Page:Elementary Text-book of Physics (Anthony, 1897).djvu/442

428 If $$n$$ be any even whole number, the values of $$x$$ given by this equation represent points on the screen $$mn$$ at which the waves from $$A$$ and $$B$$ meet in the same phase and unite to produce light. If $$n$$ be any odd whole number, the corresponding values of $$x$$ represent points where the waves meet in opposite phases, and therefore produce darkness. It appears, therefore, that starting from $$s,$$ for which $$n = 0,$$ we shall have darkness at distances $$\frac{\tfrac{1}{2}\lambda}{b}, \frac{\tfrac{3}{2}\lambda}{b}, \frac{\tfrac{5}{2}\lambda}{b},$$ etc., and light at distances $$0, \frac{\lambda c}{b}, \frac{2 \lambda c}{b}, \frac{3 \lambda c}{b},$$ etc.

From equation (113) we have $$n = \frac{2bx}{c\lambda}\cdot$$ Since $$\tfrac{1}{2}n\lambda$$ is the number of wave lengths that the wave front from $$B$$ falls behind that from $$A, \tfrac{1}{2}nT,$$ where $$T$$ represents the period of one vibration, is the time that must elapse after the wave from $$A$$ produces a certain displacement before that from $$B$$ produces a similar displacement. The expression $$\frac{2\pi \tfrac{1}{2}nT}{T} = n\pi$$ is, therefore, the difference in epoch of the two wave systems. Substituting $$n\pi$$ for $$\epsilon$$ in equation (17), we have $$S = s + s' = a(2 + 2 \cos n\pi)^{\frac{1}{2}} \cos \left( \frac{2\pi t}{T} - \tan^{-1} \frac{\sin n\pi}{1 + \cos n\pi} \right)\cdot$$ Now the intensity of light for a vibration of any given period is proportional to the mean energy of the vibratory motion, and this can be shown to be proportional to the square of the amplitude. Substituting in the expression for the amplitude the value of $$n$$ and squaring, we have $$A^2 = a^2 \left( 2 + 2 \cos \frac{2bx}{c\lambda} \pi \right)$$ in which $$A^2$$ is proportional to the intensity of the illumination at distances $$x$$ from $$s.$$ When $$\frac{2bx}{c\lambda}\pi = 0,$$ its cosine is 1, and $$A^2$$ is a maximum and equal to $$4A^2 .$$ As $$x$$ increases $$A^2$$ diminishes, until $$\frac{2bx}{c\lambda}\pi = \pi,$$ in which case $$A^2 = 0.$$ $$A^2$$ then increases until it becomes again a maximum, when $$\frac{2bx}{c\lambda}\pi = 2\pi.$$ In short, if $$AB$$ (Fig. 119) represent the line $$mn$$ of Fig. 118, the ordinates to a sinuous curve like $$abc$$ will represent the intensities of the light along that line.