If we now touch the vibrating string A´ B lightly with the finger, or with a feather at the middle point C, Fig. 40, it will give out a more acute but fainter sound than before, and while the extent of its vibrations is diminished, their frequency is doubled. In like manner, if we touch the string A´´ B´´, Fig. 40, at a point C, so that A´´ C is one-third of A´´ B´´, the note will be still more acute, and correspond to thrice the number of vibrations. All this might have been expected; but the wonderful part of the experiment is, that the vibrating string A´ B´ divides itself at C into two parts A´ C, C B´, the part A´ C vibrating round A and C as fixed points, and the part C B´ round C and B´, but always so that the part A´ C is at the same distance on the one side of the axis A´ B´ as at A m C, while the part C B is on the other side, as at C n B. Hence the point C, being always pulled by equal and opposite forces, remains at rest as if it were absolutely fixed. This stationary point is called a node, and the vibrating portions A´m C, C n B´ loops. The very same is true of the string A´´ B´´, the points C and D being stationary points; and upon the same principle a string may be divided into any number of vibrating portions. In order to prove that the string is actually vibrating in these equal subdivisions, we have only to place a piece of light paper with a notch in it on different parts of the string. At the nodes C and D it will remain perfectly at rest, while at m or n in the middle of the loops it will be thrown off or violently agitated.
The acute sounds given out by each of the vibrating portions are called harmonic sounds, and they accompany the fundamental sound of the string in the very same manner as we have already seen that the eye sees the accidental or harmonic colours while it is affected with the fundamental colour.
The subdivision of the string, and consequently the production of harmonic sounds, may be effected without touching the string at all, and by means of a sympathetic action conveyed by the air. If a string A B, for example, Fig. 40, is at rest, and if a shorter string A´´ C, one third of its length, fixed at the two points A´´ and C, is set vibrating in the same room, the string A B will be set vibrating in three loops like A´´ B´´, giving out the same harmonic sounds as the small string A´´ C.
It is owing to this property of sounding bodies that singers with great power of voice are able to break into pieces a large tumbler glass, by singing close to it its proper fundamental note; and it is from the same sympathetic communication of vibrations that two pendulum clocks fixed to the same wall, or two watches lying upon the same table, will take the same rate of going, though they would not agree with one another if placed in separate apartments. Mr. Ellicott even observed that the pendulum of the one clock will stop that of the other, and that the stopped pendulum will, after a certain time, resume its vibrations, and in its turn stop the vibrations of the other pendulum.
The production of musical sounds by the vibrations of a column of air in a pipe is familiar to every person, but the extraordinary mechanism by which it is effected is known principally to philosophers. A column of air in a pipe may be set vibrating by blowing over the open end of it, as is done in Pan’s pipes; or by blowing over a hole in its side, as in the flute; or by blowing through an aperture called a reed, with a flexible tongue, as in the clarionet. In order to understand the nature of this vibration, let AB, Fig. 41, be a pipe or tube, and let us place in it a spiral spring AB, in which the coil or spire are at equal distances, each end of the spiral being fixed to the end of the tube. This elastic spring may be supposed to represent the air in the pipe, which is of equal density throughout. If we take hold of the spring at m, and push the point m towards A and towards B in succession, it will give us a good idea of the vibration of an elastic column of air. When m is pushed towards A, the spiral spring will be compressed or condensed, as shown at m A, No. 2, while at the other end it will be dilated or rarefied, as shown at m B, and in the middle of the tube it will have the same degree of compression as in No. 1. When the string is drawn to the other end of the tube B, the spring will be, as in No. 3, condensed at the end B, and dilated at the end A. Now when a column of air vibrates in a pipe AB, the whole of it rushes alternately from B to A, as in No. 2, and from A to B as in No. 3, being condensed at the end A, No. 2, and dilated or rarefied at the end B, while in No. 3 it is rarefied at A and condensed at B, preserving its natural density at the middle point between A and B. In the case of the spring the ends AB are alternately pushed outwards and pulled inwards by the spring, the end A being pushed outwards in No. 2, and B pulled inwards, while in No. 3 A is pulled inwards and B pushed outwards.
Fig. 41.
That the air vibrating in a pipe is actually in the state now described, may be shown by boring small holes in the pipe, and putting over them pieces of a fine membrane. The membrane opposite to the middle part between A and B where the particles of the air have the greatest motion, will be violently agitated, while at points nearer the ends A and B it will be less and less affected.
Fig. 42.
