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I• is well known that to obtain very s•nall side lobe radiation the amplitude distribution across a radiating surface should be such that the maximum mnplitude is at the center and the minimum at the edges. The Gaussian distribution gives no side lobes, but requires an infinitely large radiator. This paper gives the analysis of the clampededge disk, vibrating in its first normal mode. It shows that the dynamic curve approximates the Gaussian form and that the sound pressure distribution has very small side lobes, the ampli•rle of the iirst side h•l}e being 33 decibels below that of the axial lobe.For a disk of a given diameter the width of the axial lobe may be decreased by raising the frequency of the first normal mode, which requires a corresponding increase in the disk thickness. At very high frequencies the disk thickness may become prohibitive. In such cases a thin disk can be forced to vibrate in the shape of the first normal mode by a proper radial distribution of the driving force. HIS analysis of the radiation from a clamped-edge disk grew out of the development of transducers for use in ultrasonic guidance devices for the blind. 1 Such devices make use of echo-ranging techniques, requiring that both the radiator and the microphone have directional characteristics with a sharp central beam and compromise is therefore necessary. The fundamental requiremen-t is that the radiator be circular, with maximum amplitude of displacement at the center falling off to a very small (or zero) value at the edge. The cosine-squared distribution and certain power series distributions have been suggested as possible alternatives. The very small side lobes. A directional characteristic * clamped-edge disk, driven at the frequency of its of this sort may be obtained by mounting a free-edge circular disk in the focal plane of a parabolic reflector and driving it at the frequency of one of its circular normal modes. • The resulting characteristic resembles that obtained from a rigid piston in an infinite baffle, the first side lobe having an amplitude approximately 13 percent of the amplitude of the central beam. While side lobes of this magnitude have been acceptable in the guidance device, smaller side lobes might result in improved performance, and in some applications might even be essential. Accordingly, the possibility of reducing the side lobes was investigated.It is well known that a circular radiating surface, having a Gaussian amplitude distribution, will produce a directional characteristic having a central beam and zero side lobes. Such a distribution is not a physical possibility, however, since it requires a radiator of infinite radius. A first normal mode, satisfies the above requirement and has a further advantage in that it does not require a complicated driving mechanism.In order to investigate the performance of theclamped-edge disk in more detail, the differential equation of the vibrating disk was solved subject to the boundary conditions of zero amplitude and zero slope at the edge of the disk. The equat...

In the attempt to obtain a sharp beam of supersonic sound from a parabolic horn, it became apparent that the sharpness of the beam was a function of the mode of vibration of the diaphragm exciting the horn. Experiments showed that the sharpest beams with the smallest side lobes were obtained when the diaphragm was radiating most of the energy toward the side walls of the horn rather than toward the mouth. Experiments with a free-edge disk for a diaphragm connected the wide angle radiation with the two nodal circle mode of vibration of the disk. Comparison between directional patterns measured with the disk in a flat baffle and calculated patterns confirmed the correlation between the wide angle radiation and the two nodal circle mode. Further analysis indicated that similar radiation patterns could also be obtained from one and three nodal circle modes of vibration. In the analysis, the differential equation for the vibration of a thin plate is solved for a free-edge circular disk assuming circular symmetry and considering a driving force applied at an arbitrary circle concentric with the disk. The dynamic curve of the disk is given for any frequency in the form of a series of the normal modes of the disk, and the directional pattern of the radiation from the disk mounted in an infinite baffle is calculated. The calculations have been so systematized and tabulated that the dynamic curve and directional pattern for any disk and frequency can be calculated quite readily. The analysis is sufficient to predict the disk diameter and thickness required for the one, two, or three nodal circle modes and for any frequency, when the disk is mounted in a flat baffle and when the maximum radiation is placed at an arbitrary angle from the axis. On the basis of the analytical predictions, a disk has been made to give the wide angle radiation required with the disk vibrating with one nodal circle instead of two nodal circles as discovered experimentally.

Direct-radiator loudspeakers are often mounted with the back of the diaphragm working into a completely enclosed space. Conventional theory states that when the maximum linear dimension of such an enclosure is small compared with the wave-length, the pressure is uniform throughout, and the acoustical impedance presented to the loudspeaker is --j/w(V/od), where V is the enclosed volume. Although it has not been clearly established how small an enclosure must be before it is "small compared with the wave-length," the foregoing expression is generally used, at low audiofrequencies, to calculate the acoustical impedance of closed loudspeaker housings.It is shown here that while the acoustical impedance of a closed rectangular housing is capacitive at very low frequencies, it passes through zero as the frequency increases and becomes that of an inertance as the frequency of the first normal mode is approached. For a typical housing 11 in. X22 in. X22 in., the frequency at

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