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tensity, which will be greater still if the surface be spherical and concentric with him. Undulations of sound diverging from one focus of an elliptical shell (N. 180) 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 a certain extent. 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 (N. 189).

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 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 co-existence 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. A tuningfork affords a good example of interference. When that instrument vibrates, its two branches alternately recede from and approach one another; each communicates its vibrations to the air, and a musical note is the consequence. If the fork be held upright about a foot from the ear, and turned round its axis while vibrating, at every quarter revolution the sound will scarcely be heard, while at the intermediate points it will be strong and clear. This phenomenon arises from the interference of the undulations of air coming from the two branches of the fork. When the two branches coincide, or when they are at equal distances from the ear, the waves of air combine to reinforce each other; but at the quadrants, where the two branches are at unequal distances from the ear, the lengths of the waves differ by half an undulation, and consequently destroy one another.

SECTION XVII.

Vibration of Musical Strings

Harmonic Sounds - Nodes Vibration of Air in Wind-Instruments - Vibration of Solids - Vibrating Plates Bells - Harmony - Sounding Boards - Forced Vibrations - Speaking Machines.

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Resonance

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 pianoforte, 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. Different modes of vibration may be obtained from the same sonorous body. Suppose a vibrating string to give the lowest C of the pianoforte 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 vibrations 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. These acute sounds are called

the harmonics of the fundamental note. It is clear, from what has been stated, that the string thus vibrating could not give these harmonics spontaneously unless it divided itself at its aliquot parts into two, three, four, or more 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, or other aliquot points, according to the corresponding harmonic sound, will remain on it during its vibration, but will instantly fly off from any of the intermediate points. 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 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.

Harmonics generally co-exist with the fundamental sound in the same vibrating body. If one of the lowest strings of the pianoforte be struck, an attentive 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 their pitch becomes higher. According to the law of co-existing undulations, the whole string and each of its aliquot parts are in 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 (N. 181).

Harmony consists in an agreeable combination of sounds. When two chords perform their vibrations in the same time, they are in unison; but, when their vibrations are so related as to have a common period, after a few oscillations they produce concord. Thus, when the vibrations of two strings bear a very simple relation to each other, as where one of them makes two, three, four, &c., vibrations in the time the other makes one; or, if it accomplishes three, four, &c., vibrations while the other makes two, the result is a concord which is the more perfect the shorter the common period. In discords, on the contrary, the beats are distinctly audible, which produces a disagreeable and harsh effect, because the vibrations do not bear a simple relation to one another, as where one of two strings makes eight vibrations while the other accomplishes fifteen. The pleasure afforded by harmony is attributed by Dr. Young to the love of order, and to a predilection for a regular repetition of sensations natural to the human mind, which is gratified by the perfect regularity and rapid recurrence of the vibrations. The love of poetry and dancing he conceives to arise in some degree from the rhythm of the one and the regularity of the motions in the other.

A blast of air passing over the open end of a tube, as over the reeds in Pan's pipes; over a 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 inversely proportional to the length of the divisions, giving the fundamental note or one of its harmonics. The nodes are produced on the principle of interferences by the reflection of the longitudinal undulations of the air at the ends of the pipe, as in the musical string, only that in one case the undulations are longitudinal, and in the other transverse.

A pipe, either open or shut at both ends, when sounded, vibrates entire, or divides itself spontaneously into two, three, four, &c., segments separated by nodes. The whole column gives the fundamental note by waves or vibrations of the same length with the pipe. The first harmonic is produced by waves

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