the form b. After reflexion at A, it will resume its first form a, and so on. Now, instead of a single jerk, let the hand holding the free end execute a series of equal continuous transverse vibrations. Each complete vibration will cause a wave, ab Fig. 31, to Fig.31. B a pass along the tube from A to B, where reflexion will reverse the protuberances, so that the wave will return from B to A stern foremost. Next let the tube be again fastened at both ends, as before, and the vibrations of the hand impressed at some intermediate point, as C, Fig. 32. C Fig. 82. Two sets of waves will now start from C in the directions of the arrows. They will be reflected at A and B, and then their effects will intermingle. Let us suppose that the tube has been set in steady motion. and, on the removal of the hand, continues its vibrations without any external force acting on it. Two sets of equal waves are now moving with equal velocities from A towards B and from B towards A, and we have to determine the motion of the tube under their joint action. Suppose that a crest' a, Fig. 33, moving from A towards B, meets an equal trough' b, moving from B towards A, at the point c. An undisturbed particle of the tube situated at this point is solicited by equal forces in opposite directions, and therefore remains at Fig.33 A B rest. The two equal and opposite pulses then proceed to cross each other, but, as a moves to the right and b to the left with equal speed, there is nothing to give either of them at the point c an influence superior to that exerted in the contrary direction by the other. The particle at c therefore remains at rest under their joint influence, i.e. a node is formed at that point. If a trough had been moving from A towards B, and an equal crest from B towards A, the result would clearly have been the same: hence A node is formed at every point where two equal and opposite pulses, a crest and a trough, meet each other. 51. The annexed figure represents two series of equal waves advancing in opposite directions with equal velocities. The moment chosen is that at 1 Provided that confusion with water-waves be explicitly guarded against, there is no objection to retaining this convenient phraseology for distinguishing between opposite protuberances. which crest coincides with crest and trough with trough. The joint effect thus produced does not appear in the figure, our object at present being merely to determine the number and positions of the resulting nodes. For the sake of clearness, one set of waves is represented slightly below the other, though in fact the two are strictly coincident. Fig. 84. m Let the waves abdf...z be moving from left to right, the waves z'ts'q'...a' from right to left. The crest klm meets the trough pn'm at m. After these have crossed each other, the trough ghk and the crest rq'p will also meet at m, since km and pm are equal distances. Similarly the crest efg and the trough ts'r will meet at m. Accordingly the point m is a node, and, by exactly the same reasoning, so are a, c, e, g, k, p, r, t, &c. The distances between pairs of consecutive nodes are all equal, each being a single pulse-length, i.e. half a wave-length, of either series. Two pulse-lengths, as gk and km, give three nodes g, k, and m; three pulse-lengths four nodes, and so on. There is thus always one node more than the number of pulses. On the other hand, the fixed ends of the tube, which are the origins of the systems of reflected waves, occupy two of these nodes. Deducting them we arrive at this result : The number of nodes formed is one less than the number of the pulse-lengths (or half wave-lengths), which together make up the length of the vibrating tube. 52. We will now ascertain how the portions of the tube between consecutive nodes move while the two systems of waves are simultaneously passing along it. Let A and B, Fig. 35, be the fixed ends, as before, and let us take five nodes at the points 1, 2, 3, 4, 5. In (1), the systems of waves coincide, accordingly each point of the tube is displaced through twice as great a distance as if it had been acted on by only one system. The tube thus takes the form indicated by the strong line in the figure. In (2), one set of waves has moved half a pulse-length to the right, and the other the same distance to the left. The two systems are now in complete antagonism, the displacements being equal in amount and opposite in direction at every point. The tube is therefore momentarily in its undisturbed position. In (3), each system has moved through a pulse-length, and the maximum displacements are again produced on the tube, but in opposite directions to those of (1). In (4), where the systems have moved through a pulselength-and-a-half, the tube passes again through its undisturbed position, and, in (5), regains that which it occupied in (1), the systems of waves, meanwhile, having each traversed two pulse-lengths, or one wavelength'. Thus the tube executes one complete vibration in the time occupied by a pulse in passing along a length of the tube equal to twice one of its own ventral segments. In other words, the tube's rate of vibration varies as the number of segments into which it is divided. It moves most slowly in the form shown in Fig. 26 with but a single segment; twice 'The reader will find that Fig. 35 is rendered more readily intelligible by drawing the two systems of waves in different colours, and the successive positions of the tube in black. |