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M. Savart's Violins. Nodal Surfaces and Lines.
law of form and motion, which it is beyond our power to investigate. In that case its molecules must have lateral as well vertical motions, and its vibrations must be partly longitudinal and partly twisting, in a way easier imagined than described.
d. If the discs be dissimilar in form, as well as unequal in dimension, the vibrations of the connecting rod will of course be very complicated.
144. These principles have been applied by M. Savart, and apparently with success, to the improvement of violins, and the construction of these delicate instruments on scientific and experimental grounds.
145. In the vibrations of solids of any figure and dimension, and also in those of masses of air or any other elastic fluid, there will, generally, be nodal surfaces, all the points of which are at rest.
Where the surfaces out-crop or intersect the external surface of the mass, there will be a nodal line.
a. It appears from what we have said, that the motions of the molecules of a rod, which commmunicates the vibrations of one disc to another, or, more generally, which vibrates longitudinally by any exciting cause, are not of necessity analogous to those of the air in a cylindrical pipe; at least not to that simple case of the latter vibrations, which we have heretofore considered in Chapter III., Part II. The several transverse sections of such a rod, in the act of vibration, do not necessarily merely advance and recede longitudinally, but may become curves of double curvature; in short, such a rod may be considered, as an assemblage of vibrating discs, ranged along a common axis, along which they may, it is true, be also carried backwards and forwards with a vibratory motion, while at the same time their flexure is changing from convex to concave, and vice versa.
Nodal Surfaces and Lines.
Now it may happen that a point, or a line, (straight or curved,) in any one of such discs, may be advancing in the direction of the axis in consequence of the bodily motion of the whole disc, while, in virtue of its flexure in the act of changing its figure, it may be receding; and this advance or recess may so balance each other, that the point or line shall be at rest. If this be true at one instant, it will be so at all instants; because the vibrations have all one period, and follow the same law of increase and decrease in their phases.
Thus we have a nodal point, or a nodal line; and as each disc, by reason of the law of continuity, must have a similar one, the assemblage of such lines will mark out within the rod a nodal surface, dividing it into separate solids, whose molecules on either side of such surface are in opposite phases of their motion.
b. What is here said of rods, applies of course to solids of any figure and dimension; neither is there the slightest reason why it should not apply to vibrating masses of air, or any other elastic fluid. Any such mass may be conceived as cut up into two or more oppositely vibrating portions, pervading it according to certain laws.
146. If fine sand is strewed over the surfaces of solids, the motions of the particles in the act of marking out the nodal lines will easily distinguish such vibrations, as are executed parallel to the surface, which are called tangential vibrations, and in which, of course, the surface is not thrown into waves, from such as are at right angles to it, which are called transverse vibrations, when the surface itself leaps up and down. Vibrations, compounded of both these, where the surface both swells and falls, and shifts laterally backwards and forwards, are called oblique vibrations.
How to Distinguish Normal from Tangential Vibrations.
a. In transverse vibrations, the particles of sand dance, and are violently thrown up and down over the whole extent of the vibrating portions, till, at length, they are entirely dispersed from them.
When a large disc of glass is set vibrating vigorously by a bow, perpendicular to its plane, the grains of sand will fly up some inches from it and be scattered in all directions.
b. In tangential vibrations, they only glide along close to the surface, and meet and settle on the nodal lines, and that, sometimes, with incredible swiftness. The reason why they retreat to the nodal lines is easily understood. The amplitude of the excursions of the vibrating molecules of the surface diminishes as we approach a nodal line. Hence a particle of sand anywhere situated, if thrown by an advancing vibration towards this line, will not be thrown quite so far back by the subsequent retreating vibration, because its then situation is one less agitated. Thus the motion of each particle of sand is one of alternate advances towards the node and recesses from it, but the advances are always greater than the recesses. In consequence, it creeps along the surface, and will not rest till it has attained the node.
