Page:Elementary Text-book of Physics (Anthony, 1897).djvu/173

§134] the paper above $$AB$$, so will coincide with $$sp$$, $$s'o'$$ with $$s'p'$$, etc., and hence $$no$$ with $$np$$. But $$no$$ is a circle with $$C$$ as a centre; $$np$$ is, therefore, a circle of which the centre is $$C'$$, on a perpendicular to $$AB$$ through $$C$$, and as far below $$AB$$ as $$C$$ is above. When, therefore, a wave is reflected at a plane surface, the centres of the incident and reflected waves are on the same line perpendicular to the reflecting surface, and at equal distances from the surface on opposite sides.

134. Theoretical Velocity of Sound.—The disturbance of the parts of any elastic medium which is propagating sound is assumed, in theoretical discussions, to take place in the line of direction of the propagation of the sound, and to be such that the type of the disturbance remains unaltered during its propagation. The velocity of propagation of such a disturbance may be investigated by the following method, due to Rankine.

Let us consider, as in § 129, a portion of the elastic medium in the form of an indefinitely long cylinder. If a disturbance be set up at any cross-section of this cylinder (Fig. 51), which consists of a displacement of the matter in that cross-section in the direction of the axis of the cylinder, it will, by hypothesis, be propagated in the direction of the axis with a constant velocity $$V$$, which is to be determined. If we consider any cross-section of the cylinder which is traversed by the disturbance, the matter which passes through it at any instant will have a velocity which may vary from zero to the maximum velocity of the vibrating matter, either positively when this velocity is in the direction of propagation of the disturbance, or negatively when it is opposite to it.

If we now conceive an imaginary cross-section $$A$$ to move along the cylinder with the disturbance with the velocity $$V$$, the velocity