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divide the plate into four equal triangles, each pair of which will make their excursions on opposite sides of the plate. The nodal lines and pitch vary not only with the point where the bow is applied, but with the point by which the plate is held, which being at rest necessarily determines the direction of one of the quiescent lines. The forms assumed by the sand in square plates are very numerous, corresponding to all the various modes of vibration. The lines in circular plates are even more remarkable for their symmetry, and upon them the forms assumed by the sand may be classed in three systems. The first is the diametrical system, in which the figures consist of diameters dividing the circumference of the plate into equal parts, each of which is in a different state of vibration from those adjacent. Two diameters, for example, crossing at right angles, divide the circumference into four equal parts; three diameters divide it into six equal parts; four divide it into eight, and so on. In a metallic plate, these divisions may amount to thirty-six or forty. The next is the concentric system, where the sand arranges itself in circles, having the same centre with the plate; and the third is the compound system, where the figures assumed by the sand are compounded of the other two, producing very complicated and beautiful forms. Galileo seems to have been the first to notice the points of rest and motion in the sounding-board of a musical instrument; but to Chladni is due the whole discovery of the symmetrical forms of the nodal lines in vibrating plates (N. 184). Professor Wheatstone has shown, in a paper read before the Royal Society in 1833, that all Chladni's figures, and indeed all the nodal figures of vibrating surfaces, result from very simple modes of vibration oscillating isochronously, and superposed upon each other; the resulting figure varying with the component modes of vibration, the number of the superpositions, and the angles at which they are superposed. For example, if a square plate be vibrating so as to make the sand arrange itself in straight lines parallel to one side of the plate, and if, in addition to this, such vibrations be excited as would have caused the sand to form in lines perpendicular to the first had the plate been at rest, the combined vibrations will make the sand form in lines from corner to corner (N. 185).
M. Savart's experiments on the vibrations of flat glass rulers are highly interesting. Let a lamina of glass 27in.56 long, 0·59
of an inch broad, and 0.06 of an inch in thickness, be held by the edges in the middle, with its flat surface horizontal. If this surface be strewed with sand, and set in longitudinal vibration by rubbing its under surface with a wet cloth, the sand on the upper surface will arrange itself in lines parallel to the ends of the lamina, always in one or other of two systems (N. 186). Although the same one of the two systems will always be produced by the same plate of glass, yet among different plates of the preceding dimensions, even though cut from the same sheet side by side, one will invariably exhibit one system, and the other the other, without any visible reason for the difference. Now, if the positions of these quiescent lines be marked on the upper surface, and if the plate be turned so that the lower surface becomes the upper one, the sand being strewed, and vibrations excited as before, the nodal lines will still be parallel to the ends of the lamina, but their positions will be intermediate between those of the upper surface (N. 187). Thus it appears that all the motions of one half of the thickness of the lamina, or ruler, are exactly contrary to those of the corresponding points of the other half. If the thickness of the lamina be increased, the other dimensions remaining the same, the sound will not vary, but the number of nodal lines will be less. When the breadth of the lamina exceeds the 0·6 of an inch, the nodal lines become curved, and are different on the two surfaces. A great variety of forms are produced by increasing the breadth and changing the form of the surface; but in all it appears that the motions in one half of the thickness are opposed to those in the other half.
M. Savart also found, by placing small paper rings round a cylindrical tube or rod, so as to rest upon it at one point only, that, when the tube or rod is continually turned on its axis in the same direction, the rings slide along during the vibrations, till they come to a quiescent point, where they rest. By tracing these nodal lines he discovered that they twist in a spiral or corkscrew round rods and cylinders, making one or more turns according to the length; but at certain points, varying in number according to the mode of vibration of the rod, the screw stops, and recommences on the other side, though it is turned in a contrary direction; that is, on one side it is a right-handed screw, on the other a left (N. 188). The nodal lines in the
interior surface of the tube are perfectly similar to those in the exterior, but they occupy intermediate positions. If a small ivory ball be put within the tube, it will follow these nodal lines when the tube is made to revolve on its axis.
All solids which ring when struck, such as bells, drinking glasses, gongs, &c., have their shape momentarily and forcibly changed by the blow, and from their elasticity, or tendency to resume their natural form, a series of undulations take place, owing to the alternate condensations and rarefactions of the particles of solid matter. These have also their harmonic tones, and consequently nodes. Indeed, generally, when a rigid system of any form whatever vibrates either transversely or longitudinally, it divides itself into a certain number of parts which perform their vibrations without disturbing one another. These parts are at every instant in alternate states of undulation; and, as the points or lines where they join partake of both, they remain at rest, because the opposing motions destroy one another.
