I now stop the motion, and by a sudden jerk raise a hump upon the tube, which runs along it as a pulse toward its fixed end; here the hump reverses itself, and runs back to my hand. At the fixed end of the tube, in obedience to the law of reflection, the pulse reversed both its position and the direction of its motion. Supposing c, Fig. 36, to be the fixed end of the tube, and a the end held in the hand: if the pulse on reaching c have the position shown in (1), after reflection it will have the position shown in (2). The arrows mark the direction of progression. The time required for the pulse to pass from the hand to the fixed end and back is exactly that required to accomplish one complete vibration of the tube as a whole. It is indeed the addition of such impulses which causes the tube to continue to vibrate as a whole.
Fig. 37.
If, instead of a single jerk, a succession of jerks be imparted, thereby sending a series of pulses along the tube, every one of them will be reflected above, and we have now to inquire how the direct and reflected pulses behave toward each other.
Let the time required by the pulse 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. 37, 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 c b wishes to move toward 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 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.
The term “pulse” is here used advisedly, instead 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.
Fig. 38.
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. 38. In two-thirds of a second it will have reached the position b b′ (2), Fig. 38. 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 b b′ wishes to move upward, and c b′ to move downward. The action of both upon the point b′ being in opposite directions, that point will remain fixed. And from it, as if it were a fixed point, the pulse b b′ will 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 at the same moment the pulse reflected from b′ reaches it. The pulses will neutralize 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).