Page:Elementary Text-book of Physics (Anthony, 1897).djvu/411

§325] which makes any wave front the resultant of an infinite number of elementary waves proceeding from the various points of the same wave front in one of its earlier positions. It can, however, easily be shown that when the wave lengths are small, the disturbance at any point $$P$$ (Fig. 93) is due almost wholly to a very small portion of the approaching wave. Let us consider first the case of an isotropic medium, in which light moves in all directions with the same velocity. Let $$mn$$ be the front of a linear wave perpendicular to the plane of the paper, moving from left to right or towards $$P.$$ Draw $$PA$$ perpendicular to the wave front, and draw $$Pa, Pb,$$ etc., at such obliquities that $$Pa$$ shall exceed $$PA$$ by half a wave length, $$Pb$$ exceed $$Pa$$ by half a wave length, etc. We will designate the wave length by $$\lambda.$$

It is evident that the total effect at $$P$$ will be the sum of the effects due to the small portions $$Aa, Ab,$$ etc., called half-period elements. Since $$Pa$$ is half a wave length greater than $$PA,$$ and $$Pb$$ half a wave length greater than $$Pa,$$ each point of ah is half a wave length farther from $$P$$ than some point in $$Aa;$$ hence elementary waves from $$ab$$ will meet at $$P$$ waves from $$Aa$$ in the opposite phase. It appears, therefore, that the effects at $$P$$ of the portions $$ab$$ and $$Aa$$ are opposite in sign, and tend to annul each other. The same is true of $$bc$$ and $$cd.$$ But the effects of $$Aa$$ and $$ab$$ may be considered as proportional to their lengths. Hence, by computing the lengths, we can determine the resultant effect at $$P.$$ Let $$AP = x.$$ From the construction we have