The diffraction of a scalar plane wave by an aperture in an infinite plane screen is examined theoretically. The wave function at an arbitrary point of space is expressed in terms of the discontinuity in its normal derivative at the screen, where the boundary condition is that of vanishing wave function. An integral equation for the discontinuity in normal derivative (or the residual function which measures its deviation from the simple distribution appropriate to a completely infinite screen) is the result of applying the boundary condition to the space wave function. Utilizing the integral equation (whose solution is generally unobtainable), the diffracted spherical wave amplitude at large distances from the aperture is cast into a form which is stationary with respect to small variations (relative to the correct values) of the residual functions arising from a pair of incident waves. An homogeneous expression for the amplitude is exhibited wherein the part independent of the residual functions defines a Kirchoff approximation. The connection with another stationary form of the amplitude, involving a pair of aperture wave functions, is examined. A variational expression for the plane wave transmission cross section of the aperture is based on the amplitude observed in the direction of incidence. The variational formulation is applied for a wave incident normally on a circular aperture. By comparison with the exact results available for this problem, it appears that use of simple residual functions in the variational formulation yields approximate, yet accurate expressions for the diffracted amplitude and transmission cross section over a wide range of frequencies.

• Received 13 January 1949

DOI:https://doi.org/10.1103/PhysRev.75.1423