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waves which run back. In such a tube since the wave is prevented from spreading out, the vibrations do not become less energetic as the wave advances, except from back reflections and from friction against the walls of the tube, and, therefore, the sound is heard with only slightly diminished intensity at the distant end.

When one end of a rod or wire of metal or of a long uniform beam of wood is struck the sound is carried along the rod or beam just as in a speaking tube with very little loss through waves sent out sidewise, and is therefore very distinctly heard at the farther end.

In ear trumpets, by the constraint of the smooth walls of the tube, the wave entering the wide end is gradually diminished in area till it emerges at the small end conveying all the energy that entered at the large end. Thus if the area of the large end is 100 times that of the small end the energy per cubic centimeter in the emergent wave is 100 times as great as in the wave which entered the trumpet, neglecting the loss by friction, etc.

303. Megaphone and Speaking Trumpet.—In the megaphone sound waves coming from the speaker instead of spreading out in all directions from the mouth are limited by the walls of the instrument, so that the wave emerging at the wide end has the whole energy of the voice. It will be shown in connection with the diffraction of light that when light waves pass through an opening which is not more than a wave length in diameter, the waves spread out in every direction from the opening, while if the opening is much larger, the waves, on account of interference, will not spread out so much but will travel straight forward, illuminating a spot directly opposite the opening. For precisely the same reason sound waves coming directly from the mouth spread out in every direction while waves from the larger opening do not spread out so much, and produce a more intense effect directly ahead.

Precisely how this effect is caused by the interference of waves will be better understood after studying the diffraction of light.

304. Pitch. The pitch of a sound depends on the frequency of the vibrations.-This is well shown by Savart's wheel. If a card is held so that it is struck by the teeth of a rotating cogged

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wheel a sound is given out which rises steadily in pitch as the speed of the wheel increases. If a device indicating the number

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of revolutions is attached to the wheel the number of taps per second producing a sound of a given pitch is readily determined.

FIG. 169.-Siren.

Another instrument by which the number of vibrations may be determined is the siren devised by Cagniard de la Tour, shown in figure 169. A disc having a circular row of equidistant holes is mounted on an axis so that it can rotate almost in contact with the upper surface of a flat circular box in which holes are made exactly corresponding to those in the disc, so that as the disc rotates the holes are alternately opened and closed as many times in each revolution as there are holes in the series. The box is connected by a tube with a bellows and the puffs of air that come through the holes of the disc as it rotates give rise to a tone which is higher in pitch the faster

the disc rotates. A revolution counter is attached to the axle so

that the speed may easily be determined.

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The holes of the disc are inclined one way and those in the upper plate of the box are oppositely inclined, so that the blast of air through the holes causes the rotation to take place automatically, the speed being controlled by the strength of the blast and a brake if necessary.

For finding the number of vibrations of a tuning-fork the graphic method may be used, illustrated in figure 170. A point or stylus is fixed to one prong of a tuning-fork which is mounted so

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that the stylus just touches a sheet of smoked paper stretched over a cylindrical drum. The axle of the drum is a coarse screw by which the drum is moved slowly lengthwise as it rotates. If the fork is set vibrating, on rotating the drum a wavy curve will be drawn in helical form around the drum, each wave corresponding to a vibration of the fork. To find the number per second a second curve may be simultaneously drawn alongside of the first by a tuning-fork whose frequency of vibration is known; or a small electric marker connected with a clock may be mounted with its point touching the drum close beside the stylus of the fork, so that its marks made every second lie close to the curve

drawn by the fork. The number of vibrations of the fork per second are found by counting the undulations in the curve between two consecutive time marks.

These various methods of experiment show that the pitch. of a sound depends only on the frequency of vibration, and that it makes no difference whether the sound comes from a tuningfork or from the puffs of a siren or the taps of Savart's wheel, all will have the same pitch if the frequency is the same.

305. Doppler's Principle. The pitch of a sound as heard depends on the number of waves that reach the ear per second. Consequently if the ear is moving toward the sounding body the apparent pitch will be raised, since more waves per second will meet the ear; and conversely if the ear is moving away from the sounding body it will receive fewer waves per second than if it were at rest and the pitch will appear lower.

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Let E (Fig. 171) represent the position of the ear and S that of a sounding body making n vibrations per second, and let the distance from A to E be V, the distance that sound travels per second. Then between A and E there are n sound waves which will reach the ear in one second if it remains at E, but if in one second the ear advances a distance v to E' it will meet in addition the waves between E and E'. Let x be the number of waves between E and E' and n' the number reaching the ear per second, then

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the sign being plus or minus according as the ear has a velocity v toward, or away from, the sounding body.

A similar change in pitch is observed when the sounding body is moving toward or away from the observer. But in this case the formula is somewhat different as the wave length of the sound is changed in consequence of the motion.

Let S, the source of sound, have a velocity v toward the observer at E. In one second as it advances from S to S' it gives out n waves. The first of these waves leaving it at S has reached A, having advanced a distance V equal to the velocity

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of sound in the medium, by the time that the sounding body giving out the nth wave has reached S'. All n waves therefore V-v lie between A and S', and the wave length, ', is

n

But the number of waves that will reach E per second will be the number of wave lengths that are contained in the distance

that the waves travel per second, or n'

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V
X

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where the minus sign is to be taken when the velocity of the sounding body is toward the hearer.

Doppler's principle explains the sudden lowering in pitch observed in a locomotive whistle as it passes, and why a bicycle bell on an approaching wheel is heard of a higher pitch than when it is receding. It has also a most interesting application to light waves (§906).

RESONATORS AND ANALYSIS OF SOUND.

306. Quality of Sound.-The ear readily observes the difference in quality or timbre between the sound of a violin and that of a flute or between notes of an organ and those of a piano though of the same pitch.

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