Investigation of Wave Processes by the Model Method Using Ultrasonic Waves

S. I. Krechmer and S. N. Rzhevkin, Moscow

There exist a number of methods for investigating wave processes with the aid of models. The simplest method consists in observing the propagation of waves on the surface of a liquid. This method excellently illustrates refraction, diffraction, and other wave phenomena, and is widely used for demonstration purposes[^1]. For studying the propagation of waves in rooms and for clarifying a number of questions in architectural acoustics, this method is also successfully applied and gives rather visual characteristics of wave propagation upon reflection from surfaces of complex form[^2],[^3]. However, there is no doubt that the entirely different nature of waves on a surface (transverse waves) and waves in air (longitudinal waves) does not permit the analogy to be developed far in the direction of quantitative conclusions. Moreover, in the water-wave method there are a number of serious inconveniences arising, first, from the large attenuation of surface waves and from the dependence of their velocity on wavelength[^4]. Since the attenuation increases rapidly as the wavelength decreases, it is necessary to use the longest possible waves (with wavelength of the order of 1 cm), which leads, first, to indistinct photographic images of wave patterns and, second, to the necessity of making models of very large dimensions (of the order of 1 m). The presence of strong dispersion of surface waves makes the study of wave patterns for the case of impulse propagation extremely unclear, since the form of the impulse changes as it propagates, and a distinct pattern cannot be obtained. The use of drops as a source of waves therefore always gives very uncertain results. The presence of capillary rise at the boundary of solid bodies distorts both the wave and the optical pattern and makes it impossible to draw conclusions about the details of the phenomena of diffraction and absorption of sound. This creates a serious limitation on the applicability of the method.

Conversely, the investigation of impulsive waves in air is excellently accomplished by the Schlieren method of Toepler or directly by observing the “shadow” of these waves[^5],[^6]. In view of the absence of noticeable dispersion, the patterns of the fronts of wave impulses in air prove to be

are quite sharp and make it possible to investigate the finest details of reflections from surfaces of very complex form and to draw important conclusions in architectural acoustics[^7]. The limitation of this method lies in the difficulty of obtaining a pattern of sinusoidal waves of a definite frequency, since for them the method is insufficiently sensitive. In view of this, the study of the dependence of various processes (for example, the phenomena of diffraction, reflection, and absorption) on wavelength becomes impossible. The wave patterns of pulses evidently represent only certain integral characteristics of waves for the continuous frequency spectrum to which the given pulse is equivalent, and from them one cannot draw conclusions concerning waves of a definite length.

The use of ultrasonic waves for model investigations of wave processes offers a number of advantages in comparison with other methods. As shown by the investigations carried out by us at the Physical Institute of the Academy of Sciences, the application of ultrasonic waves in a liquid makes it possible to study wave processes of a definite frequency and, consequently, permits one (by changing the frequency) to study the frequency dependence of various processes. A number of examples of such an investigation will be given below. The fact that ultrasonic waves in a liquid are longitudinal allows one, with greater justification, to seek in them analogies with waves in air than is the case for transverse waves on a surface. The analogy must, of course, extend to the purely geometrical properties of the phenomena of reflection and diffraction and to their dependence on wavelength. As for the absorption of sound, here the question of similarity of phenomena is considerably complicated, and one can speak only of illustrating the phenomena, without attempting for the present to give quantitative laws. It is convenient to take the wavelengths of ultrasound in a liquid to be of the order of millimeters. The use of such waves is especially simple technically; moreover, for them it is possible without difficulty to obtain stroboscopic photographs, using Kerr cells or other methods, and thus to study the pattern of traveling waves. With such wavelengths the dimensions of the models can be made quite miniature. At the boundary of solid bodies it is possible to obtain perfectly distinct and undistorted wave patterns. Some difficulty here is presented by thermal inhomogeneities, which appear as a result of convection currents in the liquid, arising from the heat liberated by the quartz oscillator and as a result of absorption of the ultrasonic waves.

Experimental method. For the study of various cases of propagation of ultrasonic waves it was expedient to make use of an optical arrangement according to Toepler’s method[^5],[^6], making it possible to observe the pattern of propagation of ultrasonic waves in a liquid poured into a vessel of large volume with transparent plane-parallel walls.

