Sources of Sound

When a single particle is acted on by a number of distinct forces, each of which would cause it to perform S.H.M., the question arises as to the resultant motion. A number of impor tant cases require consideration:—I. Vibrations of the Same Fre quency and in the same straight line. It may be shown that such a system of vibrations is reduced to a single resultant by means of a vector polygon, the angles representing relative phases, in exactly the same manner as a system of forces acting at a point.

II.

Two Vibrations of nearly the same frequency and acting in the same straight line: beats. This case has very important applica tions in sound and is of frequent occurrence. If two vibrations of the same amplitude and nearly equal frequency act together the resultant amplitude will at first be double the single amplitude. As the higher frequency vibration gains on the lower, however, thereby changing the relative phase, a point will be reached when they are in opposition and will neutralize each other (amplitude zero). This cyclic process goes on so long as both vibrations are maintained. If the two frequencies are N and N+aN, the re sultant amplitude will vary between 2a and o, the time interval between two successive maxima being 27 / ON , i.e., the frequency, of the beats will be aN, the difference between the two, nearly equal, frequencies. The phenomenon of beats is observed when two notes nearly in tune are heard together—a periodic rise and fall of intensity being noticed. As the beats between two notes increase in frequency a sensation of roughness or discord (dis sonance) sets in, and ultimately two independent notes of the diatonic scale are recognised. The beat effect has direct applica tion in the comparison of one sound frequency (e.g., that of a vibrating string) with a standard of frequency such as a tuning fork. The well-known heterodyne of radio-telegraphy or telephony is a beat effect between two sets of radio-frequency (e.g., cycles per second) oscillations, slightly mistuned. The beat-note is adjusted to have any frequency, say 500 periods per second, which falls within the audible range. III. Vibrations at right angles. In this case we are dealing with the motion of a particle on a plane surface. When the imposed vibrations have the same frequency, it is readily shown that the particle traces an elliptical path, which may vary from a straight line, when the phases differ by 18o°, to a circle when the phases differ by 90° and the amplitudes are equal. (a). Frequencies nearly equal. In this case

the particle follows a path which slowly changes through the various forms, straight line, ellipse, circle, etc., due to the slowly changing phase difference. The frequency of performance of a complete cycle of figures will, of course, be aN, the differ ence between the two nearly equal frequencies. (b). Frequencies Commensurate. 2:1, 3:1, 4:1, etc. Here again the particle traces out a curve having a certain number of loops—this number being equal to the ratio of frequencies. For nearly commensurate fre quencies the curve slowly changes as the phase difference varies. Numerous mechanical devices have been designed for drawing automatically the various "harmonic " curves obtained by compounding two or more harmonic vibrations. These generally consist of two compound pendulums controlling a common writing point, and each capable of performing simple harmonic motion in one or possibly two directions simultaneously. Beau tiful and fascinating designs may be traced in this way.

An optical method of exhibiting a small difference of frequency between two tuning forks was devised by J. A. Lissajous (1822 188o). A small mirror is attached to one prong of each fork, one of which vibrates in a vertical plane and the other horizontally. A narrow pencil of light is reflected successively from the two mirrors and falls on a screen. When the forks vibrate together the spot of light on the screen traces out the "resultant" curve of the two vibrations. If the two forks have the same frequency the figure is a stationary ellipse; if they differ slightly in frequency, the ellipse assumes successively the various forms including a straight line and a circle. These figures are known as Lissajous figures. The same principle may, of course, be applied to any two systems vibrating at right angles with approximately equal, or commensurate, frequencies. Recently Dye has employed the cathode-ray oscillograph for the harmonic comparison of very high electrical frequencies (of the order p.p.s.).

Fourier's Theorem.—Summation of any number of simple harmonic-vibrations of commensurate frequencies. Synthesis and analysis of complex wave-forms of vibration. This very important theorem due to J. B. J. Fourier (Theorie de la Chaleur. Paris, 1822) asserts that any single-valued periodic function whatever can be expressed as a summation of simple harmonic terms having frequencies which are multiples of that of the given function. (See