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Communication of the Vibrations of a Violin-String to the Wood.
which are stretched from end to end of it are divided into unequal parts by the bridge, A, on which they all press strongly, and at the same time rest in small notches, so as not to slip laterally on it. The portion, B, of the string which lies towards the handle, C, of the instrument, is free, and is set in vibration by the bow in its own plane; but that on the other side of the bridge, D, is loaded with a mass of horn or whalebone, E, to which, all the other strings are also attached, and which, being only tied to the woodwork, cannot propagate the vibrations of any one string sounding separately, by reason of the contradictory and unequal tensions of the other three. Thus the bridge is in fact acted on only by the vibrations of that part, ABC, of the string which is crossed by the bow, as if it terminated abruptly at its point of pressure, A.
These vibrations constantly tend, therefore, to tilt the bridge laterally backwards and forwards, and to press up and down alternately the two little prominences or feet, F, G, by which it rests on the belly of the violin. It, therefore, sets the wood of the upper face in a state of regular vibration, and this again is communicated to the back through its sides, a peg set up in the inside of the fiddle and through, called the soul of the fiddle, or its soundingpost.
In consequence, if sand be strewed over the upper surface, it will assume a regular arrangement in nodal lines when the bow is drawn; and the same subdivision is also observed in the wood of the under surface, if the sounding post be exactly placed in the centre of symmetry of the nodal figures.
c. The experiment can hardly be made, however, with a common fiddle, by reason of the convexity of its surface, on which sand will not rest; but if one be constructed with plane boards, or if, abandoning the fiddle, a string be stretched on a strong frame. over a bridge, which is made to rest on the centre of a regularly formed plate or disc of metal or wood, strewed with sand, the surface thus set in vibration by the string will be seen to divide itself by regular nodal figures.
Joint Vibrations of a Plate and String as a System. Sounding-Boards.
138. If the tension or length of the string, thus placed in vibratory communication with a plate, be changed, so as to vary the note it speaks, the nodal figures on the plate undergo a corresponding variation, and the plate still vibrates in unison with the string; or, which is the same thing, the two, together with the interposed bridge, form a vibrating system, in which, though the vibrations of the several parts are necessarily very different in their nature and extent, yet they have all the same periods.
This important fact, deduced by Savart from his experiments, confirms the results of Chladni's experiments on the sounds of such thin plates, and shows that they are not, like those of strings, confined to certain fixed harmonics, but, according to the forms of their nodal lines, and the proportions of the vibrating areas in opposite states of excursion, may assume any assigned period; in other words, given the vibrating plate and the pitch, a nodal figure may be described on it, which shall correspond to that pitch, and the plate (with more or less readiness, however,) is always susceptible of such a vibration as shall yield that note and produce that nodal figure. How far this proposition is general, and with what limitations it is to be understood, we shall soon see.
139. Meanwhile this remark, it will be observed, furnishes a complete explanation of the effect of sounding-boards in musical instruments.
It is not, as some have supposed, that there exist in them fibres in every state of tension, some of which are therefore ready to vibrate in unison with any proposed sound, and, therefore, reinforce it. Such a cause could at best produce but a very feeble effect.
It is the whole board which vibrates as part of a system with every note, and (as vibrations may be superposed to any extent) the same sounding-board may at once form a part of any number
Communication of the Longitudinal Vibrations of Rods to Solids.
of systems, and vibrate in unison with every note of a chord. Still some modes will always be more difficult than others, sounding-board will be perfectly indifferent to all sounds.
140. The longitudinal vibrations of a rod of glass, excited by rubbing it with a wet cloth, may also be used to excite vibrations in a given point of a solid perpendicular to its surface, by applying its end to it, or cementing it to the solid by mastic.
In this way Chladni applied it to draw forth the sounds of glass vessels, (which, when hemispherical, of sufficient size, and of even thickness, are remarkably rich and melodious,) in an instrument which he called the Euphone, exhibited by him in Paris and Brussels. The principle of this instrument was at the time concealed; but the enigma was subsequently solved by M. Blanc, who on his part independently made the same remark, and applied it to a similar purpose.
141. If the solid, (a circular glass disc for instance,) to which such a vibrating rod or tube is fastened, be of small comparative dimensions, its vibrations are commanded by those of the rod, and the sound yielded will be that of the rod alone; and vice versa, if the disc be large, and the rod small, the note sounded will be that of the disc, which will entirely command the rod; but in the intermediate cases, the note will be neither that of the disc nor the rod separately, but the two will vibrate together as a system, each yielding. somewhat to the other.
a. This is a case exactly analogous to that of a reed-pipe, in which the reed and column of air mutually influence each other's note. See art. 88.
Vibrations Communicated between two Plates by a Rod.
This mutual influence of propagated motion, by which two periodically recurring impulses affect each other's period, and force themselves into synchronism, extends to cases where at first sight it would be hardly suspected.
b. Thus Ellicott observed that two clocks fastened to the same board, or even standing on the same stone pavement, beat constantly together, though when separated their ratios were found to differ very considerably; and Breguet has since made the same remark on watches.
c. Thus also two organ-pipes vibrating side by side, if very nearly in unison, will under certain circumstances force themselves into exact concord, as has been observed by Hudlestone, and lately recalled to notice by some experiments made in Copenhagen.
a. The experiment with the disked tuning-fork and pipe, related in art. 93, may here again be referred to.
142. The longitudinal vibrations of a rod have also been used by M. Savart, to communicate vibrations from one solid to another; as, for instance, from the upper to the under of two circular discs, cemented at their centres to the two ends of the rod, at right angles to their planes, as at fig. 95.
143. If the two discs be of the same dimensions and materials, so as to yield, when separately vibrating, the same note; the vibrations of one of them, (the upper for instance,) excited by a bow, will be exactly imitated by the other; and sand strewed over both will arrange itself in precisely the same forms in both discs, and that, into whatever number of vibrating segments that immediately excited be made to subdivide itself.
State of Vibration of the connecting Rod..
But if the discs separately do not agree in their tones, the system may yield a tone intermediate, and each being differently forced from its natural pitch, the nodal figures on them will no longer correspond.
a. The state of vibration, into which the molecules of the connecting rod are thrown in such cases, deserves a nearer examination.
For simplicity, let us suppose the discs equal, the rod cylindrical, and the vibration of the system such, that each disc shall subdivide itself into four quadranta segments. In this case it is clear that, as the form assumed at any instant by the upper disc is undulated or wrinkled, as represented in fig. 96, the section of the rod in immediate contact with it, and which obeys all its motions, must assume a similar form, and so of all the rest. Thus, if we conceive the rod split into infinitesimal columns, parallel to its axis, all the columns in two opposite quadrants will be ascending, while those in the other two are descending; and thus the two corresponding opposite quadrants of the lower plate will be drawn upwards, while the alternate ones are forced downwards, giving a similar distortion to its figure, and disposing it to a similar vibration only.
b. It will depend on the length of the rod, and the time taken by an undulation to run over its length, compared with that of a vibration of either disc, whether the phases of vibration in the two discs shall be the same at the same instant or not. It may happen that, for instance, the quadrant, DB, of the upper disc shall have completed its downward motion, and begun to return, before the pulsation propagated through the rod has arrived at the lower disc; and in that case the corresponding quadrants of the two discs will be always in opposite phases of their periodic motion. But the nodal lines will of necessity correspond in both.
c. When the two discs are unequal, the propagation of the pulses through the rod must of course cease to be uniform, and each section of it down its whole length will have its own peculiar