The apparatus used for the work was arranged according to the scheme shown in Fig. 1. Rays from the stroboscopic illuminator \(C\) are collected by the condenser \(K\) in the slit \(Щ\) and, diverging, enter-

are first passed into the long-focus lens \(L_1\), arranged so that farther on the light passes through the slit as a parallel beam, i.e., the principal focus of the lens \(L_1\) lies precisely in the plane of the slit. The parallel beam of light passes through a tank with plane-parallel glass walls, in which ultrasonic waves from the emitter \(I\) propagate, and is collected by a second long-focus lens \(L_2\). In the principal focus of the lens \(L_2\) there is thus obtained a real image of the slit, which is covered by a narrow screen \(E\) in the form of a wire of suitable thickness. Behind the screen is a camera \(F\), focused on the middle plane of the tank

Fig. 1.

in which the beam of ultrasonic waves propagates. The long-focus lenses \(L_1\) and \(L_2\) consisted of two halves of an achromat and had a principal focal length of \(750\text{--}1100\ \mathrm{mm}\) with a diameter of \(110\ \mathrm{mm}\).

The wire screen was mounted on a rider for motion along the optical axis, and in addition could be moved perpendicular to the optical axis by means of a micrometer screw. By changing the width of the screen wire, or simply moving it perpendicular to the optical axis (i.e., blocking diffraction spectra of different orders), one can obtain different degrees of “sensitivity” of the optical arrangement and pick out more or less fine details. This device must in practice often be used in order to obtain the clearest picture in one or another region of the field of view.

Without entering into an analysis of the complex diffraction pattern arising from ultrasonic waves,\(^1\) one may reason on the basis of geometrical optics. In passing through a homogeneous medium all regular light rays (the zero-order spectrum) will be blocked, and the field of view will be dark. If, however, an optical inhomogeneity is encountered in the path of the rays, scattered (diffracted) rays will arise which will not be focused on the screen, will pass by it, and will give in the focus

\(^1\) A detailed analysis of these phenomena was made by Lucas and Biquard, Brillouin, and others, § 9, 10 (the question is analyzed in the review article by Hiedemann, Advances in Physical Sciences, 16, 586, 1936).

of the camera lens an image of all the inhomogeneities arising in the liquid poured into the tank, in the form of bright spots on a dark background. If, between the lenses \(L_1\) and \(L_2\), a parallel beam of light passes through a liquid in which ultrasonic waves are propagating, producing a periodic change of density and, consequently, of the refractive index, moving with the speed of sound (corresponding to the given liquid), then under stroboscopic illumination we shall see a series of bright bands on a dark background, separated by distances corresponding to the period of the inhomogeneity, i.e., the wavelength \(\lambda\). With constant illumination, however, we shall see a certain general brightening of the region in which the ultrasonic wave is propagating. Standing ultrasonic waves in a liquid, under stroboscopic illumination, give bright bands located at a distance of a whole wavelength, since this will be the period of the inhomogeneity; with constant illumination the bright bands will be located at a distance of half a wavelength, since at the places of nodes, separated by a distance \(\frac{\lambda}{2}\), the strongest scattering will occur on average. This simplified interpretation of diffraction phenomena on ultrasonic waves is valuable in that it makes it possible to visualize clearly the formation of images of the fronts of sound waves.

Fig. 2.

Fig. 3.

The stroboscopic illumination required to obtain a stationary picture of traveling waves was produced with the aid of a Kerr cell connected to a generator exciting the emitter of ultrasonic waves. In addition to the high-frequency voltage, a constant voltage of such magnitude was applied to the Kerr cell that the cell opened the light once per period and so that the duration of illumination did not exceed 0.1 of a period. Figure 2 explains how the intensity of the light is modulated when a high-frequency voltage is applied to the Kerr cell. The curve \(K\) in the drawing shows the form of the characteristic of the Kerr cell. Figure 3 shows the dependence of the light intensity on time during a period.

To avoid the establishment of a system of standing waves, in the tank on

at its end, opposite the emitter, a “trap” for ultrasonic waves was arranged, consisting of a system of walls placed at such angles that the ultrasonic wave could return to the middle part of the vessel only after multiple reflections, as a result of which it was completely damped (the shape of the bath is shown in Fig. 1).

