of anything which could be called harsh or piercing. As compared with a pianoforte note of the same pitch the fork-tone is wanting in richness and vivacity, and produces an impression of greater depth, so that one is at first inclined to think the fork employed must be an Octave too low.

It is a direct inference from the general theory of the nature of quality that simple tones can differ only in pitch and intensity. Accordingly we find that tuning-forks of the same pitch, mounted on resonance-boxes and set gently vibrating by a resined bow, exhibit, whatever be their forms and sizes, differences of loudness only. When made to sound with equal intensity by suitable bowing, their tones are absolutely undistinguishable from each other.

2. Sounds of vibrating strings.

49. Sounding strings vibrate so rapidly that their movements cannot be followed directly by the eye. It will be well, therefore, to examine how the slower and more easily controllable vibrations of non-sounding strings are performed, before treating the proper subject of this section. Take a flexible caoutchouc tube ten or fifteen feet long and fasten its ends to two fixed objects separated from each other by that distance. The tube can be conveniently set in periodic vibration by impressing a swaying

movement upon it with the hand near either extremity, in suitable time. According to the rapidity of the motion thus communicated, the tube will take up different forms of vibration. The simplest of these is shown in Fig. 26. A and B being its fixed extremities, the tube vibrates as a whole between the two extreme positions AaB and AbB.

The tube may also vibrate in the form shown in Fig. 27, where AabB and AcdB are its extreme positions.

Fig.27

B

A

In this instance the middle point of the tube, C, remains at rest, the loops on either side of it moving independently, as though the tube were fastened at C as well as at A and B. For this reason the point C is called a node, from the Latin nodus, a knot. The loops AC and CB are termed ventral segments.

Fig. 28 shows a form of vibration with two nodes, at C and D, dividing the distance AB into three

equal ventral segments. We may also obtain forms with three, four, five, &c., nodes, dividing the tube into four, five, six, &c., equal ventral segments, re

Fig.28.

spectively. 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 suddenly jerking the hand transversely

Fig. 29.

to it through a small 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 reflexion, and so on. At each reflexion 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 way back it will have

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

α

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.

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.