Rayleigh obtained general verification of this theory by means of a speaking trumpet and high pitched sounds—a considerable concentration of sound on the axis being obtained. Boyle, using a Langevin type of piezoelectric quartz disc (see p. 12) showed that the sound distribution (the polar diagram) in water was in good agreement with theory. In one example R = 7.65 cm., N = 135,000 p.p.sec., c= i.5 X cm./sec. for water, X= i•I cms. whence a = approximately. This angle was verified experimentally. The application of such a high frequency directional sound beam has already been mentioned.
Double Sources.—The "piston" radiator to which we have referred is a single source of sound—it radiates from one face only, the anterior face being suitably screened. A vibrating dia phragm mounted on a ring radiates from both faces, the radiation from one face being at any instant in opposite phase to that from the other face. Such a vibrator is described as a double source. As we have seen (see p. 13) a double source, such as a diaphragm on a ring, situated in an open space, air or water, is inaudible from any point in its equatorial plane, maximum sound being received at right angles to this plane.
Nondirectional Source Used with Mirrors, Trumpets, Zone Plates, etc.—As in the case of piston sources of large area, a "directed" source, employing for example a mirror or a trumpet, can exhibit good directional properties only if the dimensions of the mirror, etc., are not small compared with the wave-length of the sound. This condition involves bulky apparatus if the wave length is appreciable, the alternative being the use of high fre quency sounds of short wave-length. If the beam is to be narrow, it is also desirable that the area of the source at the focus of a mirror for example should not be large. This involves a further difficulty when a large sound-output is required, for the source must then have a very large amplitude, a condition which leads to inefficient transmission both in air and in water. In air, serious degradation of energy may take place at large amplitudes (see Hart, Roy. Soc. Proc., A. 105, 1924, p. 8o) and in water "cavitation" troubles arise (see Boyle, Roy. Soc. Proc., Canada, III, 1922, p. 157). The use of mirrors, trumpets, zone plates, etc., are consequently more suited to the directional reception of high frequency sounds—where such troubles do not arise. Concave
mirrors of say 1 to 2 feet diameter and 6 to 12 inches focal length are fairly efficient reflectors of high pitched sounds like a watch tick or the notes from a Galton's whistle.
Multiple Sources.—As in the case of a large piston vibrator, a line of small "point," sources, suitably arranged, will have definite directional properties. Suppose we have "m" equidistant sources in a straight line vibrating with the same phase, ampli tude, and frequency. We require to know the polar distribution of amplitude in any plane passing through the line of m sources. (See fig. 8a.) It is readily proved that the resultant amplitude r at a distant point P is given by if d is the spacing-distance of the sources each of amplitude i/m, and a is the orientation of the point P with respect to the line source. An important case arises when d=X/ 2, i.e., when the sources are spaced half a wave-length apart. Then This is zero whenever sin (7r/2.m cos a) is zero, i.e., whenever cosa.m/2 is an integer. Suppose, for example there are 6 sources spaced X/2 apart, then m.cosa./2 may be 1, 2, or 3, whence a= 48', 42' and o° giving the directions OP of zero am plitude. The primary maximum occurs when a= 90° and second ary maxima at a= 60° and 30° approx. The polar distribution from o to 7r is shown in fig. 8b the value of r being plotted radi ally from 0 in the direction a. This distribution is the same in all planes which include the line of sources. When d has other values, the polar distribution includes secondary maxima which may approach the primary in magnitude (see paper by H. Stenzel, Elecktrische Nachrichten Technik, Band 4, Heft 6, PP- 253). It will be seen that a vertical line of sources spaced X/2 apart will give a definite concentration of energy in a horizontal plane at right angles to the line of sources. Such a concentration is, for example, of considerable importance in a fog signalling device, where a maximum energy concentration is required in a horizontal plane surrounding a light-vessel. It may similarly be shown that a number of equidistant sources arranged on a circle and vibrating in phase will give a primary maximum of inten sity on the axis, with a number of zero and secondary maxima positions as in the case of a disc.