For photographing the observed wave-field pattern we used a Zeiss telephoto lens mounted on a \(13 \times 18\ \text{cm}\) camera. The telephoto lens made it possible to obtain a sharp image of the wave pattern in the bath at natural size or even larger. As a result of a number of trials, a design of the ultrasonic emitter was developed that made it possible to avoid secondary radiation from the emitter itself and to obtain a plane ultrasonic wave. The piezoquartz plate was placed between a lead block and a thin, tightly stretched membrane insulated from it \((0.02\ \text{mm})\), and this entire system could be rotated about the normal to the plate, which, as will be seen from the further exposition, is very important for obtaining a homogeneous plane wavefront. As the liquid in which we studied the propagation of ultrasonic waves, vaseline oil was chosen, since it is completely transparent and is an excellent insulator.

The radiation from a piezoquartz plate cut perpendicular to the electric axis and excited at the oscillation frequency corresponding to its thickness should have had the form of plane waves in the shape of a series of parallel bands. Observation showed, however, that quartz plates for the most part give nonuniform radiation with a number of secondary rays (Fig. 4).

This is explained, as it has been possible to show\(^{11}\), by the fact that not only longitudinal waves arise in the plate, giving a plane front, but, in addition, bending waves; moreover, a series of oscillatory zones separated by nodal lines (Chladni figures) is formed on the plate. Under these conditions the plate radiates as a complex grating and gives lateral diffraction beams. This becomes quite obvious if one compares Fig. 4 with Fig. 14, which shows the pattern of the passage of waves through a diffraction grating. To reduce the influence of the lateral beams, which distort the observed wave patterns, we used the special device mentioned above, allowing the entire quartz holder to be rotated about the normal to the quartz plate. Since in rectangular plates the nodal lines are for the most part parallel to the edges, by turning the plate so that the edges stand at an angle to the optical axis of the apparatus, we give the diffracted sound beams such a direction that their images are not obtained. Only the principal plane wave remains (Fig. 5). This method proved very successful. We succeeded in obtaining sufficiently homogeneous wave beams with a plane front for wavelengths from 1 to 3 mm.

Wave patterns for the simplest cases. Under constant illumination, a traveling wave reveals its presence only by a general brightening of the background due to the appearance of scattered on...

to the sound grating of light rays, but individual sound waves are not visible. If a wave is reflected from a plane plate at normal incidence, standing waves are formed, which give a pattern of parallel bands at distances \(\frac{\lambda}{2}\), i.e., twice as small as in stroboscopy. This occurs because brightening is obtained at each pressure antinode, i.e., at intervals of half a wave.

At incidence at an angle to the normal, as is known, standing waves arise parallel to the reflecting surface, with the distance between antinodes equal to \(\frac{\lambda}{2}\cdot\frac{1}{\cos \varphi}\), where \(\varphi\) is the angle of incidence. In the photograph in Fig. 6 the standing waves parallel to the horizontal reflecting surface of the oil are clearly visible. The incident wave travels from the right, from below; the reflected wave goes to the left, downward (see the arrows in the figure).

In Fig. 7 a photograph is shown (with stroboscopic illumination) for the case of waves passing through a glass plate 5 mm thick. Since the velocity of sound in glass is \(c_1 = 5000\ \text{m/sec}\), and in oil \(c_2 = 1500\ \text{m/sec}\), then at an angle of incidence greater than

\[ \varphi = \arcsin \frac{c_2}{c_1}, \]

total internal reflection from the glass must occur. Fig. 8 shows precisely the case of total internal reflection from the glass plate at an angle of incidence of about \(30^\circ\), beginning with which the transmitted wave disappears. The reflected ray is clearly visible; there is no transmitted ray. At smaller angles of incidence the transmitted ray is quite distinctly visible.

Reflecting diffraction grating. At normal incidence on a metallic surface with grooves (spacing between grooves \(d = 4.5\) mm), with the plane of incidence perpendicular to the grooves, strong diffracted rays are obtained. In Fig. 9 diffraction was photographed for \(\lambda = 2.9\) mm \(\left(\frac{\lambda}{d} = 0.645\right)\). The first-order diffracted ray is clearly visible. Its angle of inclination to the normal, measured in the figure, is \(40^\circ\), which corresponds to the calculation for a diffraction grating by the formula \(\sin \varphi = \frac{\lambda}{d}\). In Fig. 10 \(\lambda = 1.96\) mm \(\left(\frac{\lambda}{d} = 0.436\right)\), and here we already see two diffracted rays at angles of \(26^\circ\) and \(59^\circ\) to the normal. Calculation gives in this case the angles \(25.6^\circ\) and \(60^\circ\). In the case \(\frac{\lambda}{d} > 1\), diffraction cannot take place at normal incidence. At oblique incidence (at an angle \(\varphi\)) a ray diffracted by a grating with spacing \(d\) can appear (at an angle \(\varphi'\)) if the condition is satisfied:

\[ \sin \varphi + \sin \varphi' = n \frac{\lambda}{d}. \]

To the article by S. I. Kreimer and S. N. Rzhevkin

Fig. 4

Fig. 5

Fig. 6

To the article by S. I. Krechmer and S. N. Rzhevkin

Fig. 7.

