DIRECT AND REFLECTED PULSES. 93 has its definite time of vibration dependent on its length, weight, thickness and tension, and my impulses must synchronise with that time. I stop the motion, and now by a sudden jerk I raise a bump upon the tube, which runs along it as a pulse to wards its fixed end, where the If, instead of imparting a C single jerk to the end of the (1) FIG. 33. (2) tube, I impart a succession of jerks, thereby sending a series of pulses along the tube, every one of them will be reflected above, and we have now to enquire how the direct and reflected undulations behave towards each other. с FIG. 34. с C I start a pulse along the tube. Let the time required for it to pass from my hand to the fixed end be one second; at the end of half a second it occupies the position a b (1), fig. 34, its foremost point having reached the middle of the tube. At the end of a whole second it would have the position b c (2), its foremost point having reached the fixed end c of the tube. At the moment when reflection begins at c, let another jerk be imparted at a; the reflected pulse from c moving with the same velocity as this direct one from a, the foremost points of both will arrive at the centre b (3) at the same moment. What must occur? The hump a b wishes to move on to c, and to do so must move the point b to the right. The hump cb wishes to move towards a, and to do so must move the point b to the left. The point b, urged by equal forces in two opposite directions at the same time, will not move in either direction. Under these circumstances, the two halves a b, b c of the tube will oscillate as if they were independent of each other (4). Thus by the combination of two progressive pulses, the one direct (2) (3) (4) b and the other reflected, we produce two stationary pulses on the tube a c. The vibrating parts a b and b c are called ventral segments; the point of no vibration b is called a node. STATIONARY AND PROGRESSIVE WAVES. 95 I use the term 'pulse' here advisedly, lest I should introduce confusion into your thoughts by the employment of the more usual term wave. For a wave embraces two of these pulses. It embraces both the hump and the depression which follows the hump. The length of a wave therefore, is twice that of a ventral segment. Supposing the jerks to be so timed as to cause each hump to be one-third of the tube's length. At the end of one-third of a second from starting the pulse will be in the position a b (1), fig. 35. In two-thirds of a second it will have reached the position b b' (2), fig. 35. At this moment let a new pulse be started at a; after the lapse of an entire second from the commencement we shall have two humps upon the tube, one occupying the position a b (3), the other the position b'c (3). It is here manifest that the end of the reflected pulse from c, and the end of the direct one from a, will reach the point b' at the same moment. We shall therefore have the state of things represented in (4), where bb' wishes to move upwards, and cb' to move downwards. The action of both upon the point l' being in opposite directions, that point will remain fixed, and from it, as if it were a fixed point, the pulse bb' will appear to be reflected, while the segment b'c will oscillate as an independent string. Supposing that at the moment b b' (4) begins to be reflected at b', we start another pulse from a, it will reach b (5) at the same moment the pulse reflected from b' reaches it. The pulses will neutralise each other at b, and we shall have there a second node. Thus, by properly timing our jerks, we divide the rope into three ventral segments, separated from each other by two nodal points. As long as the agitation continues the tube will vibrate as in (6).* * There is no theoretic limit to the number of nodes and ventral segments that may be thus produced. By quickening the impulses, I divide the tube into four ventral * 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 we may employ a hook turned by a wheel, or whirling table, and a string of catgut 10 or 12 feet long, with silvered beads strung along it, as a vibrating chord. Attaching one end of the string to the hook, the other end to a freely moving swivel connected with a fixed stand, on turning the wheel and properly regulating both the tension and the rapidity of rotation, the beaded chord may be caused to rotate as a whole, and to divide itself successively into 2, 3, 4, or 5 ventral segments. When the whole chord is enveloped in a cylinder of light from the electric lamp every bead describes a brilliant circle, and a very splendid experiment is the result. NODES AND VENTRAL SEGMENTS. 97 segments separated by three nodes; here again I have five ventral segments and four nodes. With this particular tube the hand may be caused to vibrate sufficiently quickly to produce ten ventral segments, as shown in fig. 35 (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 easily performed. It is quite plain, that any other oscillating body whose vibrations are of sufficient power, and whose periods are of the proper kind, may be substituted for the hand. Fixing, for example, one end of a tolerably heavy rod in a vice, or, better still, screwing one end into an anvil or other heavy block, and attaching to the free end of the rod the end of our india-rubber tube; by varying the length of the rod its vibrations may be caused to synchronise with the various vibrations of the tube, and thus the latter may be caused to swing as a whole, or to divide itself into any number of oscillating parts. 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 to you. To make the connexion of both classes of vibrations more obvious, I will vary our last experiments. Before you is a piece of india-rubber tubing, 10 or 12 feet long, stretched from a to c, fig. 36, and made fast to two pins at c and a. The tube is blackened, and behind it is placed a surface of H |