Page:Elementary Text-book of Physics (Anthony, 1897).djvu/165

§ 130] suppose the layer $$a$$, from any cause whatever, to execute regular vibrations, this movement will be transmitted to the succeeding layers with the velocity given by the formula, and, in time, each layer of particles in the cylinder will be executing vibrations similar to those of $$a$$. If the vibrations of $$a$$ be performed in the time $$t$$, the motion will be transmitted during one complete vibration of $$a$$ to a distance $$s = vt$$, where $$v$$ is the velocity of propagation, say to $$a'$$, during two complete vibrations of $$a$$ to a distance $$2s = 2vt$$, or to $$a$$, during three complete vibrations to $$a'$$, and so on. It is evident that the layer $$a'$$ begins its first vibration at the instant that $$a$$ begins its second vibration, $$a''$$ begins its first vibration at the instant that $$a'$$ begins its second and $$a$$ its third vibration. The layer midway between $$a$$ and $$a'$$ evidently begins its vibration just as $$a$$ completes the first half of its vibration, and therefore moves forward while $$a$$ moves backward. This condition of things existing in the cylinder constitutes a wave motion. While $$a$$ moves forward, the portions near it are compressed. While it moves backward, they are dilated. Whatever the condition at $$a$$, the same condition will exist at the same instant at $$a', a$$, etc. The distance $$a' = a'a$$ is called a wave length; it is the distance from any one particle to the next one of which the vibrations are in the same phase (§ 21). If the condition at $$a$$ and $$a'$$ be one of condensation, it is evident that at $$d$$, midway between $$a$$ and $$a'$$, there must be a rarefaction. In the wave length $$aa'$$ exist all intermediate conditions of condensation and rarefaction. These conditions must follow each other along the cylinder with the velocity of the transmitted motion, and they constitute a progressive wave moving with this velocity. If the vibratory motion with which $$a$$ is endowed be communicated by a sounding body, the wave is a sound-wave. If, instead of a cylinder of the substance, we have an indefinite medium in the midst of which the sounding body is placed, the motion is transmitted in all directions as spherical waves about the sounding body as a centre.

130. Mode of Propagation of Wave Motion.—The mode of transmission of wave motions was first shown by Huygens, and