The Directional Characteristics of a Clamped-Edge Disk

Abstract

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...