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Concave Ultrasonic Transducers*
L. D. Rozenberg
1. Recently, various devices intended for obtaining powerful ultrasonic beams have begun to find application. Such beams are used both for physical and biological experiments and for industrial purposes (dispersion, sterilization, coagulation, etc.).
The very technique of obtaining powerful beams is of interest. First of all, they can be obtained by the methods of classical optics, i.e., with the aid of concave mirrors, lenses, etc.¹ However, such a solution is complicated by the need to have (as in optics) powerful “point” sources of ultrasound, which so far are still very inconvenient to handle and therefore are very little used (for example, Hartmann’s jet generator). When using the most widespread piezoquartz transducers, the power taken from a unit surface is usually limited: in air—by the mechanical strength of quartz, and in liquids—by the occurrence of cavitation.
Obtaining high power is therefore reduced to increasing the radiating surface, i.e., to departing from the more advantageous form of a point source.
There exists, however, another path, essentially different from optical methods and based on the possibility of realizing powerful coherent primary “self-luminous” sources of sound. In fact, if the transducer (piezoelectric or magnetostrictive) is uniform over its entire surface, then all points of its surface will radiate in phase, and the shape of the wave front emitted near the radiating surface will correspond to the shape of that surface. Giving the latter the shape of a concave section of a sphere, one can obtain a spherical wave converging toward the center of curvature, i.e., focus the sound energy by means of the transducer itself. In this case the surface of the transducer, and consequently its full power, can theoretically be made arbitrarily large; the limit is the possibility of obtaining a homogeneous quartz plate of large diameter.
Fig. 1.
* Although the article deals with quartz transducers, almost all the arguments and conclusions are also valid for magnetostrictive transducers.
The degree of concentration of energy near the focus depends, as will be shown below, on the ratio between the wavelength and the radius of curvature of the radiator \(f\), on the one hand, and the magnitude of the aperture angle of the radiator \(a_m\) (Fig. 1), on the other hand.
- Grützmacher1 was the first to point out the possibility of concentrating ultrasound by means of a concave quartz radiator. By cementing a circular quartz plate of radius \(76\) mm, ground in the form of a section of a sphere of radius \(f = 100\) mm, to a round plate, he obtained in oil, at a frequency of \(370\) kc/s (wavelength in oil \(\lambda = 3.5\) mm), an increase in the sound intensity at the center of curvature by a factor of 160 in comparison with the intensity at the surface of the radiator.
If such a radiator is placed so that the center of curvature coincides with the free surface of the oil, one can observe an oil fountain arising in the region of the center of curvature (focus) of the radiator as a consequence of the large magnitude of the radiation pressure (the so-called acoustic wind).
The height of this fountain is an indirect indicator of the sound intensity at the focus of the radiator. Thus, in Grützmacher’s experiments the height of the fountain reached \(40\) cm.
The next investigation was carried out by S. S. Tumansky,2 who established that the presence of an air cushion adjacent to the convex side of the quartz plate increases the height of the fountain on average by \(1.7\)–\(3\) times.
In addition, Tumansky investigated the dependence of the fountain height on the voltage applied to the quartz for four plates of different sizes. The results he obtained make it possible to establish the essential fact of proportionality between the height of the fountain and the electrical power supplied to the quartz.
Indeed, the height of the fountain must be proportional to the magnitude of the radiation pressure, which in turn is proportional to the flux density of acoustic energy. Since the area of the cross section of the fountain is determined by the size of the focal spot and therefore does not depend on the magnitude of the pressure, the latter may be considered proportional to the total power radiated by the quartz. But this power must be proportional to the electrical power supplied to the quartz, or to the square of the voltage on the quartz. Thus, one may expect a quadratic dependence between the height of the fountain and the voltage applied to the quartz.
In Fig. 2, the voltage on the quartz is plotted along the abscissa, and the height of the fountain along the ordinate. The scales on both axes are logarithmic. The straight line corresponds to proportionality between the height and the square of the voltage. As is seen from Fig. 2, the points lie well along the straight line, with only slight scatter, confirming the assumptions stated above.
Fig. 2.
Tumansky succeeded in obtaining fountains up to \(70\) cm high with a quartz diameter of \(65\) mm, focal distance \(48\) mm, and frequency \(640\) kc/s (wavelength in oil \(\lambda = 2.8\) mm).
Six years later, in 1945, a paper by Labey$^{4}$ appeared, in which the results were described of a comparative experimental study of five quartz plates with a surface of $1\ \mathrm{sq.\ m}$ and radii of curvature $f=\infty$, 25, 8, 7, and 4 cm.
Despite the large number of curves, the results obtained are very unclear and even strange. Apparently, they should not be trusted.
