An Exact Algebraic Evaluation of Path-Length Difference for Two-Source Interference

Abstract

When studying wave interference, one often wants to know the difference in path length for two waves arriving at a common point P but coming from adjacent sources. For example, in many contexts interference maxima occur where this path-length difference is an integer multiple of the wavelength. The standard approximation for the path-length difference is path-lengthdifference≈dsinθ, where d is the distance between the sources and θ is the angle between the perpendicular bisector of d and the line connecting P to the midpoint of d. A common derivation of Eq. (1) begins with the seemingly paradoxical approximation that two paths that meet at a common point can be treated as parallel. In this paper we present an alternative derivation that first finds a simple, exact expression for the path-length difference, valid even when the paths clearly are not parallel. We then show the circumstances under which Eq. (1) is a useful approximation to the exact expression and finally determine an upper limit to the error inherent in using Eq. (1) in place of the exact expression. No math is required beyond the Pythagorean theorem and simple algebra.