A Fast Method of Deriving the Kirchhoff Formula for Moving Surfaces

The Kirchhoff formula for a moving surface is very useful in many wave propagation problems, particularly in the prediction of noise from rotating machinery. Several publications in the last two decades have presented derivations of the Kirchhoff formula for moving surfaces in both time and frequency domains. Here we present a method originally developed by Farassat and Myers in time domain that is both simple and direct. It is based on generalized function theory and the useful concept of imbedding the problem in the unbounded three-dimensional space. We derive an inhomogeneous wave equation with the source terms that involve Dirac delta functions with their supports on the moving data surface. This wave equation is then solved using the simple free space Green's function of the wave equation resulting in the Kirchhoff formula. The algebraic manipulations are minimal and simple. We do not need the Green's theorem in four dimensions and there is no ambiguity in the interpretation of any terms in the final formulas. Furthermore, this method also gives the simplest derivation of the classical Kirchhoff formula which has a fairly lengthy derivation in physics and applied mathematics books. The Farassat-Myers method can be used easily in frequency domain.