Focusing of Ultrasound in Water by a Concave Mirror

Fox and Griffin investigated the focusing of ultrasound by a concave mirror. In the first of their papers,^1 proceeding from the usual classical expressions valid for focusing systems with small aperture angles, the authors calculate the gain coefficient of a concave mirror—

la, by the latter meaning the ratio of the intensity at the center of the focal spot to the intensity in the plane incident wave. As was to be expected, the result obtained by them is identical with the classical expression for the intensity at the center of the Airy disk, well known in optics.

In addition to the maximum amplification coefficient, the authors introduce the concept of an average amplification coefficient, by the latter meaning the ratio of the energy flux passing through the Airy disk to the intensity in the incident wave.

Calculations are made both for a circular and for a rectangular aperture. The complicated expressions obtained in the latter case are tabulated (which, in fact, is the only valuable part of this work).

In the experimental work² the ultrasonic beam was emitted by a quartz plate measuring \(12 \times 15\) mm at a frequency of \(4.25\) MHz, with a total radiation power of about \(2\) W. As a concave mirror an ordinary watch glass was used, with a diameter of \(34\) mm and a radius of curvature of \(68\) mm. When the focus of the mirror coincided with the surface of the water, a fountain approximately \(10\) cm high arose.

Measurements were made in several ways: by the cover-glass method and with a spherical radiometer; in addition, the power radiated by the quartz was calculated from the voltage applied to the quartz and the quality factor of the quartz, measured from the electrical side. In measuring the intensity in the plane wave emitted by the quartz, all three methods gave results in good agreement. In measuring the amplification coefficient, results were obtained that agreed well with the calculated values given in the first paper. Thus, for example, the largest amplification coefficient from the experiments proved to be \(70\), while the calculated value was \(74\).

To determine the law of energy distribution in the focal spot, the following experiment was performed: the intensity in the focus was observed while part of the surface of the mirror aperture was covered by an opaque screen. The results obtained also agreed well with the calculated data, which is indirect confirmation of the correctness of the derived distribution law.

As an extrapolation, it was found that at the highest voltages applied to the quartz in this work (\(1200\) V), the pressure amplitude in the focus reached \(41\) atm. No cavitation was observed in this case, which is in complete agreement with the work of Muller and Willard³, who likewise did not observe cavitation at a pressure amplitude of \(120\) atm at a frequency of \(5\) MHz.

The appendix to the paper gives a derivation of the magnitude of the power radiated by the quartz, from the applied voltage and the mechanical quality factor.

L. D. Rozenberg

CITED LITERATURE

1. V. Griffing and F. Fox, J.A.S.A. 21, 348 (1949).
2. F. Fox and V. Griffing, J.A.S.A. 21, 352 (1949).
3. I. F. Muller and G. W. Willard, J.A.S.A. 20, 589 (1948).