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be greater still if the surface be spherical and concentric with him. Undulations of sound diverging from one focus of an elliptical shell converge in the other after reflection; consequently a sound from the one will be heard in the other as if it were close to the ear. The rolling noise of thunder has been attributed to reverberation between different clouds, which may possibly be the case to some degree; but Sir John Herschel is of opinion, that an intensely prolonged peal is probably owing to a combination of sounds, because the velocity of electricity being incomparably greater than that of sound, the thunder may be regarded as originating in every point of a flash of lightning at the same instant. The sound from the nearest point will arrive first, and if the flash run in a direct line from a person, the noise will come later and later from the remote points of its path in a continued roar. Should the direction of the flash be inclined, the succession of sounds will be more rapid and intense, and if the lightning describe a circular curve round a person, the sound will arrive from every point at the same instant with a stunning crash. In like manner, the subterranean noises heard during earthquakes, like distant thunder, may arise from the consecutive arrival at the ear of undulations propagated at the same instant from nearer and more remote points; or,
if they originate in the same point, the sound may come by different routes through strata of different densities.
Sounds under water are heard very distinctly in the air immediately above, but the intensity decays with great rapidity as the observer goes farther off, and is altogether inaudible at the distance of two or three hundred yards: so that waves of sound, like those of light, in passing from a dense to a rare medium, are not only refracted but suffer total reflection at very oblique incidences.
The laws of interference extend also to sound. It is clear that two equal and similar musical strings will be in unison if they communicate the same number of vibrations to the air in the same time. But if two such strings be so nearly in unison that one performs a hundred vibrations in a second, and the other a hundred and one in the same period,—during the first few vibrations, the two resulting sounds will combine to form one of double the intensity of either, because the aërial waves will sensibly coincide in time and place, but the one will gradually gain on the other, till, at the fiftieth vibration, it will be half an oscillation in advance; then the waves of air which produce the sound being sensibly equal, but the receding part of the one coinciding with the advancing part of the other, they will destroy one another, and
occasion an instant of silence. The sound will be renewed immediately after, and will gradually increase till the hundredth vibration, when the two waves will combine to produce a sound double the intensity of either. These intervals of silence and greatest intensity, called beats, will recur every second, but if the notes differ much from one another, the alternations will resemble a rattle; and if the strings be in perfect unison, there will be no beats, since there will be no interference. Thus by interference is meant the coexistence of two undulations, in which the lengths of the waves are the same; and as the magnitude of an undulation may be diminished by the addition of another transmitted in the same direction, it follows, that one undulation may be absolutely destroyed by another, when waves of the same length are transmitted in the same direction, provided that the maxima of the undulations are equal, and that one follows the other by half the length of a wave.
WHEN the particles of elastic bodies are suddenly disturbed by an impulse, they return to their natural position by a series of isochronous vibrations, whose rapidity, force, and permanency depend upon the elasticity, the form, and the mode of aggregation which unites the particles of
the body. These oscillations are communicated to the air, and on account of its elasticity they excite alternate condensations and dilatations in the strata of the fluid nearest to the vibrating body from thence they are propagated to a distance. A string or wire stretched between two pins, when drawn aside and suddenly let go, will vibrate till its own rigidity and the resistance of the air reduce it to rest. These oscillations may be rotatory, in every plane, or confined to one plane, according as the motion is communicated. In the piano-forte, where the strings are struck by a hammer at one extremity, the vibrations probably consist of a bulge running to and fro from end to end. The vibrations of sonorous bodies, however, are compound. Suppose a vibrating string to give the lowest C of the piano-forte, which is the fundamental note of the string; if it be lightly touched exactly in the middle, so as to retain that point at rest, each half will then vibrate twice as fast as the whole, but in opposite directions; the ventral or bulging segments will be alternately above and below the natural position of the string, and the resulting note will be the octave above C. When a point at a third of the length of the string is kept at rest, the vibration will be three times as fast as those of the whole string, and will give the twelfth above C. When the point of rest
is one-fourth of the whole, the oscillations will be four times as fast as those of the fundamental note, and will give the double octave, and so on. Now, if the whole string vibrate freely, a good ear will not only hear the fundamental note, but will detect all the others sounding along with it, though with less and less intensity as the pitch becomes higher. These acute sounds, being connected with the fundamental note by the laws of harmony, are called its harmonics. It is clear, from what has been stated, that the string thus vibrating freely could not give all these harmonics at once, unless it divided itself spontaneously at its aliquot parts into segments in opposite states of vibration, separated by points actually at rest. In proof of this, pieces of paper placed on the string at the half, third, fourth, and other aliquot points, will remain on it during its vibration, but will instantly fly off from any of the intermediate points. Thus, according to the law of co-existing undulations, the whole string and each of its aliquot parts arein different and independent states of vibration at the same time; and as all the resulting notes are heard simultaneously, not only the air, but the ear also, vibrates in unison with each at the same instant. The points of rest, called the nodal points of the string, are a mere consequence of the law of interferences. For if a rope fastened at one end be moved to and fro at the other extremity, so as