obtained by adding the corresponding ordinates of the two component curves. In the first two cases one component has the same wave length as the resultant, while the other has half that wave length. These two components have the same amplitude in the first case as in the second, but their relative phases are different in the two cases and hence the resultant curves are different. In the lower curve one component has one third the wave length of the resultant, while the component having half the wave length is absent or has zero amplitude.

315. Musical Tones. In tones suitable for music the upper partial tones almost exactly fall into the harmonic series, starting with the fundamental; that is, their frequencies are very nearly exact multiples of the frequency of the fundamental.

But the partial tones given out by bells and plates when struck do not correspond, even approximately, to the lower terms of the harmonic series, and are quite unsuited for music.

316. Koenig Resonators and Manometric Flames.-The French acoustician Koenig made use of resonators in which the front part was cylindrical and could

be pushed in or drawn out so that each could easily be adjusted in pitch.

FIG. 177.-Resonator and manometric flame.

To observe the vibration of the resonators he employed manometric flames. A small, flat, disc-shaped box or capsule of wood, was divided into two chambers by a thin membrane, such as goldbeater's skin. The cavity on one side of the diaphragm was connected by a short tube with a resonator, while the cavity on the other side had two openings, through one of which illuminating gas was admitted, while the other was connected with a fine jet where the gas burned in a small flame. The vibrations of the air in the resonator were transmitted through the diaphragm in the manometric capsule to the illuminating gas causing the flame to dance.

The image of such a flame viewed in a rotating mirror is drawn out in a band of light which when the flame is in oscillation shows serrations like saw teeth as shown in figure

178; the particular form of the serrations revealing the mode of vibration of the flame.

A

317. Sensitive Flames.-Under some conditions a gas flame may be very sensitive to sound. A small cylindrical jet is required having an aperture about .5 mm. in diameter, and the pressure must be such as to produce a long flame just on the brink of roaring. pressure of about 9 inches of water is commonly required. Such a flame is sensitive to the vibrations of exceedingly short waves of sound, and breaks into a shorter flaring or roaring flame when a bunch of keys is jingled in its vicinity or a sharp hiss given or a very high-pitched whistle sounded.

By means of sensitive flames sound waves so short as to be quite inaudible may be detected, and their interference and reflection studied.

INTERFERENCE AND BEATS.

318. Superposition of Waves.-When the same portion of fluid is traversed by two waves, the motion of the particle will be the resultant of the two and may thus become very complicated.

In case of surface waves in a liquid the eye can readily observe a series of short waves running over longer ones and preserving their motion as though over an undisturbed medium.

319. Interference of Waves. Suppose A and B are the centers of two series of waves of the same wave length and amplitude. Then there will be certain points, as at C, where waves from the two sources act together so as to produce great disturbance. Suppose the lines in the diagram represent the crests of waves, then at the points C the crest of one wave is

superposed on another, while at C' the troughs of two waves come together; a half period later the crests will be at C' and the troughs at C. There will result therefore along the line CC' a series of waves of double the amplitude of the original waves. At the points D, however, the crest of a wave from one center coincides with the trough of a wave from the other, therefore there will be at least a partial neutralization at points along the line DD'. If the waves coming together at D have equal amplitudes and equal wave lengths and are also simple harmonic waves they would separately produce at D equal and opposite displacements at every instant and will therefore completely neutralize each other.

This interaction of two sets of waves by which at certain points one is more or less com

A.

B.

D

D'

FIG. 179.-Interference of waves.

pletely neutralized by the other is known as interference.

320. Energy is Not Lost in Interference.-When there is complete interference at any point there is no motion of the medium and no energy at that point, but the energy of the two interfering waves is not lost or destroyed but appears at neighboring points (such as C, Fig.. 179) where the amplitude of the component waves are added. For at these points the energy of the resultant vibration is four times what it would be if one of the trains of waves were suppressed. There results from the interaction of the two wave systems a different distribution of energy, but the total energy remains unchanged.

321. Interference of Sound Waves.-The interference of sound waves is well shown in the following experiment. The sound waves from a tuning-fork (Fig. 180) enter a suitable receiver which is connected to an ear-piece by means of two tubes, one of which has a sliding portion by which its length can be varied. When the tubes are adjusted to be of equal length the sound of the fork is distinctly heard by the observer at E. As the sliding tube is drawn out, making one tube longer than the other, the sound grows fainter and reaches a minimum when one tube is

longer than the other by a half wave length of the sound waves sent out by the fork, for in this case the waves reach the ear through the two tubes in opposite phases and interfere. If the slider is drawn out still farther the sound increases in strength reaching a maximum when one tube is just a whole wave length longer than the other.

If two similar organ pipes of the same pitch are mounted on a rather small air chest, as shown in figure 181, and sounded simultaneously they will usually sound in opposite phases, owing to an oscillation of the air in the air chest itself. The sound waves coming from one pipe will thus interfere with those from the other and the fundamental tones will be almost completely neutralized, the higher harmonics will, however, still be heard.

Ear

FIG. 180.

FIG. 181. Interference
between organ pipes.

322. Beats. If two organ pipes sounding together are not exactly of the same pitch the sound comes in pulses or throbs called beats. For in this case one pipe is giving out more vibrations per second than the other and, consequently, the relative phases of the two are constantly changing, as shown in the following figure where the dotted curves represent the waves from the two pipes, one of which is supposed to give out eleven vibrations for every ten of the other. The full line represents the resultant motion. It is clear that while one pipe is gaining one complete vibration on the other, there will be an instant when the waves are in opposite phase and interfere and another instant when they will be in the same phase and strengthen each other. There will therefore be one hundred beats in the time in which one pipe has made one hundred more vibrations

than the other. Or if one pipe makes m vibrations per second and the other n, the number of beats per second is m - n.

Beats are easily heard when two adjoining notes on the piano or organ are simultaneously struck, and the lower the notes are on the scale the slower will be the beats. Beats do not occur, however, between notes that are very different in pitch. For example, no beats would be heard in case of two simple tones, one making 200 vibrations per second and the other 300 vibrations, though if one made 2000 vibrations and the

FIG. 182.-Formation of beats.

other 2100 there would be 100 beats per second, heard as a distinctly jarring roughness. The explanation of this was given by Helmholtz (§358).

In tuning two strings or two forks to unison they are adjusted until no beats are heard. If it is required that two tuning-forks shall be very accurately of the same pitch they may be tuned to make the same number of beats per second with a third fork, as it is easier to count accurately three or four beats per second than to distinguish between no beats at all and very slow beating.

STANDING WAVES AND VIBRATING BODIES.

323. Transverse Vibration of a Cord.-Take a long flexible rubber tube or other elastic cord fixed at one end, as at P (Fig. 183), and, holding it slightly taut, let the hand give a sudden movement to one side and back again.

A wave is set up as at A which runs the length of the cord, is reflected at P, and returns on the opposite side of the cord as shown at B. On reaching the hand it is reflected to A and the motion is repeated. Thus the cord makes a complete vibration and returns to its original form in the time in which a wave runs the length of the cord and returns.

If the wave is not sent out by so sharp a movement it may take the form shown in the lower part of the figure where the cord simply swings to and fro or vibrates sidewise.