« PreviousContinue »
Vibration of Musical Strings
Harmonic Sounds - Nodes - Vibration of Air in Wind-Instruments Vibration of Solids - Vibrating Plates Bells - Harmony-Sounding Boards - Forced Vibrations Resonance Speaking Machines.
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
half as long as the tube, the second harmonic by waves a third as long, and so on. The harmonic segments in an open and shut pipe are the same in number, but differently placed. shut pipe the two ends are nodes, but in an open pipe there is half a segment at each extremity, because the air at these points is neither rarefied nor condensed, being in contact with that which is external. If one of the ends of the open pipe be closed, its fundamental note will be an octave lower: the air will now divide itself into three, five, seven, &c., segments; and the wave producing its fundamental note will be twice as long as the pipe, so that it will be doubled back (N. 182). All these notes may be produced separately by varying the intensity of the blast. Blowing steadily and gently, the fundamental note will sound; when the force of the blast is increased the note will all at once start up an octave; when the intensity of the wind is augmented the twelfth will be heard; and, by continuing to increase the force of the blast, the other harmonics may be obtained, but no force of wind will produce a note intermediate between these. The harmonics of a flute may be obtained in this manner, from the lowest C or D upwards, without altering the fingering, merely by increasing the intensity of the blast and altering the form of the lips. Pipes of the same dimensions, whether of lead, glass, or wood, give the same tone as to pitch under the same circumstances, which shows that the air alone produces the sound.
Metal springs fastened at one end, when forcibly bent, endeavour to return to rest by a series of vibrations, which give very pleasing tones, as in musical boxes. Various musical instruments have been constructed, consisting of metallic springs thrown into vibration by a current of air. Among the most perfect of these are Mr. Wheatstone's Symphonion, Concertina, and Æolian Organ, instruments of different effects and capabilities, but all possessing considerable execution and expression.
The Syren is an ingenious instrument, devised by M. Cagniard de la Tour, for ascertaining the number of pulsations in a second, corresponding to each pitch: the notes are produced by jets of air passing through small apertures, arranged at regular distances in a circle on the side of a box, before which a disc revolves pierced with the same number of holes. During a revolution of the disc the currents are alternately intercepted and allowed to
pass as many times as there are apertures in it, and a sound is produced whose pitch depends on the velocity of rotation.
A glass or metallic rod, when struck at one end, or rubbed in the direction of its length with a wet finger, vibrates longitudinally, like a column of air, by the alternate condensation and expansion of its constituent particles, producing 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, 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 elevated while the other is depressed; and, as these two motions meet in the nodal lines, they neutralise 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. 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. If a bow be drawn across one of the angles of a square plate of glass or metal held firmly by the centre, the sand will arrange itself in two straight lines parallel to the sides of the plate, and crossing in the centre so as to divide it into four equal squares, whose motions will be contrary to each other. Two of the diagonal squares will make their excursions on one side of the plate, while the other two make their vibrations on the other side of it. This mode of vibration produces the lowest tone of the plate (N. 183). If the plate be still held by the centre, and the bow applied to the middle of one of the sides, the vibrations will be more rapid, and the tone will be a fifth higher than in the preceding case: now the sand will arrange itself from corner to corner, and will