(45^\circ\text{C}\), and \(60\)—\(790\) mm Hg; they found that collisions with molecules or atoms of Ar, He, and \(N_2\) do not substantially affect the vibrations of ethylene molecules. Collisions with hydrogen molecules, however, prove to be approximately 10 times more effective in exciting transitions than collisions with the molecules of ethylene itself.

Air and oxygen of different humidity. Knudsen[^214,^216] investigated the absorption of sound in air and \(O_2\) of different humidity at frequencies of 3, 6, and 10 kHz by the echo method. He found that the absorption is greater than the classical value; it was further found that

the absorption maximum shifts into the region of higher frequencies when the humidity is increased. Kneser \(^{206, 209, 210}\) explained the increase in absorption by the finite time required for the establishment of equilibrium of the vibrational heat capacity of oxygen molecules, and by its change in the presence of water vapor. If \(h\) denotes the ratio of the number of water-vapor molecules to the number of air or oxygen molecules, then, in Kneser’s opinion, the expression \(k_1=\alpha h^2\) gives excellent agreement with Knudsen’s experimental data; a more exact expression is given below. In this expression \(\alpha\) denotes an individual constant which has different values for air and for oxygen. Since

\[ \frac{k_0}{k_1}=\chi \ll 1,\quad \beta \approx \frac{1}{k_1}, \]

this equation determines the dependence of the relaxation time on humidity. The quadratic dependence between \(h\) and \(k_1\) indicates that two water-vapor molecules are active in the conversion of the vibrational energy of \(O_2\) into the energy of translational motion. The dependence of the velocity of sound on the humidity of air was also investigated in other works \(^{188, 196, 219, 311, 312}\). Knudsen pointed out the importance of these data for architectural acoustics and sound communication. In the latter case it proves possible, under known weather conditions, to choose a signal frequency that provides the maximum communication range. Kao \(^{192}\) found that in dry air free of \(CO_2\), between 40 and 140 kHz, dispersion is absent (with an accuracy up to \(1^0/_{00}\)), which agrees with theoretical ideas.

Oxygen with different humidity and different impurities. Kneser and Knudsen \(^{211}\), using the echo method developed by Knudsen, investigated the absorption of sound at acoustic frequencies in \(O_2\) containing different amounts of the impurities \(H_2\), He, CO, \(CO_2\), \(O_3\), \(H_2S\), \(C_2H_2\), \(C_6H_6\), \(C_2H_5OH\), \(HCCl_3\), \(CCl_4\), \(CS_2\), \(NH_3\), and \(H_2O\). When oxygen was contaminated with \(HCCl_3\), \(O_3\), \(CO_2\), CO, or He, no absorption maxima were observed, but in all cases an increase in absorption with increasing \(h\) was found. In the remaining cases the change in the relaxation time (especially for water vapor) could be represented by the expression: \(k_1=\alpha h\), where \(\alpha\) is a constant characterizing the contaminating gas. Since, according to the simplified absorption formula, \(\omega_{\max}=k_1\), then at a given frequency the concentration of maximum absorption \(h_m\) is determined by the expression \(h_m=\frac{\omega}{\alpha}\). The authors calculated from the maximum absorption at 6 kHz (since at this frequency the observations were the most reliable) numerical values of \(\alpha\). Substituting these values of \(\alpha\) into the simplified absorption formula in place of \(k_1\), and multiplying by the corresponding \(h\), they obtained calculated absorption curves that agreed well with the measured ones. In the case of a water-vapor impurity this

the calculation proved unsatisfactory. To determine a suitable relation for the case of water vapor, they made measurements in a mixture of \(O_2—H_2O\) over a wide range of frequencies, from which they determined \(h_m\). As a check, they carried out measurements in a mixture of \(O_2—NH_3\). The results of the measurements are shown in Fig. 17.

The linear dependence of \(h_m\) on the frequency in the case of \(NH_3\) is striking. For water vapor a dotted curve was obtained, which can be described by the equation: \(k_1=\alpha h+\beta h^2\). However, this expression is not sufficiently good either, but nevertheless from the course of \(h_m\) one may conclude that collisions of one oxygen molecule with two molecules of water vapor are of substantial importance. The satisfactory agreement of the above formula with the measured values indicates, moreover, the smallness of \(k_1^0\) in completely pure oxygen, since the exact expressions should have the following form:

Fig. 17. Frequency of maximum absorption in oxygen as a function of the concentration of contaminating gases, according to Kneser and Knudsen.

\[ k_1=k_1^0+\alpha h \]

and, correspondingly:

\[ k_1=k_1^0+\alpha h+\beta h^2 . \]