The air, notwithstanding its rarity, is capable of transmitting its undulations when in contact with a body susceptible of admitting and exciting them. It is thus that sympathetic undulations are excited by a body vibrating near insulated tended strings, capable of following its undulations, either by vibrating entire, or by separating themselves into their harmonic divisions. If two chords equally stretched, of which one is twice or three times longer than the other, be placed side by side, and if the shorter be sounded, its vibrations will be communicated by the air to the other, which will be thrown into such a state of vibration that it will be spontaneously divided into segments equal in length to the shorter string. When a tuning-fork receives a blow and is made to rest upon a piano-forte during its vibration, every string which, either by its natural length or by its spontaneous subdivisions, is capable of executing corresponding vibrations, responds in a sympathetic note. The same effect will be produced by the stroke of a bell near a piano or harp. Some one or other of the notes of an organ are generally in unison with one of the panes or with the whole sash of a window, which consequently resounds when those notes are sounded. A peal of thunder has frequently the same effect. The sound of very large organ-pipes is generally inaudible till the air be set in
motion by the undulations of some of the superior accords, and then the sound becomes extremely energetic. Recurring vibrations occasionally influence each other's periods. For example, two adjacent organ-pipes nearly in unison may force themselves into concord; and two clocks, whose rates differed considerably when separate, have been known to beat together when fixed to the same wall, and one clock has forced the pendulum of another into motion, when merely standing on the same stone pavement, These forced oscillations, which correspond in their periods with those of the exciting cause, are to be traced in every department of physical science. Several instances of them have already occurred in this work. Such are the tides, which follow the sun and moon in all their motions and periods. The nutation of the earth's axis also, which corresponds with the period, and represents the motion of the nodes of the moon, is again reflected back to the moon, and may be traced in the nutation of the lunar orbit. And, lastly, the acceleration of the moon's mean motion represents the action of the planets on the earth reflected by the sun to the moon.
In consequence of the facility with which the air communicates undulations, all the phenomena of vibrating plates may be exhibited by sand strewed on paper or parchment, stretched over a harmonica glass or large bell-shaped tumbler. In order to give due tension to the paper or vellum, it must be wetted, stretched over the glass, gummed round the edges, allowed to dry, and varnished over, to prevent changes in its tension from the humidity of the atmosphere. If a circular disc of glass be held concentrically over this apparatus, with its plane parallel to the surface of the paper, and set in vibration by drawing a bow across its edge, so as to make sand on its surface take any of Chladni's figures, the sand on the paper will assume the very same form, in consequence of the vibrations of the disc being communicated to the paper by the air. When the disc is removed slowly in a horizontal direction, the forms on the paper will correspond with those on the disc, till the distance is too great for the air to convey the vibrations. If the disc while vibrating be gradually more and more inclined to the horizon, the figures on the paper will vary by degrees; and, when the vibrating disc is perpendicular to the horizon, the sand on the paper will form into straight lines parallel to the surface of the
disc, by creeping along it instead of dancing up and down. If the disc be made to turn round its vertical diameter while vibrating, the nodal lines on the paper will revolve, and exactly follow the motion of the disc. It appears, from this experiment, that the motions of the aërial molecules in every part of a spherical wave, propagated from a vibrating body as a centre, are parallel to each other, and not divergent like the radii of a circle. When a slow air is played on a flute near this apparatus, each note calls up a particular form in the sand, which the next note effaces, to establish its own. The motion of the sand will even detect sounds that are inaudible. By the vibrations of sand on a drumhead the besieged have discovered the direction in which a counter-mine was working. M. Savart, who made these beautiful experiments, employed this apparatus to discover nodal lines in masses of air. He found that the air of a room, when thrown into undulations by the continued sound of an organ-pipe, or by any other means, divides itself into masses separated by nodal curves of double curvature, such as spirals, on each side of which the air is in opposite states of vibration. He even traced these quiescent lines going out at an open window, and for a considerable distance in the open air. The sand is violently agitated where the undulations of the air are greatest, and remains at rest in the nodal lines. M. Savart observed, that when he moved his head away from a quiescent line towards the right the sound appeared to come from the right, and when he moved it towards the left the sound seemed to come from the left, because the molecules of air are in different states of motion on each side of the quiescent line.
A musical string gives a very feeble sound when vibrating alone, on account of the small quantity of air set in motion; but when attached to a sounding-board, as in the harp and pianoforte, it communicates its undulations to that surface, and from thence to every part of the instrument; so that the whole system vibrates isochronously, and by exposing an extensive undulating surface, which transmits its undulations to a great mass of air, the sound is much reinforced. The intensity is greatest when the vibrations of the string or sounding body are perpendicular to the sounding-board, and least when they are in the same plane with it. The sounding-board of the piano-forte is better disposed than that of any other stringed instrument, because the