VIBRATION. We have had in many articles to consider the effects of vibratory motions, but we have not yet given the explanation of the simple vibration, so as to enable a student with no very extensive knowledge of mathematics to form some conception of its character. The theory of the vibrations of the particles of an elastic fluid is the key to what is known of the phenomena of sound and light [Acousvics ; UNDULATORY THEORY]; and there is some reason to suspect, or at least those whose opinions are worthy of attention have suspected, that the causes of the sensible phenomena of heat, electricity, and magnetism will also be found in the vibrations of matter of some kind. All the particles of material bodies, even when solid, are probably in continual vibration ; and it is certain that very slight disturbances will communicate sensible amounts of vibration to considerable distances, and this through all manner of different substances, from loose earth to compact stone, and through those in every kind of state, from the abriform to the solid.
Little as may be known of most of the vibrations which are per petually occurring, nothing is more certain, from the fundamental laws of mechanics, than that every such vibration in every individual particle is either made up of one or several motions of one particular kind, or of an exceedingly close approximation to such simple motion or combination of motions. It is not merely swinging backwards and forwards which constitutes a vibration ; such a motion might certainly be so called. at the pleasure of any one, but another name must then be invented to designate that particular sort of vibration of which, and of no other, we have to speak in the first instance. The piston of a steam-engine, for example, when it is forced upwards with continually accelerated velocity until it strikes the top of the cylinder, and is then forced downwards in the same manner, does not show what is mathematically called a vibration ; but take one of those more recent constructions, in which the is checked as soon as the piston has acquired momentum enough to carry it to the top of the cylinder, so that the force is nearly spent before it begins to return, and we have something to which the term vibration is much more nearly applicable.
The simple vibration, of which we have said all others may be com pounded, is best imagined as follows :—Let a point Q revolve uniformly round a circle A q a b, and from Q draw Q P perpendicular to A a. Then P moves over A a in the manner of a simple vibration ; the whole vibration being from A to A again. At A and a the velocity of r is extinct, the whole motion of q being perpendicular to A a; but at o the velocity is greatest, P then moving as fast as q. If we measure the time t from the epoch of Q being at B, and suppose the motion of Q to be in the direction a Q A, and n to be the angular velocity of Q, we have (0 r = x, 0 A = a) x = a sin nt, while the velocity of r is n a cos n t, the acceleration of r is — nla sin n t, or — and if w be the weight of a particle at r, the pressure necessary to maintain it in this state of vibration is always directed towards 0, and is, in units of the same kind ex in.
32.1908 x to,if x and a be measured in feet, n in theoretical angular units [ANGLE], and t in seconds [VELOCITY]. If T be the number of seconds in the whole vibration from A to A again, we have n=2 x and the pressure is 1.2264 The pressure, it appears, requisite to main tain a simple vibration must be always in a given proportion to the distance of r from o, and always directed towards o ; and the relation between the pressure at a given value of x and the time of vibration is wholly independent of a, the excursion of the particle. For the mechanical reason of this property, see ISOCHRONISM. To form a more convenient expression, let N be the number of vibrations in a second, and let x be measured in hundredths of inches instead of in feet; then T=1÷s, and for x we must write x=1200, which gives for the pres sure For example, if a particle vibrate only 100 times in a second, which is not much [Acousrics], and have an excursion of one five-hundredth of an inch (sr =100, x=.2), the force of restitution at the extremity of the excursion is more than twice the weight of the particle. By this formula it is easy to get a just idea of the greatness of the molecular forces required to produce those vibrations which are constantly excited in sonorous and other bodies.