Kirchhoof's formula, its vector analogue, and other field equivalence theorems

1. lntroduction One of the purposes of this symposium is to dissuss some of the theoretical difficulties involved in the solution of electromagnetic problems. For this reason I believe that the treatment of diffraction problems is an appropriate topic. In particular, the problem of radiation from a horn (acoustic or electromagnetic) has been treated approximately with the aid of one of several formulas suggested by Huygens' physical idea about wave propagation. These formulas are not equally powerful, do not always give equally good approximations, and do not inspire the same a pm'ori confidence in the results. Of course, if no approximations were made in the formulas, it would not matter which formula was used since then the result would always be exact. However, approximations are unavoidable except when the answer is already known and when there is no need for any formula. To understand how Huygens' physical assumption that the conditions at the front of a wave determine the subsequent wave motion suggests various formulas, let us consider a source first in an infinite homogeneous medium and then inside a perfectly rigid horn. In the first case the wavefronts are closed surfaces surrounding the source. In the second case, the wavefronts are open surfaces sliding along the walls of the horn until they reach the aperture. Eventually , the wavefronts will become closed surfaces surrounding the horn as well as the source. Kirchhoff derived an explicit formula for the field outside a closed surface (8) in terms of the wave function and its normal derivative on (8) on the assumption that the source is inside (8) and that the medium outside (S) is homogeneous. In the case of the horn Kirchhoff's surface of integration must enclose the horn as well as the source since the horn introduces a discon-tinuity in the medium. This closed surface may be chosen to consist of an " aperture surface " (8.) together with the exterior surface of the horn. In the case of electromagnetic waves the present writer proved another theorem [l, 21 (the " Induction Theorem ") in which the surface of integration is solely the aperture surface (8J. An obvious analogue of this theorem for scalar waves