Thus \(k_1^0\) is very small in comparison with the measured values \(\alpha h\). The values of \(\alpha\) for each contaminating gas turn out to be very different from one another, which proves important for the further development of the molecular theory of dispersion and absorption. Since this development, despite the interesting works of Kneser and Knudsen, is far from complete, we shall confine ourselves merely to a summary of the values they calculated for the probability \(W\) that, upon collision of an oscillating oxygen molecule with a molecule of the contaminating gas, distortions of the vibrational state arise:

\(Cl_2\) and impurities. Eucken and Becker\(^{100—102}\) carried out detailed investigations of the dispersion of sound in \(Cl_2\), as well as in \(CO_2\), at differ-

ULTRASOUND

TABLE 1

Contaminating gas	$\alpha \cdot 10^{-5}$	$W \cdot 10^{4}$	Contaminating gas	$\alpha \cdot 10^{-5}$	$W \cdot 10^{4}$
$\mathrm{C_2H_5OH}$	610	85	$\mathrm{CHCl_3}$	9.6?	1.3
$\mathrm{NH_3}$	160	26	$\mathrm{CO}$	$\lesssim 6$	$\lesssim 1.3$
$\mathrm{C_6H_6}$	130	25	$\mathrm{H_2}$	4.8	0.5
$\mathrm{H_2O}$	$\left\{\begin{array}{l}11?\\ 3\text{—}5 \cdot 10^8\end{array}\right.$	24.2	$\mathrm{CO_2}$	$\lesssim 1.7$	$\lesssim 0.4$
$\mathrm{C_2H_2}$	43	8.3	$\mathrm{O_3}$	$\lesssim 2$	$\lesssim 0.32$
$\mathrm{H_2S}$	13	2.4	$\mathrm{N_2}$	$\lesssim 0.5$	$\lesssim 0.1$
$\mathrm{CCl_4}$	$\lesssim 10$	$\lesssim 2.4$	$\mathrm{He}$	$\lesssim 0.4$	$\lesssim 0.06$
			$\mathrm{O_2}$	—	$<0.06$

…contaminations; the investigation was carried out in the ultrasonic region. In these cases as well, a decrease in the time required for equilibrium to be established was observed in the presence of contaminants, which indicates a shift of the dispersion region toward high frequencies; this shift is clearly visible in Fig. 18, belonging to Eucken and Becker.

Fig. 18. Sound dispersion in chlorine with admixtures at room temperature, according to Eucken and Becker.

In the case of contamination by means of $\mathrm{CH_4}$, not a definite curve but an entire region is obtained; we shall not dwell on the reasons for this. The change in the time of establishment of equilibrium $\theta$ under various contaminations is characterized by the figures given in Table 2, calculated for a concentration of the contaminating gas equal to unity.

TABLE 2

Gas	\(\beta_{AB}\cdot 10^6\) at one atmosphere
Cl\(_2\)—Cl\(_2\)	4.2
Cl\(_2\)—N\(_2\)	5.6
Cl\(_2\)—HCl	0.013
Cl\(_2\)—CH\(_4\)	0.015
Cl\(_2\)—CH\(_4\)	0.019
Cl\(_2\)—H\(_2\)	0.040
Cl\(_2\)—He	0.086
Cl\(_2\)—CO	0.029
CO\(_2\)—CO\(_2\)	5.7
CO\(_2\)—CH\(_4\)	0.31
CO\(_2\)—CH\(_4\)	0.23
CO\(_2\)—He	0.16
CO\(_2\)—H\(_2\)	0.027
CO\(_2\)—HCl	0.014
CO\(_2\)—H\(_2\)O	0.0028

Wahlman \(^{371}\) investigated the dispersion of sound (in the ultrasonic region) in carbon dioxide with various admixtures. In the case of CO\(_2\)—H\(_2\), the time of establishment of the vibrational heat capacity of CO\(_2\) proved to be linearly related to the hydrogen content. In agreement with the data of Eucken and Becker, it was found that the probability of excitation, or correspondingly of distortion, of the vibrational state of a carbon dioxide molecule upon collision with a hydrogen molecule is 100 times greater than upon collision with another carbon dioxide molecule. In investigations of a mixture of carbon dioxide with nitrogen or argon it was found that the probability of excitation of a carbon dioxide molecule upon collision with a nitrogen molecule or an argon atom is of the same order, or even less, than upon collision with a carbon dioxide molecule. In addition to the works mentioned, devoted to the study of dispersion and absorption in gases, there also exists a number of others: \(^{126, 199, 231, 242, 243, 257, 263, 287, 288, 325 \text{ and } 362}\).

