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Longitudinal-Tangential Vibrations of Cylinders.
shall remark the very same phenomenon as in rectangular plates, viz. that the nodal points on this edge correspond nearly to the middles of the intervals between those of the opposite one.
c. If the cylinder, instead of being turned at.once half round, be turned only a little at a time, and always in the same direction, the riders will come to the points of rest constantly more and more towards one or the other end of the cylinder, according as it is turned to the right or the left; and if the locus of all the nodal points be traced by this means, it will be found to be a species of spiral line or screw, making one or more turns round the cylinder according to its length.
d. But there exists here a peculiarity, bearing an obvious relation to what we have observed already in the case of rectangular plates. The continuity of this spiral is interrupted near the middle of the cylinder, or rather it stops short at a point n, on one side of the central point, and recommences at N, a point equidistant on the other side; but in a contrary direction, so as to form on the two moieties of the length of the cylinder a right and a left-handed
e. Again, these spirals are not equally inclined to the axis in all parts of their course. They consist of portions alternately much and little inclined, having points of maximum and minimum inclination alternately at every 90° of their course round the cylinder, as in fig. 111; thus dividing the cylinder into four quadrantal portions, which are related to each other in the same manner as the upper and under faces, and the right and left sides of the vibrating parallelopipeds examined in art. 147.
149. In the case of a hollow tube, the nodal lines of the internal surface consist of spirals, in all respects. similar to those on the external surface; only that their coils run exactly along the intervals of those of the external one. So that in all cases, those points of the internal surface are most strongly agitated by the
Higher Modes of Vibration of Cylinders.
vibrations which correspond to points at rest on the outer, and vice versa.
a. The nodal lines of the internal surface ay be examined by strewing in it a little fine sand, provided its diameter be so large as not to drive all the sand into a crowded line along the bottom.
b. M. Savart has noticed a very curious phenomenon in this case. At the points of maximum inclination, the sand gathers itself up in a circular heap, and remains at perfect rest; but at those of minimum inclination, it forms a long ellipse, the borders of which keep constantly circulating in one direction; and if, instead of sand, a small globe of ivory or wax be put into the tube, it remains at these points, it is true, without shifting its place, but spins constantly in one direction round a vertical axis, so long as the vibration continues.
150. We have all along supposed that the state of vibration, into which the cylinder or tube is thrown, is that corresponding to the gravest tone it can yield by vibrations of the kind in question. M. Savart has examined its higher modes, and has pointed out other peculiarities. We will merely remark that, in these modes, the threads of the screw break off, and reverse their directions at the points of union of the several ventral segments.
General Law of the Communication of Vibratory Motion.
THE COMMUNICATION OF VIBRATIONS FROM ONE VIBRATING BODY
151. We have already seen that a rod placed between two discs, one of which is set in vibration, becomes the means of communicating its vibrations to the other.
But it may be announced as a general proposition, that whenever a vibrating body is brought into intimate contact with another, it communicates to it its own vibrations, more or less effectually as their union is more perfect, and all the parts of the body thus set in vibration by communication are agitated by motions, not merely similar in their periods, but actually parallel in their directions to those of the original source of the motion.
This similarity of the communicated to the original vibrations was proved by the experiments of M. Savart, and is best illustrated by examples.
Example 1. Let A, fig. 112, be a long flat glass ruler, or rod, cemented with mastic to the edge of a large bell-glass, such as is used for the harmonica, or musical glasses, or a large hemispherical drinking-glass, perpendicular to its circumference. Let it be very lightly supported in a horizontal position on a bit of cork at C, and then let the bell-glass be set in vibration by a bow, at a point opposite the place where the rod meets it. It will vibrate transversely, i. e. the motions of its molecules will be perpendicular to its surface; and these motions will be communicated to the rod,
Joint Vibrations of two Rods Transverse to each other.
without any change in their direction, whose vibrations will be longitudinal-tangential, as will be rendered evident by strewing its surface with sand, when the nodal lines will be formed as in art. 147; and, if the apparatus be inverted, and the sand strewed on the under side of the rod, the nodal lines will be seen to correspond to the points of greatest excursion on the other side, as in that article.
In this combination the original tone of the bell-glass is altered, and the note produced differs both from that yielded by it, or by the glass rod vibrating alone. The two vibrate as a system together; and, what is singular, the sound of the glass is considerably reinforced by the combination.
Example 2. Let A', fig. 113, be a rectangular strip of glass, firmly cemented at right angles to another strip, A, across its breadth. Let the latter be lightly supported on two bits of cork, C, C', fastened to a wooden piece, B, so as just to touch A in the places of two of its nodes when vibrating transversely. Then if A be placed horizontally, and strewed with sand, and A' be set in longitudinal-tangential vibration, either by rubbing with a wet cloth, or by any other means, A will vibrate transversely, as will be known by the dancing of the sand, and its settling on the nodes, C, C.
On the other hand, if A be held vertically, and agitated transversely by a bow, while A' is horizontal and strewed with sand, the latter will indicate longitudinal-tangential vibrations, both by the creeping of the sand, and by the difference of the nodal figures on its two faces.
Example 3. Let M, fig. 114, be a rectangular plate, mounted like A in the last example, but instead of carrying a simple plate A', let it carry a system of circular discs traversed by a lamina, as in the figure. Then, if the faces of these discs and of the lamina, M, be horizontally placed and strewed with sand, and the lamina, M, be set in longitudinal-tangential vibration, all the discs will be so too, and the sand will arrange itself in figures which, on every alternate disc, 1, 3, 5, &c. will be of one species, (such as at a for instance,) but on every other, 2, 4, 6, &c. will be of a different species, as b.
Communication of the Vibrations of a String to a Disc.
Now if the whole apparatus be inverted, so as to place the lamina, M, uppermost, and let the system of discs hang down, the then upper surfaces of the discs will exhibit the same system of nodal figures, but in the reverse order; i. e. the discs 1, 3, 5, &c. will give the figure b, and 2, 4, 6, &c. the figure a.
In this apparatus, if the connecting piece which traverses all the discs be examined, it will be found to vibrate transversely, while the discs and lamina, M, vibrate tangentially, and vice versâ.
Example 4. Let A, fig. 115, be a strong frame of wood of the form [, across the extreme edges of which is stretched a strong catgut, or other cord, and let LL' be a circular disc of glass, or metal, retained between the chord and back of the frame by the pressure of the former. Then, if the chord be set in vibration by a bow drawn transversely across it in one steady direction, the vibrations of the chord will all lie in the plane of the bow, and will be communicated in the same direction to the disc, which will execute tangential vibrations, all its molecules moving to and fro in lines parallel to the bow through the whole extent of the disc. This is easily verified by the direction in which sand strewed on it creeps.
Conceive the whole apparatus placed with the chord vertical, and projected on the plane of the horizon. If, as in fig. 115, a, FF" be the projection of the bow, the surface of the disc will be marked with nodal lines parallel to it, the sand there being left, while that in the intermediate spaces creeps along to the edges, as marked by the arrows, and runs off.
If the projection of the bow, FF, be oblique to the line joining the points of support of the disc, as in fig. 115, b, the nodal line will be curved, as there shown, but the motion of the molecules of the sand going to form it will still be parallel to FF'.
Finally, if the bow be drawn parallel to the line joining the points of support, as in fig. 115, c, the nodal line will be formed of two arcs making a cusp, but the same law of molecular motion will still hold good, as the arrows indicate.
Example 5. Let LL', fig. 116, be a rectangular lamina fastened at one end into a block, T, and at the other attached to a