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1. WHEN the prongs of a tuning-fork are squeezed between the fingers and suddenly released, they spring back not only to their original position, but to a nearly, equal distance on the other side, and swing backwards and forwards a great number of times before they finally come to rest. This is an example of vibration.

The time occupied in swinging from one side to the other and back again is called the periodic time, or the period of vibration, or simply the period; and the distance that any particle of the fork travels, first to one side and then to the other side of its position of equilibrium, is called the amplitude of vibration for this particle.

2. A tuning-fork, when well started, usually makes several thousand vibrations before coming to rest. Their amplitudes gradually decrease, and hence the sound

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emitted becomes fainter; but the periodic time remains sensibly constant, and hence the pitch does not sensibly alter. The pitch of a sound depends only on the periodic time of the vibrations which are its physical cause. Any two bodies, however different in character, if they vibrate with equal periodic times, produce sounds of the same pitch. And it is a general law that unless a vibrating body be very widely distorted from its position of equilibrium, its periodic time, and therefore its pitch, are independent of the amplitude of vibration. Most of our musical instruments give sounds varying greatly in loudness according to the force employed in producing them ; but though this force influences the loudness, it does not influence the pitch, or music would be well nigh impossible.

3. This constancy of pitch is closely connected with the following law of elastic resistance. When an elastic body is distorted from its natural form or size, the force required to distort it, or, what is equal and opposite to this, the force of restitution exerted by the body, is directly proportional to the amount of distortion. For example, if we compare the force with which a tuning fork must be squeezed to make its prongs approach by 5th of an inch with that required to make them approach byth of an inch, we shall find the former to be precisely double of the latter. When the fork is vibrating, the forces of elasticity are always urging it towards its position of equilibrium. They vanish for an instant

when it is passing through this position; they then gradually increase as it departs further from this position, and attain their maximum in the position of greatest displacement. Thus far they have been opposing and gradually destroying its motion, until, in the extreme position, it comes for an instant to rest. As it returns to the position of equilibrium they go through the same values again in backward order, and restore to it the velocity which they previously destroyed, but in the opposite direction. Similar action occurs on the other side of the position of equilibrium, and the whole motion of the fork can thus be divided into four equal parts which are reversed copies of each other.

4. The proportionality of the elastic force called out by displacement to the displacement itself is thus a fundamental law of the vibratory motions which give rise to musical sound; and we shall commence our analysis of vibratory motion by discussing the simplest conceivable case the case of a particle vibrating in a straight line under the action of a force which urges it towards the middle point of its path, and varies directly as the distance of the particle from this point. The motion of a particle under these conditions is called Simple Harmonic Motion.

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