Theoretical interpretation of the results. To interpret the different probabilities of the transition of translational energy into vibrational energy, Oldenberg \(^{260, 383}\) and Heil \(^{144a}\) made use of the impulse theorem of classical mechanics. They succeeded in giving an interpretation of the differing probability of excitation of different types of vibrations, for example, in explaining why vibrations of the deformation type are excited more easily than vibrations of the valence type. Although, in these simplified considerations, application of the impulse theorem leads to correct conclusions, calculation of more subtle quantitative relations is inaccurate, since the classical impulse theorem, strictly speaking, is not applicable to

processes under consideration. Gernes and Ramien137 and 280, under Franck’s direction, carried out work on calculating the excitation of vibrational states in collisions with slow electrons; they showed that the classical momentum theorem is not applicable in these cases. It is easy to understand the significance of these investigations for interpreting the results of studies of sound dispersion in gases if, together with Franck and Eucken110, one takes into account the importance of the mechanical disturbance of the state of the electron shell of an oscillating molecule upon collision with another molecule. In this way it is possible to determine the effectiveness of various colliding molecules as a function of the distortions that they produce in the electron shell of their partner. Of course, one must take into account here the mutual distortions of both colliding molecules, since the perturbation of the electron shell of the oscillating molecule depends essentially on the state of the electron shell of the molecule colliding with it. For example, substantial distortions should arise in impacts of unipolar particles (electrons and ions), dipoles, and also atoms and radicals capable of reacting with one another owing to the appearance of exchange forces.

Experimental data from studies of gas mixtures attest to the correctness of Franck and Eucken’s views. Thus, Eucken and Becker102 found that, for the conversion of the energy of translational motion into the energy of vibrational motion (and conversely), collisions with light molecules, under suitable conditions, may prove considerably more effective than collisions with heavy molecules, which is incompatible with the ideas of classical mechanics. Further, the probability of transition proves, as a rule, to be relatively smaller upon collision of particles possessing insignificant chemical affinity, but increases considerably (sometimes by a factor of a thousand) upon collision of particles that are in principle capable of reacting chemically with one another, even if no actual reaction occurs during the collisions. When the reactivity of the mixture Cl₂ + CO was increased by weak illumination, a considerable increase in the transition probability was observed. Kneser and Knudsen found that the transition probabilities increase as the chemical affinity increases. In agreement with the views of Franck and Eucken, it was found (Kneser and Knudsen) that excitation of O₂ is stronger upon collision with dipoles than with particles containing no dipoles.

Further development of the Franck–Eucken concepts may be achieved after the accumulation of more extensive experimental material. Knowledge of the excitation function of the vibrational energy of various molecules is very important for the problem of gas reactions, since, in London’s opinion, activation of molecules is associated with excitation of the energy of vibrational motions.

2. Speed of sound in liquids. Only a few of the available works contain accurate data on the speed of ultrasonic waves in liquids. One should mention the works of Hubbard and Loomis182–185, Freyer, Hubbard, and Andrews116, 117, 282, Bache-

...ma8, 102a and Bachem and Giedemann101. Since the data of the American authors are available in review works132, 407, we shall not dwell on them. The results of Bachem’s measurements, as well as those of Bachem and Giedemann, agree well with the results of the American authors obtained with the aid of an interferometer; the data obtained from measurements of diffraction spectra, however, differ from those indicated above. Table 3 gives data obtained with the aid of an interferometer, and also from measurements of the constant ultrasonic grating at 5 198.8 kHz; moreover, the measurements with the interferometer refer to 25°C (the figures given in parentheses characterize the relative accuracy of the measurement).

Slight discrepancies in the case of organic liquids may be due to the fact that in the optical measurements commercial preparations (“chemically pure”) were used, which are not entirely free from impurities. Hubbard and Andrews worked with preparations of exceptional purity.

A large number of measurements of the velocity of ultrasound (at a frequency of 7.32 MHz) in organic liquids was carried out by Partasarathy402, 403, who calculated from his data the compressibility of the liquids investigated (adiabatic) and the ratio of heat capacities. Absorption in liquids was measured by Babulin405.

