Sounds interfere to re-enforce each other, or to produce silence; just as the crest of one wave in water may be superposed on the crest of another, or may apparently destroy all motion by filling up its trough. The simplest mode of showing this is to hold near the car a vibrating tuning-fork and turn it slowly round its axis. In some positions, the sounds from the two branches re-enforce, in others they weaken, each other. But if, while the sound is almost inaudible, an obstacle be interposed between the ear and one of the branches, the sound is heard distinctly. Beats, which will shortly be alluded to, form another excellent instance.
To give an idea of the diminution of loudness or intensity of a sound at a distance from its source, let us consider a series of spherical waves diverging from a point. The length of a wave, as we know from the theory, does not alter as it proceeds. (Indeed, as we shall presently see, the pitch of a note depends on the length of the wave; and we know that the pitch is not altered by distance.) Hence, if we consider any one spherical wave, it will increase in radius with the velocity of 'sound, but its thickness will remain unaltered. The same disturbance is thus constantly transferred to masses of air greater and greater in proportion to the surface of the spherical wave, and therefore the amount in a given bulk (say a cubic inch) of air will he inversely proportional to this surface. But the surfaces of spheres (q.v.) are as the squares of their radii—hence the disturbance in a given mass of air, i.e., the loudness of the sound, is inversely as the square of the dis tance from the source. This follows :a once from the law of conservation of energy (see FORCE), if we neglect the portion which is constantly being frittered down into heat by fluid friction. All sounds, even in the open air, much more rapidly in rooms, are ex tinguished ultimately by conversion into an equivalent of heat. Bence sounds really in intensity at a greater rate than that of the inverse square of the distance; though there are cases on record in which sounds have been beard at distances of nearly 206 miles. But if, as in speaking-tubesand speaking-trumpets, sound be prevented from diverging in spherical waves, the intensity is diminished only by fluid friction, and thus the sound is audible at a much greater distance, but of course it is confined mainly to a particular direction.
As already remarked, the purest sounds are those given by a tuning-fork, which (by the laws of the vibration of elastic solids) vibrates according to the same law as a pendu lum. and communicates exactly the same mode of vibration to the air. If two precisely
similar tuning-forks be vibrating with equal energy beside each other, we may have either a sound of double the intensity, or anything less, to perfect silence, according to their relative phases. If the branches of both be at their greatest elongations simulta neously, we have a doubled intensity—if one be at its widest, and the other at its nar rowest. simultaneously, we have silence, for the condensation produced by one is exactly annihilated by the rarefaction produced by the other, and rice versa. But if the branches of one be loaded with a little wax, so as to make its oscillations slightly slower, it will gradually fall behind the other in its motion, and we shall have in succession every grade of intensity from the doable of either sound to silence: The effect will be a periodic swelling and dying away of the sound, and this period will be longer the more nearly the two forks vibrate in the same time. This phenomenon is called a beat, and we see at once from what precedes, that it affords an admirable criterion of a perfect unison, that is, of two notes whose pitch is the same. It is easy to see, by the same kiwi of reasoning, that if two forks have their times of vibration nearly as 1:2, 2: 3, any simple numerical ratio—there will he greater intervals between the beats according as the exact ratio is more nearly arrived at.
We must now consider, so far as can be done by elementary reasoning, the various simple modes of vibration of a stretched string, such as the cord of a violin Holding one end of a rope in the hand, the other being fixed to a wall, it is easy (after a little practice) to throw it into any of the following forms, the whole preserving its shape, but rotating round the horizontal line.
If the tension of the rope be the same in all these cases, it is easy to see that the times of rotation must be inversely as the number of equal segments into which the rope is divided; for the va rious parts will obviously have the same form; and the masses and dis• tances from the axis of rotation being proportional to their lengths, the cen trifugal forces (q.v.) will be as the squares of the lengths, and inversely as the squares of the times of rotation.