A tautological theory of diffraction

Three characteristic features of diffraction integrals, namely the quarter-cycle phase advance of the secondary sources, the proportionality of their strengths to the wave number, and the values of the obliquity factor at the limits of its domain, are explained by a one-dimensional integral identity that assumes nothing about the spatial variation of the wave function. If we subsequently assume uniform spherical primary waves, we can then relate the integrand to the wave function on the surface of integration. If that surface is a primary wavefront, the integrand becomes consistent with spherical secondary waves, but still contains an unknown obliquity factor and its derivative. If we try the standard obliquity factor, the integrand agrees exactly (not only in the high-frequency limit) with the Kirchhoff theory, and yields a near-source correction to the much-cited integrand of D.A.B. Miller (1991).