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order then to unite with the motion of the prong this downward motion of the particles must get back to the mouth when the prong is just starting down from A the second time; that is, after one complete vibration of the prong. This shows why the pipe length is one-half wave length.

394. Resonators. If the vibrating fork at the mouth of the tubes in the preceding experiments is replaced by a train of waves coming from a distant source, precisely the same analysis leads to the conclusion that the waves reflected from the bottom of the tube will reënforce the oncoming waves when the length of the tube is any odd number of quarter wave lengths in the case of a closed pipe, or any number of half wave lengths in the case of an open pipe. It is clear, therefore, that every air chamber will act as a resonator for trains of waves of a certain wave length. This is why a conch shell held to the ear is always heard to hum with a particular note. Feeble waves which produce no impression upon the unaided ear gain sufficient strength when reënforced by the shell to become audible. When the air chamber is of irregular form it is not usually possible to calculate to just what wave length it will respond, but it is always easy to determine experimentally what particular wave length it is capable of reënforcing. The resonators on which tuning forks are mounted are air chambers which are of just the right dimensions to respond to the note given out by the fork.

395. Forced vibrations; sounding boards. Let a tuning fork be struck and held in the hand. The sound will be entirely inaudible except to those quite near. Let the base of the sounding fork be pressed firmly against the table. The sound will be found to be enormously intensified. Let another sounding fork of different pitch be held against the same table. Its sound will also be reënforced. In this case, then, the table intensifies the sound of any fork which is placed against it, while an air column of a certain size could intensify only a single note.

The cause of the response in the two cases is wholly different. In the last case the vibrations of the fork are transmitted

through its base to the table top and force the latter to vibrate in its own period. The vibrating table top, on account of its large surface, sets a comparatively large mass of air into motion and therefore sends a wave of great intensity to the ear, while the fork alone, with its narrow prongs, was not able to impart much energy to the air. Vibrations like those of the table top are called forced because they can be produced with any fork, no matter what its period. Sounding boards in pianos and other stringed instruments act precisely as does the table top in this experiment; that is, they are set into forced vibrations by any note of the instrument and reënforce it accordingly.

396. Beats. Since two sound waves are able to unite so as to reënforce each other, it ought also to be possible to make them unite so as to interfere with or destroy each other. In other words, under the proper conditions the union of two sounds ought to produce silence.

Let two mounted tuning forks of the same pitch be set side by side, as in Fig. 352. Let the two forks be struck in quick succession with a soft mallet, for example, a rubber stopper on the end of a rod. The two notes will blend and produce a smooth, even tone. Then let a piece of wax or a small coin be stuck to a prong of one of the forks. This diminishes slightly the number of vibrations which this fork makes per second, since it increases its mass. Again, let the two forks be sounded together. The former smooth tone will be replaced by a throbbing or pulsating one. This is due to the alternate destruction and reënforcement of the sounds produced by the two forks. This pulsation is called the phenomenon of beats.

FIG. 352. Arrangement of forks for beats

The mechanism of the alternate destruction and reënforcement may be understood from the following. Suppose that one fork makes 256 vibrations per second (see the dotted line AC in Fig. 353), while the other makes 255 (see the heavy line AC). If at the beginning of a given second the two forks

are swinging together, so that they simultaneously send out condensations to the observer, these condensations will of course unite so as to produce a double effect upon the ear (see A', Fig. 353). Since now one fork gains one complete vibration per second over the other, at the end of the second considered the two forks A

B

will again be vibrating xxxän

Α'

B'

together, that is, sending out condensations which add their effects as before (see C'). In the middle of this second, however, the two forks are vibrating in opposite directions (see B); that is, one is sending out rarefactions while the other sends out condensations. At the ear of the observer the union of the rarefaction (backward motion of the air particles) produced by one fork with the condensation (forward motion) produced by the other results in no motion at all, provided the two motions have the same energy; that is, in the middle of the second the two sounds have united to produce silence (see B'). It will be seen from the above that the number of beats per second is equal to the difference in the vibration numbers of the two forks.