Fig. 8.

Fig. 10.

Fig. 11.

It is easy to see that in the case \(\frac{\lambda}{d}=1.3\) this condition is satisfied only for diffraction angles directed farther from the normal than the incident ray. In Fig. 11, at an angle of incidence \(\varphi=23^\circ\), the diffracted ray is clearly visible, going closer to the surface and backward at an angle of \(57^\circ\), which corresponds to the formula given above.

Diffraction of a plane wave on a cylinder. Diffraction of a plane wave on a circular cylinder (\(d=9\) mm) is shown in Fig. 12 with stroboscopic illumination for the case \(\lambda=1.76\) mm, \(\frac{d}{\lambda}=5.1\). In the photographs the spherical diffracted wave and the dark (soundless) diffraction zones behind the cylinder are clearly visible. It is distinctly seen that behind the cylinder in both photographs there is a bright zone. This shows that sound passes through here. In the photograph (Fig. 13) the pattern, at the same wavelength as in the preceding photograph, was photographed under steady illumination. In this case standing waves are visible at distances half as large as in Fig. 12, having the form of parabolas. It is interesting to note that the standing waves bend backward behind the cylinder and, as it were, pass into diffraction bands. On a more careful analysis of the photograph we notice that, approximately in the plane of the front surface of the cylinder, there occurs a transition of the dark bands of the standing waves into the light diffraction bands behind the cylinder. This circumstance is not entirely clear theoretically. Although the problem of diffraction by a cylinder is regarded as completely solved[^12], the distribution of waves in detail has not been clarified, and the model method here may be very useful in a number of concrete cases.

Diffraction grating. To observe the wave pattern we used a diffraction grating made of steel rods 2 mm in diameter, with a spacing of 3 mm between centers. This grating partly reflects and partly transmits waves. Diffraction spectra at normal incidence can appear only if the wavelength \(\lambda\) is smaller than the grating period \(d\). In Fig. 14 (\(\lambda=1.91\) mm, \(\frac{\lambda}{d}=0.64\)) diffraction waves are clearly visible on both sides of the grating. The angle of the diffraction waves with the normal is about \(40^\circ\), in accordance with the formula

\[ \sin \varphi = \frac{\lambda}{d}. \]

It is interesting to note the formation of a series of dark “rays” going midway between the directions of the direct and the diffracted waves. An entirely similar pattern is produced by the radiation of a quartz plate in the case of the occurrence of transverse waves, as was already mentioned above (Fig. 4).

Diffraction of a slit. A metal plate transmits a considerable fraction of the ultrasound, and therefore it is not rational to make a slit in such a plate, since the ultrasound will pass not only

through the slit, but also through the thickness of the metal, which confuses the wave pattern. In view of this, we used an “air partition” in the form of a flat vessel (3 mm thick) made of thin celluloid. Such an air partition practically does not transmit ultrasound, since the transmission coefficient through the oil–air boundary is only about \(10^{-7}\). In the middle of the vessel a slit \(5 \times 60\) mm was cut in both walls, and inside the vessel a celluloid-lacquer frame, with an opening of the same size and a thickness of 3 mm, was glued in. The entire vessel was immersed in a bath with oil. In this way a slit was obtained in a partition completely opaque to ultrasound. In Fig. 15, for \(\lambda = 1.2\) mm \(\left(\frac{\lambda}{d}=0.24,\right.\) where \(d = 5\) mm is the slit width), diffraction bands are clearly visible on both sides. The dark (soundless) bands go at angles of 14 and 29° to the axis of the main beam, which agrees with the calculation by the formula for diffraction minima

\[ \sin \varphi_m = m \frac{\lambda}{d}. \]

Interference from two coherent sources.
In Fig. 16 an interference pattern is photographed, obtained when a plane wave passes through two apertures (diameter 5 mm, distance between centers 12 mm). In the photograph interference bands are clearly visible, having the form of hyperbolas, with foci located at the centers of the apertures.