I. Ya. Elpiner and A. P. Sheinker$^{5}$, in order to obtain whooping-cough bacillus endotoxins, used a quartz plate 50 mm in diameter with a focal length of 100 mm at a frequency of about 600 kHz. When a flask with liquid was introduced into an oil fountain, a secondary fountain several centimeters high arose inside the flask on the surface of the liquid.
During 1947–1948, reports on a number of works in this area appeared in the literature. However, the works themselves were not published in full, and they can be judged only from brief abstracts of the corresponding reports$^{6,7,8,9,10}$.
Thus, Willard$^{6}$, by the method of diffraction of light by ultrasonic waves, observed the pattern in the focal plane of a concave quartz plate ($D=25$ mm, $f=63.5$ mm). The diameters of the diffraction rings, as he writes, agreed well with the usual calculation (?), namely, were $\lambda \dfrac{f}{D}$; $2\lambda \dfrac{f}{D}$; $3\lambda \dfrac{f}{D}$, etc.
Fine$^{7}$ investigated a family of radiators with a surface of $4\ \mathrm{sq.\ dm}$ and radii of curvature $f=\infty$; 25; 7 and 4 cm. He found that the active component of the radiation resistance depends on the radius of curvature, and that the effective mass of the crystal decreases with increasing curvature. It is unclear what is meant by the statement that the focus “... in accordance with Williams’s prediction, does not always coincide with the center of curvature” (?).
The most interesting is the work of Mueller and Willard$^{8}$, who used a crystal having $f=63.5$ mm and $D=28.6$ mm at a frequency of 5 MHz. Measurements of the radiation pressure showed that 75% of all the energy radiated by the quartz passed through the area of the Airy circle, whose radius was $r_{0}=1.5$ mm (the optical calculation gives 86%). Hence it follows that, with 90 W of supplied power, at the center of the Airy circle the flux density of sound energy (sound intensity) was $5\,000\ \mathrm{W/cm^{2}}$ ($\simeq 206$ dB), which corresponds to an acceleration of $25\cdot 10^{6}g$ and to a pressure amplitude of 120 atm. It is interesting that no cavitation was observed in this case, although at a frequency of 25 kHz the latter already occurs at a sound intensity of $0.5\ \mathrm{W/cm^{2}}$.
In Willard’s paper$^{9}$, account is taken of the circumstance that, as the angle $\alpha$ increases (see Fig. 1), the angle between the axis of the quartz and the normal to its surface changes. Unfortunately, neither the initial considerations nor the final results are given in the annotation.
Finally, O’Neil$^{10}$ gave an approximate calculation of the field near the focus, under the assumption that the normal velocities of the quartz are constant, that the aperture angle ($x_m$) is small, and that $D \simeq \lambda$. The ratio of the sound intensity at the center of the focal spot to the intensity at the surface of the radiator has the form $(kh)^2$, where $k=\dfrac{2\pi}{\lambda}$, and $h$ is the depth of the radiator (see Fig. 1). This calculation gives satisfactory agreement with the experiments of Mueller and Willard$^{8}$, if one takes into account that in reality the normal velocities fall from the center of the quartz plate to its edges.
A. I. Gubanov$^{11}$, considering the distribution of normal velocities over the surface of the plate to be constant, wrote the expression for the pressure near the focus in the form of an integral. For the ratio of the pressure at the center of the focal spot to the pressure at the surface of the radiator, this integral becomes the expression
\[ K_{p}=\frac{S}{r_{1}\lambda}. \]
(where \(S\) is the surface of the radiating portion of the sphere, and \(r_1\) is its radius of curvature), identically equal to the expression
\[ K_p = kf(1-\cos \alpha_m), \]
obtained by the author of the present review (see \({}^{1}\), formula (6.28)).
Gubanov drew attention to the fact that taking into account attenuation in the medium, for a quartz plate of a specified curvature, leads to the existence of an optimum frequency, since with increasing frequency \(K_p\) increases and, at the same time, propagation losses increase.
If it is assumed that the attenuation in the medium follows the law
\[ e^{-\frac{\alpha' r_1}{\lambda^2}}, \]
where \(r_1\) is the distance from the surface of the emitter to the focus, then the optimum wavelength will be
\[ \lambda_{\mathrm{opt}}=\sqrt{2\alpha' r_1}. \]
In this case
\[ \frac{\alpha' r_1}{\lambda_{\mathrm{opt}}^2}=0.5, \]
and
\[ e^{-0.5}=0.6, \]
i.e., at the optimum frequency the sound pressure at the focus will decrease, owing to attenuation, by 40%.
Gubanov obtained the pressure distribution in the focal plane only for small aperture angles; in this case (as, indeed, was to be expected) the familiar classical distribution is obtained. Finally, an expression is given for the pressure on the axis, suitable for any aperture angles, and the size of the focal spot “in depth,” i.e. along the axis, is given.
