the guitar, and the piano, the strings are fastened to posts in a sounding board or wooden resonator box.

212. Beats. When two sounding bodies that have nearly the same time of vibration are sounded together, the alternate interference and coincidence of the sound waves produce rhythmical variations in the intensity of the sound. These are called beats and can be readily heard.

If the waves sent out by the two sources are represented by the curves in the upper part of Fig. 194, we can see that one body must vibrate ten times while the other vibrates eleven. This means

F

FIG. 194

that at every tenth wave of one, and every eleventh wave of the other, the waves will interfere, while at times midway between the waves will assist each other. The curve resulting from these two sets of waves is shown in the heavy line. The beat would come between F and G, at that part of the curve which swings the greatest distance from the straight line AB, while at A or B, where the interference is nearly complete, there would be very little sound. The heavy curve is constructed as follows: draw a set of vertical lines across the straight line AB; then any point e on the curve will be found by making de equal to the sum or difference of ab and ac, depending upon whether they are upon the same or opposite sides of AB. It is evident that in order to make a curve that will represent several beats between two musical sounds, a much more extended drawing would be required.

If two tuning forks, giving 128 and 129 vibrations per second respectively, are sounded together, they will be in

the same phase once per second and in opposite phases a half second later. This combination gives one beat per second. If a fork giving 130 vibrations is sounded with the one giving 128, they will be in the same phase twice per second and in opposite phases a quarter second later, and there will be two beats per second. By vibrating together forks with greater differences in the number of their vibrations, it is seen that the number of beats between two sounds is equal to the difference between the numbers of their vibrations per second. This can be demonstrated by the use of two forks that vibrate in the same time. By pressing a piece of beeswax upon one prong of one of these forks, it can be made to give fewer vibrations, and a beat will be heard when the forks are sounded together. By increasing the load, pressing a shot into the wax if needed, a greater number of beats per second will be produced.

213. Properties of Musical Tones. In order that a vibrating body may produce a musical tone, its vibrations must be rapid, continuous, and isochronous. A musical tone may be a simple tone, in which the vibrations are all alike, or it may be a compound tone, formed of a combination of two or more vibrations. A noise differs from a musical tone in being formed by a mixture of a great variety of vibrations that cannot be resolved into simple ones. A tuning fork gives a simple tone; a piano string, a compound tone; and the fall of a pile of lumber, a noise.

The principal characteristics of musical tones are intensity, pitch, and quality.

214. Intensity. The intensity of a sound depends upon three things: the amplitude of the vibration producing it, the area, and the distance at which the sound is heard.

(a) Amplitude. -When a tuning fork is struck, the energy which it can impart to the air will depend upon the extent of its vibrations. If these are slight, only a weak tone is produced. Strike a harder blow, and the amplitude increases, the energy the fork can give to the air is greater, and the sound is louder. The relation between amplitude and intensity may be readily shown by substituting a tuning fork for the whalebone in the demonstration in § 190. Make a number of traces on the glass when the fork is sounding at different intensities, and compare them.

(b) Area. A small tuning fork, on being put into vibration, sets only a small quantity of air in motion and gives a sound having but little intensity; but if the prongs are broad, the amount of air put in motion is greater and the sound is louder.

(c) Distance. Since the sounding body is sending out waves in every direction, the sound wave is the outside of a spherical shell of which the body is the center. The shell of molecules to be vibrated becomes larger and larger as the sound wave passes out from the center, and hence the that can be imparted to each air molecule becomes less and less. The intensity of sound depends directly on the amount of this energy.

FIG. 195

energy

B

Suppose a source of sound is at the point A, Fig. 195. Suppose this to be the center of a spherical shell of which the radius is AB: the wave of sound produced by the sounding body will be received all over the surface of the sphere. If now this sphere is replaced by a larger one, of which the radius is AC, the same sound wave will be received over the larger surface and the intensity of

the sound on each unit of area of this surface will be decreased.

The areas of these surfaces are directly proportional to the squares of their radii; hence we may write the intensity of sound varies inversely as the square of its distance from the sounding body.

If the waves of sound can be kept in one direction, as by being reflected from the inner surface of a tube, the intensity at any point will be greater and they will go much farther. Speaking tubes are made on this principle.

215. Pitch. The pitch of a tone depends upon the number of vibrations made per second by the vibrating body that produces it, the pitch being relatively high when the vibrations are rapid, and low when they are slow.

Since the velocity of sound is the product of the wave length and the number of vibrations per second, it is evident that the wave length is greater for tones of a low pitch than for those of a high pitch. It is also evident that as the period (or length of time of one vibration) becomes greater the pitch becomes lower.

When the corner of a stiff card is drawn across the cover of a cloth-covered book, a certain sound will be heard. On moving the card more rapidly, the pitch of the sound produced is made higher than at first.

The speed at which a circular or "buzz" saw is running can be judged by the pitch of the tone which it gives when sawing a log. A knot in the log lessens the speed and lowers the pitch of the tone.

216. The Siren. - The name siren is given to an instrument used to determine the number of vibrations required to produce tones of different pitch, as well as to show the re

lation between them. A simple form and the method of using it are described in the following:

Demonstration.

Cut out a disk of bristol board, or, better, of

some thin metal, 30 cm. in diameter. From the center describe four concentric circles, with diameters of 28, 24, 20, and 16 cm. respectively. Divide these circles into 32, 24, 20, and 16 parts respectively, and drill holes 6 mm. in diameter through the disk at these points.

Fit the disk to a rotating machine. Into each end of a rubber tube fit a glass tube; and holding one end directly opposite to one row of holes, put the disk in rotation and blow through the tube. If the rotation is begun very slowly, the separate puffs of air can be heard as they go through the holes in the disk and are then cut off; but if the speed is increased, the puffs will link themselves into a musical tone and the pitch will continue to rise as long as the speed is increased.

FIG. 196. Siren

Rotate the wheel with uniform velocity and blow through the holes in the different circles, beginning with the smallest and going to the largest. Is the result pleasing? Describe it. Compare with the effect produced by blowing through all four at once.

A little practice will enable one to turn the handle of the wheel uniformly. By counting the number of turns given to the handle per minute the number of rotations of the siren wheel can be found. The product of the number of rotations of the siren wheel per second by the number of holes in the circle used will give the number of vibrations per second for the tone produced.

217. Doppler's Principle. When both the sounding body and the ear that hears the sound are stationary, the number of waves that strike the ear per second is the same