TABLE 3

Liquid	Water	Toluene	Benzene
Measurements with an interferometer	1498.1±1	1306.1±1	1300.6±1
Measurements of the constant ultrasonic grating	1497.5±1.5 ±(0.3)	1303.3±1.3 (1303.36±0.04)	1299.6±1.3 (1299.6±0.5)

Liquid	Carbon tetrachloride	m-xylene
Measurements with an interferometer	919.5±1.0
Measurements of the constant ultrasonic grating	921±1.0 (921.2±0.2)	1322.0±1.3 (1321.96±0.13)

The relative accuracy achieved in measurements of the ultrasonic grating (5·10−5) is sufficient for studying the compressibility of dilute electrolyte solutions. The works of Tsalaya321, continued by Bachem8, 102a, deserve brief discussion. The statement of the problem is as follows: Gucker135, proceeding from Debye’s theory, derived an expression characterizing the dependence of the compressibility of electro-

lytic solutions on concentration. The original formula, connected with the free energy of an ionic solution, was obtained by Debye; according to the assumptions made in its derivation, it is applicable to the limiting cases of very strong dilutions. Hücker’s calculations led to a formula of the form:

\[ \beta=\beta_1-Ac+Bc^{\frac{3}{2}}, \]

where \(\beta\) is the compressibility of the solution, \(\beta_1\) is the compressibility of the pure solvent, \(c\) is the concentration in moles per liter. \(A\) and \(B\) are constants characterizing the electrolyte, calculated from quantities some of which are determined experimentally, while others are calculated theoretically.

Fig. 19. Dependence of the speed of sound on concentration, according to Bachem.

theoretically. Hücker found that the compressibility measurements at his disposal, relating to solutions with a concentration of 0.5 mole per liter and higher, agree well with this formula, but the values of the constants \(A\) and \(B\) differ from the theoretical ones. Bachem continued Hücker’s investigations to solutions whose dilution was approximately 10 times greater. In this region the same behavior of the compressibility was found as in more concentrated solutions. As for the compressibility of solutions with concentration less than 0.1 mole per liter, for which the theory should be most valid, no definite data are available up to the present time, since investigation of this region requires a further increase in the relative accuracy of the measurements. Bachem, by his precise measurements, was further able to establish,

that the compressibility of nonelectrolyte solutions (for example, a sugar solution) gives a fundamentally different dependence of compressibility on concentration than do electrolyte solutions; Gucker, who had insufficiently accurate compressibility measurements at his disposal, came to the conclusion that the course of the compressibility of both types of solutions coincides. Fig. 19 shows the relative change in the velocity of sound (expressed in promilles) as a function of concentration (referred to the velocity of sound in pure water).

In addition to the above-mentioned exact measurements of the velocity of sound in liquids, there is a large number of other, less accurate works \(^{6, 20, 47, 52—55, 60, 67, 70, 103, 104, 107, 244, 278, 324, 341, 379, 382}\). Of great interest is Berr’s \(^{13}\) measurement of the velocity of sound in liquid oxygen (99.3% purity), boiling at a temperature of \(-183.6^\circ\mathrm{C}\) under a pressure of \(705—720\) mm. At a frequency of 7500 kHz the velocity of sound, measured from diffraction spectra, proved to be equal to 903 m per second.

Up to the present time no dispersion of sound in liquids has been detected. The results of measurements of absorption are insufficiently accurate and too few in number for any final judgment to be made on this question. The very interesting problem, indicated by Debye \(^{91, 258, 259, 393}\), of measuring the mass of electrolyte ions by investigating the propagation of ultrasounds in a liquid has likewise not yet received experimental resolution.

V. Actions of Ultrasonic Waves

As was already indicated at the beginning of the article, the works of Wood and Loomis \(^{374—377}\) proved the existence of mechanical, thermal, emulsifying, chemical, and biological actions of ultrasonic waves of great intensity. Instead of enumerating these effects (see the surveys by Grossmann \(^{132}\), Malov \(^{407}\), and Schmitt \(^{393}\)), we shall confine ourselves to considering the processes that cause the above-mentioned actions.

1. Formation and action of gas bubbles. When ultrasonic waves of great intensity pass through a liquid, the formation of gas bubbles is observed, caused by the fact that particles of air dissolved in the liquid collect in regions of minimal motion, where they unite into large bubbles. In addition, the pressure changes that arise in the formation of rarefactions also play a role in the release of dissolved air particles. In a number of works \(^{142, 190, 342}\) it was pointed out that biological effects arise only under the condition that the possibility of formation of these gas bubbles is not excluded. The theoretical investigations of Schmitt \(^{338}\) testify to the great effectiveness of such bubbles. Schmitt calculated the radial pulsations of gas bubbles under the action of alternating sound pressure. He showed that for each frequency there exists a bubble diameter at which resonance phenomena arise. Bubbles having this or a somewhat smaller diameter can create

significant pressures, exceeding normal pressure by many thousands of times, which also explains the strong mechanical action of ultra-acoustic waves on bodies of small dimensions.