The distance from the center of the focal spot to the zero of pressure along the axis proves to be equal (in our notation) to
\[ \frac{\lambda}{1-\cos \alpha_m}. \]
For small aperture angles it becomes
\[ 2\lambda \frac{f^2}{R^2}, \]
where \(R\) is the radius of the aperture of the emitter*).
- As is seen from the review given above, almost all the calculations performed pertain to various special cases and do not give a general picture of the phenomena, quite apart from the fact that most authors calculate only the value of the pressure, whereas the primary interest is the energy flux through the focal spot and the proportional magnitude of the radiation pressure.
Most of the quantities of interest to us can be obtained from the general formulas given by the author of the present review\({}^{1}\).
If we have a converging spherical wave of radius \(f\) (see Fig. 1), then the ratios of the pressure and the axial velocity at the center of the focal spot
*) In Gubanov, through an oversight, the factor 2 was omitted.
with respect to the pressure and the normal velocity on the surface of the wave front have the form
\[ K_p = kf \int_0^{\alpha_m} \Phi(\alpha)\sin \alpha\, d\alpha, \tag{1} \]
\[ K_v = kf \int_0^{\alpha_m} \Phi(\alpha)\sin \alpha \cos \alpha\, d\alpha, \tag{2} \]
where \(\Phi(\alpha)\) is the law of distribution of amplitudes over the surface of the wave front, and \(k=\dfrac{2\pi}{\lambda}\).
If it is assumed that the amplitudes (or the normal velocities) are uniformly distributed over the surface of the wave front, which can be realized by means of a quartz mosaic arranged on a concave spherical surface, then
\[ K_p = kf(1-\cos \alpha_m), \tag{3} \]
\[ K_v = kf\frac{\sin^2 \alpha_m}{2} \tag{4} \]
and, correspondingly, the energy flux at the center of the focal spot is
\[ K_u = k^2 f^2 \frac{\sin^2 \alpha_m}{2}(1-\cos \alpha_m). \tag{5} \]
Often, however, it is more convenient to compare different radiators with equal pupil radii. Using the relation evident from Fig. 1,
\[ \sin \alpha_m = \frac{R}{f}, \]
expressions (3), (4), and (5) may be rewritten in the form
\[ K_p = kR\,\frac{1-\cos \alpha_m}{\sin \alpha_m}, \tag{3a} \]
\[ K_v = kR\,\frac{\sin \alpha_m}{2}, \tag{4a} \]
\[ K_u = k^2 R^3\,\frac{1-\cos \alpha_m}{2}. \tag{5a} \]
The expression for the pressure gain obtained by Gubanov, as has already been pointed out, is identical with (3), while O’Neill’s expression
\[ K_u = k^2 h^2 = k^2 f^2 (1-\cos \alpha_m)^2 \]
is valid only for small aperture angles, when \(\cos \alpha_m \simeq 1\), and it may be assumed that expressions (1) and (2) coincide, while
\[ K_u = K_p^2 = k^2 f^2 (1-\cos \alpha_m)^2. \]
If the concave quartz plate is ground from a solid piece, and the normal velocities on the surface of the radiator cannot be regarded as constant, then the expressions given above are valid only for small aperture angles (see below). In particular, this applies to the plate \((D = 27\ \mathrm{mm},\ f = 63\ \mathrm{mm},\ \alpha_m = 11^\circ)\), with which Willard worked\(^6\). But for small aperture angles the focal spot should have a classical law of pressure distribution, i.e. the diameters of the diffraction rings should be equal to
\[ 1.22\,\frac{\lambda f}{R};\quad 2.23\,\frac{\lambda f}{R};\quad 3.24\,\frac{\lambda f}{R}, \]
and not
\[ \frac{\lambda f}{R};\quad 2\,\frac{\lambda f}{R};\quad 3\,\frac{\lambda f}{R}, \]
as indicated by Willard.
As the aperture angle increases, the rings are compressed, and at \(\alpha_m = \pi/2\) their diameters become equal to
\[ \lambda;\quad 2\lambda;\quad 3\lambda,\ \text{etc.} \]
The law of pressure distribution in this case takes the form
\[ \frac{\sin kr}{kr}. \]
Fig. 3.
- For large aperture angles one must take into account the circumstance that a quartz plate cut in such a way that the \(X\)-axis coincides with the axis of the radiator oscillates along this axis, but what is essential for radiation is the normal component of the velocity, which, as is easy to see from Fig. 3, is equal to \(v_x \cos \alpha\). Therefore, if \(v_x\) is considered constant at all points, then the pressure changes over the surface of the plate according to the law \(\Phi(\alpha)=\cos\alpha\).