147. The tangential vibrations of long flat rods or rulers of glass, as investigated by Savart, are extremely complicated, and offer most singular phenomena, some of which we shall now describe.
a. If we take a rectangular lamina of glass 27-56 inches long, 0.59 inches broad, and 0.06 inches thick, and holding it by the edges in the middle between the finger and thumb, with its flat face horizontal, strewed with sand, and, at the same time set it in longitudinal vibration, either by rubbing its under side near either end with a bit of wet cloth, by tapping it on the end with light blows, or by rubbing lengthwise a very small cylinder of glass, cemented on to its end in the middle of its breadth, and parallel to its length; in whatever way the vibration be communicated, we shall see the sand on its upper surface arrange itself in parallel lines, at right
Longitudinal-Tangential Vibrations of Rectangular Plates.
angles to its longer dimension, and always, in one or the other of the two systems, represented in figs. 97 and 98.
Now it is very remarkable that, although the same one of these two systems will always be produced by the same plate of glass, yet among different plates of the above dimensions, even though cut from the same sheet, side by side, one will invariably exhibit one system, and the other the other, without any visible reason for the difference.
Moreover, in the system, fig. 97, the disposition of the nodal lines is unsymmetrical, one of them, a, being nearer to one end, and the closer pair, f, f', not being situated in the middle; and this too is peculiar to the plate; for wherever it be rubbed, whichever end be struck, still the line a will always be formed nearest to the same extremity.
b. Now let the positions of the nodal lines be marked on the upper surface, and then let the plate be turned till the lower surface becomes the upper, and this being sanded, let the vibrations again be excited just as before. The nodal lines will now be formed quite differently, and will fall on the points just intermediate between those of the other surface; i. e. on the points of greatest excursion of its vibrating molecules. In a word if n, n, n, n, &c. in fig. 99 or 100, represent the places of the nodes on the one surface, then will n', n', &c. be those of the other.
Thus, all the motions of one half the thickness of the lamina are exactly contrary to those of the corresponding points of the other half. This property, indeed, is general, whatever be the material, length, breadth, or thickness of the lamina.
c. If, the other dimensions remaining, the thickness be increased, the sound will remain the same, but the number of nodal lines will be less. This fact alone is sufficient to prove an essential difference between the vibrating portions of such a plate, and the ventral segments of an organ-pipe harmonically subdivided.
d. If the breadth of a plate of the above length be greater than 0.6 inches, the nodal lines cease to be straight, and ranged across the plate at right angles to the sides. They pass into curves, and
Longitudinal-Tangential Vibrations of Cylinders.
when the breadth is increased to 1.57 inches, they assume the forms in figs. 101 and 102, the former representing the lines on the upper, the latter those on the under surface.
e. If the breadth be enlarged to 2:36 inches, the figures on the two faces will be as in figs. 103, 104.
f. If the dimensions be so varied as to convert the plate into a square, the nodal figures will assume the forms in figs. 105 and 106.
g. If the form of the plate pass into circular or triangular, the same mode of vibration (longitudinal-tangential) being preserved, still the opposite sides of the plate will present different nodal figures, as in 107, 108 and 109, 110.
148. In the longitudinal-tangential vibrations of cylindrical tubes or rods, there are two nodal lines which run spirally round the cylinder in opposite directions.
a. To examine the vibrations of cylinders, as sand will not lie on their convex surfaces, M. Savart employed the ingenious artifice used by Sauveur to exhibit the harmonic nodes of a vibrating string. For this purpose, the latter set astride on the string a small bit of paper cut into the form of an inverted V. But in this case it is found to answer better to encircle the vibrating cylinder with a narrow ring of paper, whose internal diameter is three or four times that of the cylinder, and which therefore hangs quite loosely on it.
b. If a cylinder of glass about 6 feet long be encircled by several such rings or riders, and, being held horizontally by the middle, as lightly as possible, be rubbed in the direction of its length with a very wet cloth, it will yield a musical sound, and all the riders will glide rapidly along it to their nearest nodal points on the upper surface where they will rest. Now let all these points be marked, and then let the cylinder be turned so as to bring the opposite portion of its circumference uppermost and horizontal, and let the vibration be again excited in the same manner. Then we