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to transmit a succession of equal waves along it, they will be successively reflected when they arrive at the other end of the rope by the fixed point, and in returning they will occasionally interfere with the advancing waves; and as these opposite undulations will at certain points destroy one another, the point of the rope in which this happens will remain at rest. Thus a series of nodes and ventral segments will be produced, whose number will depend upon the tension and the frequency of the alternate motions communicated to the moveable end. So, when a string fixed at both ends is put in motion by a sudden blow at any point of it, the primitive impulse divides itself into two pulses running opposite ways, which are each totally reflected at the extremities, and, running back again along the whole length, are again reflected at the other ends; and thus they will continue to run backwards and forwards, crossing one another at each traverse, and occasionally interfering so as to produce nodes; so that the motion of a string fastened at both ends consists of a wave or pulse, continually doubled back on itself by reflection at the fixed extremities.
A blast of air passing over the open end of a tube, as over the reeds in Pan's pipes; over hole in one side, as in the flute; or through the aperture called a reed, with a flexible tongue, as in the clarinet, puts the internal column of air into
longitudinal vibrations by the alternate condensations and rarefactions of its particles; at the same time the column spontaneously divides itself into nodes, between which the air also vibrates longitudinally, but with a rapidity proportional to the number of divisions, giving the fundamental note and all its harmonics. The nodes are produced on the principle of interferences, by the reflection of the longitudinal undulations, at the closed end or ends of the pipe, as in the musical string, only that in the one case the undulations are longitudinal, and in the other transverse. Glass and metallic rods, when struck at one end, or rubbed in the direction of their length with a wet finger, vibrate longitudinally, like a column of air, by the alternate condensation and expansion of their constituent particles, which produces a clear and beautiful musical note of a high pitch, on account of the rapidity with which these substances transmit sound. Rods, surfaces, and in general all undulating bodies, resolve themselves into nodes ; but in surfaces, the parts which remain at rest during their vibrations are lines, which are curved or plane according to the substance, its form, and the mode of vibration. If a little fine dry sand be strewed over the surface of a plate of glass or metal, ground smooth at the edges, and if undulations be excited by drawing the bow of a violin
across its edge, it will emit a musical sound, and the sand will immediately arrange itself in the nodal lines, where alone it will accumulate and remain at rest, because the segments of the surface on each side will be in different states of vibration, the one being always elevated while the other is depressed, and as these two motions meet in the nodal lines, they neutralize one another. These lines vary in form and position with the part where the bow is drawn across, and the point by which the plate is held being at rest, must necessarily be in a nodal line; the smallest variation in the pitch changes the nodal lines. A sound may thus be detected though inaudible. The motion of the sand shows in what direction the vibrations take place : if they be perpendicular to the surface, the sand will be violently tossed up and down, till it finds the points of rest; if they be tangential, the sand will only creep along the surface to the nodal lines. Sometimes the undulations are oblique, or compounded of both the preceding. The air of a room, when thrown into undulations by the continued sound of an organ-pipe, or 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.
All solids which ring when struck, 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.
All bodies have a tendency to impart their undulations both to the air and to substances in contact with them. 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 piano-forte, 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.
It is evident that the sounding-board and the whole instrument are agitated at once by all the superposed vibrations excited by the simultaneous or consecutive notes that are sounded, each having its perfect effect independently of the rest. 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. When a tuning-fork receives a blow, and is made to rest upon a pianoforte, 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. 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 resound when these 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 its sound becomes extremely energetic. Recurring vibrations occasionally influence each other's periods.