Substituting this value into expressions (1) and (2), we obtain
\[ K_p = kf\,\frac{\sin^2 \alpha_m}{2}, \tag{6} \]
\[ K_v = kf\,\frac{1-\cos^3 \alpha_m}{3}, \tag{7} \]
\[ K_u = k^2 f^2\,\frac{\sin^2 \alpha_m}{6}\,(1-\cos^3 \alpha_m) \tag{8} \]
or, passing to the radius of the pupil of the radiator,
\[ K_p = kR\,\frac{\sin \alpha_m}{2}, \tag{6a} \]
\[ K_v = kR\,\frac{1-\cos^3 \alpha_m}{3\sin \alpha_m}, \tag{7a} \]
\[ K_u = k^2R^2\,\frac{1-\cos^3 \alpha_m}{6}. \tag{8a} \]
As an example, let us calculate plate No. 43 from the experiments of Tumansky\(^3\):
\[ D = 68\ \mathrm{mm},\quad f = 48\ \mathrm{mm},\quad \lambda = 2.8\ \mathrm{mm}. \]
\[ \sin \alpha_m = 0.71,\quad \alpha_m \simeq 45^\circ. \]
Substituting these values into formulas (6), (7), and (8), we obtain
\[ K_p = 27.5,\quad K_v = 21.5,\quad K_u = 580. \]
L. D. ROZENBERG
Calculation by O’Neill’s formula gives an exaggerated value \(K_u = 750\).
As for the law of distribution of pressures and velocities, it can be obtained from the more general expressions given by the author of the present review\(^1\):
\[ \frac{p}{p_0} = kf \int_{0}^{\alpha_m} e^{ik\eta \cos \alpha} I_0(kr \sin \alpha)\,\Phi(\alpha)\,\sin \alpha\,d\alpha, \tag{9} \]
\[ \frac{v}{v_0} = kf \int_{0}^{\alpha_m} e^{ik\eta \cos \alpha} I_0(kr \sin \alpha)\,\Phi(\alpha)\,\sin \alpha \cos \alpha\,d\alpha, \tag{10} \]
where \(r\) and \(\eta\) are the cylindrical coordinates of the point under consideration, with
— — — polished quartz
——— quartz mosaic
Fig. 4.
\(\eta\) measured from the focal plane, and \(r\)—normal to the axis; \(v_0\) and \(p_0\) are the normal velocity and pressure at the surface of the radiator on its axis.
In particular, for the pressure distribution in the focal plane \((\eta = 0)\), it is easy to obtain from (9) and (6a)
\[ \frac{p}{p_{\max}} = \frac{2J_1(kr\sin\alpha_m)}{kr\sin\alpha_m}, \tag{11} \]
where \(p_{\max}\) is the pressure at the center of the focal spot.
The diameters of the diffraction rings in this case will be
\[ 1.22\frac{\lambda}{\sin\alpha_m};\quad 2.23\frac{\lambda}{\sin\alpha_m};\quad 3.24\frac{\lambda}{\sin\alpha_m};\ \ldots \]
But, since \(\sin\alpha_m = \frac{R}{f}\), these expressions turn out to coincide exactly with the classical ones obtained for small aperture angles:
\[ 1.22\frac{\lambda f}{R}\quad \text{etc.} \]
In conclusion, Fig. 4 shows the dependences of the quantities \(\frac{K_p}{kR}\), \(\frac{K_0}{kR}\), \(\frac{K_u}{kR}\) on the aperture angle for the cases of a quartz mosaic and a solid ground quartz plate. It is seen from the curves that, for aperture angles less than \(30\text{--}35^\circ\), practically both types of transducers are equivalent. At larger angles, however, a substantial difference is noticeable; for example, at \(\alpha_m = \pi/2\), \(K_p\) differs by a factor of 2, and \(K_u\) by a factor of 3. Hence the advisability of using concave quartz mosaics is clear, since they make it possible to use large aperture angles rationally. True, their fabrication and especially fitting present great technical difficulties.
References
- L. D. Rozenberg, Sound Focusing Systems, Publishing House of the Academy of Sciences of the USSR, Moscow–Leningrad, 1949.
- J. Gieutzmacher, Zeits. f. Physik 96, 342 (1935).
- S. S. Tumanskiĭ, ZhTF 7, 2047 (1937).
- L. W. Labaw, JASA 16, 237 (1945).
- I. E. El’piner and A. P. Sheĭnker, Bulletin of Experimental Biology and Medicine, No. 7, 51 (1946).
- G. W. Willard, JASA 19, 733 (A) (1947).
- L. Fein, JASA 20, 583 (A) (1948).
- G. F. Müller and G. W. Willard, JASA 20, 589 (A) (1948).
- G. W. Willard, JASA 20, 589 (A) (1948).
- H. T. O’Neil, JASA 21, 60 (A) (1949).
- A. I. Gubanov, ZhTF 19, 30 (1949).
- J. W. Strutt (Rayleigh), The Wave Theory of Light, GTTI, Moscow–Leningrad, 1940.