Page:Elementary Text-book of Physics (Anthony, 1897).djvu/414

400 wave, and some other explanation must be given for the rectilinear propagation of light in such a wave. For consider a plane wave advancing toward a point $$P,$$ and describe on it a series of circles, the distances of which from $$P$$ differ by half a wave-length. These circles cut a line in the surface drawn from $$A$$ (Fig. 93), the foot of the perpendicular to the wave-front from $$P,$$ in the points $${a, b, c,}$$ etc., and the rings enclosed between them are half-period elements. The areas of these rings are $${2\pi \overline{A a^2}, 2\pi (\overline{Ab^2} - \overline{Aa^2}), 2\pi (\overline{Ac^2} - \overline{Ab^2})},$$ etc. If $$\lambda$$ be very small, they each become equal to $$2\pi^2 l.$$ Hence if their effects at $$P$$ depend only on their areas, they would annul one another, and no light would reach $$P.$$ We are therefore forced to assume that the effect of each area in sending light to $$P$$ diminishes as the obliquity increases, so that the iirst area is more efficient than the second, the second than the third, and so on. The effectiveness of the areas diminishes at first slowly, and afterwards more rapidly, the more distant areas having nearly the same efficiency. Eepresenting the efficiency of the areas by $${m_{1}, m_{2}, m_{3}},$$ etc., and remembering that the even areas oppose the action of the odd ones, we may write the total efficiency in the form of $$\tfrac{1}{2}m_{1} + \tfrac{1}{2} (m_{1} - m_{2}) - \tfrac{1}{2}(m_{2} - m_{3}) + \tfrac{1}{2}(m_{3} - m_{4}) - \dots$$ Each of the terms in parenthesis is very nearly equal to zero, and the efficiency at $$P$$ is therefore nearly half of that of the central area. The light therefore appears to reach $$P$$ from a small area around $$A.$$

It is important to note that the deductions of this section apply only where $$A$$ is small in relation to $$x,$$ so that $$\lambda^2$$ may be neglected in comparison with $$x\lambda.$$ With sound-waves this is not true, and if a computation similar to that given above for light-waves be made for sound, not omitting $$\lambda^2,$$ it will be seen why there are no definite straight lines of sound and no sharp acoustic shadows.

326. Principle of Least Time.—The above are only particular cases of a law of very general application, that light in going from one point to another follows the path that requires least time. The reason is that values in the vicinity of a minimum change slowly, and there will be a number of points in the neighborhood of that point from which the light-waves are propagated to the given point