Two point source interference

When waves are produced in phase from two point sources, they create and interference pattern. There is constructive interference along the perpendicular bisector of the line segment joining the sources because the waves arrive at the points on that line at the same time. This area is called an antinode. A little further out, there is the first nodal line, where interference is completely destructive. The pattern continues: antinodes and nodal lines alternate in a hyperbolic shape. The number of nodal lines increases when wavelength decreases or when the distance separating the sources decreases.

On any point on a nodal line, there is an extra distance that the wave from one source must travel compared to the wave from the other source. This extra distance, called the path length difference, is given by

p.d.=(n12)λ\displaystyle \text{p.d.} = \left ( n - \frac{1}{2} \right ) \lambda,

where n\displaystyle n represents the ordinal nodal line (first, second, third, etc., from the central antinode).

The angle θn\displaystyle \theta_{n} represents the angle between the central antinode and the nth nodal line. If we make some assumptions (incorrect assumptions, but the error they contribute is insignificant), we have

sinθn=p.d.d=(n12)λd\displaystyle \sin{\theta}_{n} = \frac{\text{p.d.}}{d} = \left ( n - \frac{1}{2} \right ) \frac{\lambda}{d}.

By considering similar triangles in this setup, we get a third equation:

λ=xndL(n12)\displaystyle \lambda = \frac{x_{n} d}{L \left ( n - \frac{1}{2} \right )}.