tube into four, five, six, &c., equal ventral segments, respectively. The stiffness of very short portions of the tube alone imposes a limit on the subdividing process. Let us examine the mechanical causes to which these effects are due.

50. If we unfasten one end of the tube, and, holding it in the hand as in Fig. 29, raise a hump upon it, by moving the hand suddenly through a small

Fig. 29.

B

distance, the hump will run along the tube until it reaches its fixed extremity B; it will then be reflected and run back to A, where it will undergo a second reflection, and so on. At each reflection the hump will have its convexity reversed. Thus, if while travelling from A towards B its form was that of a, Fig. 30, on its return it will have the form b. After

reflection 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

vibrations. Each complete vibration will cause a wave ab Fig. 31, consisting of crest b, and trough a,

Fig. 31.
b

B

α

to pass along the tube from A to B, where reflection will turn crest into trough and trough into crest; 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. Fig. 32.

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 intermingled. We will suppose that the tube has been set in steady motion, and, the hand being removed, 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 their joint effect in fixing the form of vibration in which the tube swings.

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. The point c is now

solicited by a and b in opposite directions and with equal energy, and therefore remains at rest. The

A

Fig.33

B

two 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 an influence upon the point c, where they first met, superior to that exercised in the contrary direction by the other. Thus c remains at rest under their joint influence, and a node is therefore formed at that point. If a trough had been moving from A towards B, and an equal crest from B towards A, the effect would clearly have been the same.

A node must therefore be 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 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.
Ն

m

n

n

Let the waves abdf...z be moving from left to right, the waves z't's'n'...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 rap 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 in excess of 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 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 under the

action of the two systems of waves passing along it. Let AB, 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