FIG. 353. Graphical illustration of beats

To test this conclusion, let more wax or a heavier coin be added to the weighted prong; the number of beats per second will be increased. Diminishing the weight will reduce the number of beats per second.

In tuning a piano the double and triple strings are brought into unison by tuning so as to eliminate beats.

397. Interference of sound waves by reflection. Let a thin cork about an inch in diameter be attached to one end of a brass rod from one to two meters long. Let this rod be clamped firmly in the middle, as in Fig. 354. Let a piece of glass tubing a meter or more long and from an inch to an inch and a half in diameter be slipped over the cork, as shown. Let the end of the rod be stroked longitudinally with a well-resined cloth. A loud, shrill note will be produced.

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This note is due to the fact that the slipping of the resined cloth over the surface of the rod sets the latter into longitudinal vibrations, 30 that its ends impart alternate condensations and rarefactions to the layers of air in contact with them. As soon as this note is started

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FIG. 354. Interference of advancing and retreating trains of sound waves the cork dust inside the tube will be seen to be intensely agitated. If the effect is not marked at first, a slight slipping of the glass tube forward or back will bring it out. Upon examination it will be seen that the agitation of the cork dust is not uniform, but at regular intervals throughout the tube there will be regions of complete rest, n¡, Ng, Ng, etc., separated by regions of intense motion.

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The points of rest correspond to the positions in which the reflected train of sound waves returning from the end of the tube neutralizes the effect of the advancing train passing down the tube from the vibrating rod. The points of rest are called nodes, the intermediate portions loops or antinodes. The distance between these nodes is one-half wave length, for

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FIG. 355. Distance between nodes is one-half

wave length

at the instant that the first wave front a, (Fig. 355) reaches the end of the tube it is reflected and starts back toward R. Since at this instant the second wave front a, is just one wave length to the left of the two wave fronts must meet each other at a point n,, just one-half wave length from the end of the tube. The exactly equal and opposite motions of the particles in the two wave fronts exactly neutralize each other. Hence the point n, is a point of no motion, that is, a node. Again, at the instant that the reflected wave front a, met the advancing wave front a, at n the third wave front a, was just one wave length to the left of n,. Hence, as the first wave front a continues

to travel back toward R it meets a at n, just one-half wave length from n2 and produces there a second node. Similarly, a third node is produced at n,, one-half wave length to the left of n,, etc. Thus the distance between two nodes must always be just one half the wave length of the waves in the train.

In the preceding discussion it has been tacitly assumed that the two oppositely moving waves are able to pass through each other without either of them being modified by the presence of the other. That two opposite motions are, in fact, transferred in just this manner through a medium consisting of elastic particles may be beautifully shown by the following experiment

with the row of balls

used in § 380.

Let the ball at one end of the row be raised a distance of, say, 2 inches and the ball at the other

FIG. 356. Nodes and loops in a cord

Black line denotes advancing train; dotted line, reflected train

end raised a distance of 4 inches. Then let both balls be dropped simultaneously against the row. The two opposite motions will pass through each other in the row altogether without modification, the larger motion appearing at the end opposite to that at which it started, and the smaller likewise.

Another and more complete analogy to the condition existing within the tube of Fig. 354 may be had by simply vibrating one end of a two- or three-meter rope, as in Fig. 356. The trains of advancing and reflected waves which continuously travel through each other up and down the rope will unite so as to form a series of nodes and loops. The nodes at c and e are the points at which the advancing and reflected waves are always urging the cord equally in opposite directions. The distance between them is one half the wave length of the train sent down the rope by the hand.

QUESTIONS AND PROBLEMS

1. Account for the sound produced by blowing across the mouth of an empty bottle. The bottle may be tuned to different pitches by adding more or less water. Explain.

2. Explain the roaring sound heard when a sea shell, a tumbler, or an empty tin can is held to the ear.

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