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140 COMPOSITION OF VIBRATIONS. [VI. § 70.
tions to which those of a fork give rise. Hence a simple tone is due to vibrations executed according to the pendulum-law.
Such vibrations will, therefore, give rise to waves of condensation and rarefaction whose associated wave-form is that drawn in Fig. 17 bis, p. 40. It will be convenient to call the vibrations to which a simple tone is due simple vibrations; and the associated waves simple waves. We proceed to examine the modes of vibration corresponding to composite sounds.
Let us first take the case of a clang consisting of but tivo simple tones, the fundamental and its first overtone. A particle of air engaged in transmitting this sound is simultaneously acted upon by two sets of forces which, if they acted separately, would cause it to perform two simple vibrations whose periods were to each other in the ratio of 1 to 2. We have to investigate what its motion will be under the joint action of these forces. The problem before us is the composition of two vibrations executed in the same straight line.
Suppose that a series of particles originally at rest in the dotted line through O, Fig. 44, and capable of vibrating in paths perpendicular to it, are under simultaneous forces which, if acting separately, would give rise to the transmission of the parts of waves drawn in the figure. Let a and b be the