WAVES AND VIBRATIONS. Since the direct cause of a sound is the reception into the ear of waves in the air, it is necessary to analyze the nature of these waves. We may have an irregu lar, isolated di4urbanee, whieh is analogous to a "hump" passing along a stretched rope, or to the effect of dropping several stones at random intervals into a pool of water; or we may have a regular continuous succession of waves identical in all respects, which is called a "train of waves." The simplest kind of train of waves is what is called a "simple harmonic" train, such as is produced in any medium by a simple vibration of the body which is causing the waves. (Vibrations of a pendulum are simple harmonic.) Such a train of waves is character ized by its "wave-number" and "a mplitude ;" the wave-number being the number of individual waves which pass a given fixed point in one second, while the amplitude is the extent of the path of vibration of any particle of the medium through which the waves are passing. The velocity of waves of a definite character, e.g., compressional ones, in any definite homogeneous medium depends upon the properties of the me dium itself, not on the wave-number or ampli tude of the waves. So, if 2 is the wave-length, i.e., the distance from one point in the medium to the next point, measured in the direction of advance of the waves, where the conditions are identical with those at the first point, and if ..\" is the wave-number, the yelocity of the waves I' is given by the formula V=X Consequently, if A' is known, 2 can be calculated, and vice versa and the characteristics of the simple harmonic train of waves limy he said to be its wave-length and its amplitude. If sev eral trains of waves are passing through the same medium at the same time, the resulting waves—called a "complex" train—is simply the sum of the individual waves, the motion of any particle of the medium the geometrical sum of the motions which it would have, owing to each of the separate trains of waves. (This is rigidly true only if the amplitudes of these separate trains are very small compared with their wave-lengths, as in general they are.)
This is shown in Fig. 2, where A and B are two sets of simple harmonic waves which form the resultant wave C. This wave is obtained by taking the algebraic sum of the motion of the particles. The point b" is obtained by taking a"b", equal to the sum of a b and a'b'. c" d" is the sum of c d and c'd', the latter, as it occurs be low the axis, considered as having a negative sign. Conversely, it may be shown that. any com plex train of waves may be analyzed into simple harmonic trains. Therefore, eomplex trains of waves may differ in several ways: 1. The num ber of the component simple harmonic trains. 2. Their wave-numbers and amplitudes. 3. Their relative "phases," for two waves are in different phase if the maximum displacement due to one train does not coincide in position with that due to the other; or, looked at in another way, the component trains may have been started at irregular intervals. Since waves are due to the vibrations of some elastic body (e.g., a tuning-fork, the air in an organ-pipe or horn), it is necessary next to analyze the nature of vibrations. We may have an irregular vibra tion, consisting of only a few to and fro motions, then a sudden change into another vibration of a different eharaeter, the whole motion lasting only a short time, e.g., when a piece of still' paper is torn or when a scratching pen is used in writing; or we may have a regular continuous periodic vibration. The simplest possible peri odic vibration is like that of a simple pendulum, and it is called "simple harmonic." It is char acterized by a definite number of vibrations per second, i.e., its "frequency," and by the extent of the swing, i.e., its "amplitude." If a second pendulum is suspended from the bob of the first, and a third front the bob of the second, the vi bration of the third and lowest bob is no longer simple harmonic in general. its vibration is called "complex:" and it is evident that it is the sum of the vibrations of the separate pen dulums. Complex vibrations may, therefore, differ in the number of the component vibrations, and in their frequencies, amplitudes, and rela tive phases.