Practical applications of the wave-model method.
The wave-field patterns obtained by the described method are so detailed that they make it possible to analyze a number of complex cases of diffraction, reflection, or scattering of sound that occur in practice and, for the most part, are not amenable to theoretical calculation. In this way we succeeded in clarifying the pattern of reflection and scattering of sound from surfaces with a series of cylindrical or conical recesses (caissons), arranged in the form of a regular grating and representing a model of the planned ceiling of the Palace of Soviets. It was found that, at wavelengths comparable with the grating spacing, there is a fairly strong regular reflection from the surface. At wavelengths smaller than the grating spacing, distinct diffraction phenomena appear, and the regular reflection decreases sharply. We made an attempt to take quantitative account of the reflection coefficient from a surface of complex shape. For this purpose, photographs of reflection were made under constant illumination. In this case the structure of the waves is not visible, but there appears a general brightening of the dark background, which may approximately be considered proportional to the strength of the sound. By photometry of the darkening of the negative in the region of the incident and reflected waves, we obtained for the surface with cylindrical recesses at \(\lambda = 2.65\) mm (the ratio of \(\lambda\) to the grating spacing \(d\), \(\frac{\lambda}{d}=0.68\)) the coeffi-

To the article by S. I. Krechmer and S. N. Rzhevkin

Fig. 13.

Fig. 14.

coefficient of reflection is 0.36, while for \(\frac{\lambda}{d}=0.51\) the coefficient of reflection is 0.06. Thus, with decreasing wavelength, regular reflection drops sharply.

These results should be regarded as only approximate. To obtain more reliable data on the reflection coefficient, special investigations are, of course, required in order to determine the degree of blackening as a function of the intensity of the ultrasound.

Fig. 16.

Measurement of the degree of blackening for individual wave lines cannot yield a conclusion about the intensity of the ultrasound, since although with an increase in intensity the degree of blackening at first increases, subsequently, in the middle of the dark band, a lightening begins. For example, in Figs. 6 and 11 the darker bands in the region of the more intense (incident) waves appear lighter than in the region of the less intense (reflected and transmitted) waves. In the middle of the dark bands an “inverted,” lighter stripe is formed. This phenomenon was theoretically explained by Rytov1.

Significant data were also obtained for the reflection of waves from the model of the cornice of the Palace of Soviets, which runs around the entire Great Hall and has a width of about \(4\ \text{m}\). It was found that for \(f>400\ \text{Hz}\), a clearly regular reflection in the form of a directed beam will already be obtained from the cornice in air. At lower frequencies, scattering in the form of spherical waves will predominate.

Very interesting data were obtained in the investigation of strongly absorbing surfaces. We found that a metal mesh (card tape) is a complete absorber of ultrasonic waves in a liquid. When waves propagate parallel to the absorbing surface, an interesting phenomenon of bending of the wave-

wave fronts, which indicates the presence of an energy flux toward the absorber. The theory of this phenomenon has recently been given by N. N. Andreev[^14].

Undoubtedly, the method of wave models using ultrasonic waves, which even in its first steps has yielded a number of interesting results, with further development may prove very useful for a whole series of investigations of both applied and scientific character.

References

1. See, for example, the textbook by R. Pohl, Introduction to Mechanics and Acoustics.
2. A. Davis and G. Kaye, Acoustics of Buildings, London, 1927.
3. A. Davis, Modern Acoustics, London, 1934.
4. A. Davis, Proc. Phys. Soc. London, 33, 234, 1926.
5. Töpler, Beobachtungen nach einer neuen optischen Methode, Bonn, Max Cohen u. Sohn, 1864; Ostwald Klassiker, 1864.
6. Töpler, Pogg. Ann., 127, 556, 1866; 128, 176, 1866; 131, 33, 180, 1867; Ann. d. Physik, 14, 838, 1904.
7. F. Osswald, Akustische Z., 1, 167, 1936.
8. R. Lucas and P. Biquard, C. R., 194, 2132, 1932; 195, 121, 1932.
9. P. Debye and F. Sears, Proc. Nat. Acad. Sci., 18, 409, 1932.
10. Brillouin, Diffraction de la lumière par de Ultrasons, Paris, 1933.
11. S. N. Rzhevkin, Reports of the Academy of Sciences (in press).
12. P. Epstein, Encycl. d. mathem. Wissensch., Bd. V/3.
13. S. M. Rytov, Proceedings of the Academy of Sciences, No. 2, 1937.
14. N. N. Andreev, Proceedings of the Academy of Sciences, 1936.

1. Rytov. ↩