1. Cavitation. Already in the work of Boyle and Taylor ⁶⁹ the distinction between the formation of bubbles of gas dissolved in a liquid and the phenomena of cavitation known from hydrodynamics was clearly emphasized. Cavitation is understood as the formation of hollow spaces in a liquid, ruptures of the liquid. Of course, particles of gas contained in the liquid can also diffuse into these cavities and form large bubbles there. Cavitation phenomena in liquids permeated by ultrasonic waves were discovered by Kundt and Lehmann ²²⁴. For the formation of cavitations, considerably greater intensities are required than for the release of ordinary gas bubbles. Cavitations are very unstable and are easily destroyed when the conditions that caused their formation change, for example, when the pressure or temperature changes in the surrounding layers of the liquid. Rayleigh calculated that the local pressures arising upon the destruction of cavitations can reach thousands of atmospheres. From the study of erosion phenomena, especially from the work of Föttinger ¹⁰⁹, it is known that cavitations possess a strong destructive action. Föttinger already pointed out that, owing to the developing local pressures and changes of temperature, as well as processes connected with the appearance of electric charges during friction, cavitation phenomena may be accompanied by chemical processes, such as oxidation and dissociation. Indeed, almost all the observed chemical actions can be explained from this point of view. In studying oxidation processes ²⁵, ²⁶, ¹¹², ²³⁵, ³³² it was found that hydrogen peroxide is formed primarily from particles of oxygen and water. Bondy and Sollner ³⁶, ³⁷ have recently carried out a detailed investigation of the influence of cavitation on the formation of emulsions. They established that in the oil—water system the formation of emulsions under the action of ultrasonic waves occurs in those regions in which hollow spaces arise and are destroyed.

2. Mechanical effects. In certain biological investigations, where the destruction of cells whose dimensions were much smaller than the wavelength of sound was observed, the action of ultrasonic waves can be explained by simple mechanical rupture. To these same mechanical effects should be assigned the technical method developed by Klaus ⁸¹–⁸⁴ for preparing disperse systems, as well as phenomena connected with mechanical shaking, such as the explosion of unstable substances, the recrystallization of supersaturated solutions, the evaporation of superheated liquids ³⁰¹, ³⁰², ³⁹¹, ³⁹², ³⁹⁵, and also the influence of ultra-acoustic oscillations on the Barkhausen magnetic effect ¹⁷⁰.

3. Coagulation phenomena in aerosols. Extremely rapid and complete coagulation of aerosols under the action of sound waves in the lower ultrasonic region was established by Brandt and Freund ⁷³,⁴⁷. With powerful sound sources, the pro-

aggregation occurs, the resulting particles being many hundreds of times larger than the original particles, accompanied by rapid sedimentation. Independently of these authors, Pirkson \(^{262}\) described weak coagulation of tobacco smoke at higher frequencies. According to still unpublished investigations by Brandt, the explanation of this phenomenon must be sought, first, in an increase in the number of collisions and, second, in the occurrence of aerodynamic forces between particles suspended in an oscillating medium. This phenomenon is not confined only to the ultrasonic region \(^{394}\).

5. Thermal effects. Strong heating arising at high frequencies can be explained by strong absorption; at low frequencies, the heating observed at interfaces is of great importance \(^{28, 113–115, 301, 302}\). These heatings, apparently, may be connected with most of the effects observed in an ultrasonic field. Only a few of these effects cannot be produced by other means, for example by audible sounds of sufficiently high intensity. Therefore, the expediency of using ultrasonic waves in most cases is connected only with the fact that obtaining them is technically the simplest.

In addition to the works mentioned above, devoted to various actions of ultrasonic waves, there is a whole series of others \(^{19, 79, 88, 111, 120, 136, 143, 234, 246, 247, 249, 284, 317, 350, 352, 353, 378, 395, 396, 397}\).

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*

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ULTRASOUND

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1. Malov N., Advances of Physical Sciences, 9, 859, 1929*.
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* The supplement to the bibliography was compiled by N. N. Malov.

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2. W. König, Ann. d. Phys., 4, 1, 1901.
3. N. Marinesko, C. R., 201, 1187, 1935.
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16. E. Pumper, Phys. Z. d. Sowjetun., 8, 300, 1935.
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19. N. Malov, Advances in Physical Sciences, 15, 52, 1935.
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1. References 22, 168 as cited in the source. ↩

2. References 21, 22 as cited in the source. ↩

3. Reference 168 as cited in the source. ↩

4. Reference 14 as cited in the source. ↩

5. Reference 12 as cited in the source. ↩

6. Reference 154 as cited in the source. ↩