There is no theoretic limit to the number of nodes and ventral segments that may be thus produced. By the quickening of the impulses, the tube is divided into four ventral segments separated by three nodes; quickening still more we have five ventral segments and four nodes. With this particular tube the hand may be caused to vibrate sufficiently quick to produce ten ventral segments, as shown in [Fig. 38 (7)]. When the stretching force is constant, the number of ventral segments is proportional to the rapidity of the hand’s vibration. To produce 2, 3, 4, 10 ventral segments requires twice, three times, four times, ten times the rapidity of vibration necessary to make the tube swing as a whole. When the vibration is very rapid the ventral segments appear like a series of shadowy spindles, separated from each other by dark motionless nodes. The experiment is a beautiful one, and it is easily performed.
If, instead of moving the hand to-and fro, it be caused to describe a small circle, the ventral segments become “surfaces of revolution.” Instead of the hand, moreover, we may employ a hook turned by a whirling-table. Before you is a cord more rigid than the India-rubber tube, 25 feet long, with one of its ends attached to a freely-moving swivel fixed in the ceiling of the room. By turning the whirling-table to which the other end is attached, this cord may be divided into as many as 20 ventral segments, separated from each other by their appropriate nodes. In another arrangement a string of catgut 12 feet long, with silvered beads strung along it, is stretched horizontally between a vertical wheel and a free swivel fixed in a rigid stand. On turning the wheel, and properly regulating both the tension and the rapidity of rotation, the beaded cord may be caused to rotate as a whole, and to divide itself successively into 2, 3, 4, or 5 ventral segments. When we envelop the cord in a luminous beam, every spot of light on every bead describes a brilliant circle, and a very beautiful experiment is the result.
§ 4. Mechanical Illustrations of Damping Various Points of Vibrating Cord
The subject of stationary waves was first experimentally treated by the Messrs. Weber, in their excellent researches on wave-motion. It is a subject which will well repay your attention by rendering many of the most difficult phenomena of musical strings perfectly intelligible. It will make the connection of both classes of vibrations more obvious if we vary our last experiments. Before you is a piece of India-rubber tubing, 10 or 12 feet long, stretched from c to a, Fig. 39, and made fast to two pins at c and a. The tube is blackened, and behind it is placed a surface of white paper, to render its motions more visible. Encircling the tube at its centre b (1) by the thumb and forefinger of my left hand, and taking the middle of the lower half b a of the tube in my right, I pluck it aside. Not only does the lower half swing, but the upper half also is thrown into vibration. Withdrawing the hands wholly from the tube, its two halves a b and b c continue to vibrate, being separated from each other by a node b at the centre (2).
Fig. 39.
I now encircle the tube at a point b (3) one-third of its length from its lower end a, and, taking hold of a b at its centre, pluck it aside; the length b c above my hand instantly divides into two vibrating segments. Withdrawing the hands wholly, you see the entire tube divided into three ventral segments, separated from each other by two motionless nodes, b and b′ (4). I pass on to the point b (5), which marks off one-fourth of the length of the tube, encircle it, and pluck the shorter segment aside. The longer segment above my hand divides itself immediately into three vibrating parts. So that, on withdrawing the hand, the whole tube appears before you divided into four ventral segments, separated from each other by three nodes b b′ b″ (6). In precisely the same way the tube may be divided into five vibrating segments with four nodes.
This sudden division of the long upper segment of the tube, without any apparent cause, is very surprising; but if you grant me your attention for a moment, you will find that these experiments are essentially similar to those which illustrated the coalescence of direct and reflected undulations. Reverting for a moment to the latter, you observed that the to-and-fro motion of the hand through the space of a single inch was sufficient to make the middle points of the ventral segments vibrate through a foot or eighteen inches. By being properly timed the impulses accumulated, until the amplitude of the vibrating segments exceeded immensely that of the hand which produced them. The hand, in fact, constituted a nodal point, so small was its comparative motion. Indeed, it is usual, and correct, to regard the ends of the tube also as nodal points.
Consider now the case represented in (1), [Fig. 39], where the tube was encircled at its middle, the lower segment a b being thrown into the vibration corresponding to its length and tension. The circle formed by the finger and thumb permitted the tube to oscillate at the point b through the space of an inch; and the vibrations at that point acted upon the upper half b c exactly as my hand acted when it caused the tube suspended from the ceiling to swing as a whole, as in [Fig. 35.] Instead of the timid vibrations of the hand, we have now the timid vibrations of the lower half of the tube; and these, though narrowed to an inch at the place clasped by the finger and thumb, soon accumulate, and finally produce an amplitude, in the upper half, far exceeding their own. The same reasoning applies to all the other cases of subdivision. If, instead of encircling a point by the finger and thumb, and plucking the portion of the tube below it aside, that same point were taken hold of by the hand and agitated in the period proper to the lower segment of the tube, precisely the same effect would be produced. We thus reduce both effects to one and the same cause; namely, the combination of direct and reflected undulations.
And here let me add that, when the tube was divided by the timid impulses of the hand, not one of its nodes was, strictly speaking, a point of no motion; for were the nodes not capable of vibrating through a very small amplitude, the motion of the various segments of the tube could not be maintained.