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VIBRATION OF A STRETCHED STRING.

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former position. But the momentum thus acquired will not allow it to stop here. It will (if we disregard all loss of momentum by its transference to other bodies) proceed as far beyond the position of equilibrium as it was disturbed therefrom at first, and will thus continue to vibrate or oscillate through an equal space on each side of the position of equilibrium. And it would do so for ever, were it not for the action of what are called retarding forces or resistances, such as friction, resistance of the air, &c.; or, in other words, the motion is shared among gradually increasing quantities of matter until its intensity becomes inappreciably small.

59. The vibrations or oscillations thus produced are usually communicated to the surrounding air, which communicates corresponding vibrations to the ear, which recognizes them as sound, provided they be very rapid. The modes of vibration are very different in different bodies. We have already seen how a bell vibrates *; let us now inquire into the modes of vibration of a string stretched tightly between two fixed points A B, Fig. 18. On drawing the string aside with the finger, Fig. 18.

A.

as in the guitar or harp, or with a bow, as in the violin, or with a hammer, as in the piano, the force employed converts the right line AB into a curved one A a B. Now it is obvious that the line A a B is longer than A B, and the string, in order to occupy the longer path, must have its fibres or particles separated or strained somewhat further apart. The moment the disturbing force

* Natural Philosophy. Sec. 15.

ceases to act, the string, by its elasticity, recovers its position A B, but the momentum it has acquired carries it almost as far on the other side of AB as the original displacement, namely, to a b B. The friction of the air and the fixed points A B diminish the momentum of the string, so that on the rebound it reaches only to AC B, thence to A a B, the oscillations still go on, diminishing to A e B, to Afв, and, finally, to a state of rest.

Fig. 19.

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The motion of any one point in this string is seldom merely backwards and forwards in a straight line, or even in an ellipse, but more frequently it describes such curves as that shown at c, Fig. 18, or those in Fig. 19, indicating actions of extreme complexity, arising from the various unknown molecular forces of the solid.

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60. Now it must be remarked, that whether these vibrations be large or small they are performed in equal times, because the farther the string is removed from the position of rest the greater is the elastic tension, and consequently the greater the momentum and velocity with which it returns to its original position; and these two elements, the extent of the displacement and the rapidity of the return, are found to increase in exactly the same ratio, a law which applies also to the oscillation of the pendulum.

Every passage of the string over the line of rest A B, Fig. 18, is called a vibration, and the mode of comparing two or more velocities of vibration is to choose some small unit of time, such as a second, and to calculate the number of vibrations which occur in such unit. 61. During every vibration of a sounding body the air participates in its motions, and conveys to the ear

SOUNDS AND NOISES.

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a wave of sound. We have endeavoured to represent in Fig. 20 what takes place in the air during the ring

Fig. 20.

ing of a bell. Every vibration of the bell sends forth a circular wave which spreads in every direction. Now as these vibrations are isochronous, or equal-timed, all the concentric circles are equidistant, like those which proceed from the place where a stone has been dropped into water. Such waves spreading through the air, and therefore breaking upon the ear at equal intervals, constitute a musical note. In a noise, or unmusical sound, the waves follow each other like those of the sea, with no regularity either in their intervals or their intensities. The more regular they may be, the more clear or musical is the sound. The waves of sound are of course not circles but spheres, not spreading in one plane only but upwards, downwards, and on every side.

It is further necessary, to constitute a musical note, that the waves succeed each other at least sixteen times in a second, otherwise they will be each heard separately, constituting a rattle. But when they are just too rapid to be distinguished separately, they form a very low musical note; and the more rapid the higher will be the note. A shrill whistle is due to several thousand vibrations in a second.

62. Now let us suppose the number of vibrations

of a certain string are 100 in a second. On shortening the string we increase the number of vibrations per second in the very same ratio (inversely); so that it will, if shortened exactly one-half, have its vibrations exactly doubled: it will vibrate 200 times in a second, and this will yield a note exactly an octave higher than the former one *.

63. But if, instead of varying the length of the string, we vary the tension or force with which it is stretched, we get different results, for the rapidity of vibration is found to be increased as the square root of the tension. Let the string be arranged as in Fig. 21, where one end is fixed securely by a peg; the string then passes over a wedge and a pulley, and is stretched by a weight, the effective vibrating length

a

Fig. 21.

of the string being that situated between the two points of support. Let us suppose the string to vibrate 100 times in a second. Now in order to make the vibrations twice as rapid, we must increase the weight four times to make them three times more rapid, the weight must be increased the square of three, or nine times, and so on. The same arrangement will also prove the fundamental fact, that below a certain rapidity of vibration no sound is produced. If the stretching weight be very small the vibrations will be sufficiently slow for the eye to follow them, and if the number be less than 16 in a second no sound will be heard.

64. The reader may be surprised at the number of vibrations made by these strings in a second, and may

* By doubling the thickness of a cylindrical string we obtain four times the bulk or mass, and we get the number of vibrations doubled in the same space of time. In order, therefore, to study the effect of length only, we must have strings of the same thickness.

NUMBER OF VIBRATIONS DETERMINED.

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wish to know by what means these high numbers are ascertained. We will endeavour to inform him; but first it is necessary to remark, that the same note produced on any musical instrument is due to the same number of vibrations per second. Thus the tenor c, which is produced by a string vibrating 256 times in a second, as in a piano, is also produced in the flute by a column of air vibrating the same number of times per second, and also in the human voice by two chords (called the chorda vocales) contained in the upper part of the windpipe, also vibrating the same number of times per second. And however different the quality of these notes, as given by different instruments may be, they all agree in pitch, and this is determined by rapidity of vibration.

A

Fig. 22..

65. There are various methods of determining the number of vibrations per second required for the production of any note. An ingenious little instrument, called the Syren, has been invented for this purpose. It is represented in Fig. 22, in which a is a cylindrical chamber of brass. In the instrument used by the writer, this chamber is 3 inches in diameter, and 13 inch high. A pipe, B, 3 inches long, screws into an orifice in the lower part of this chamber and fits the tube of a double pair of bellows, or other machine capable of supplying an equable blast of air. The upper surface of the air chamber, which is called the table, is pierced with 25 equidistant holes arranged in a circle. A disc c, about 1

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