Plane-Wave Representations for Scalar Wave Fields

Abstract

The theory of the representation of wave fields in terms of superpositions of monochromatic plane waves is presented for fields satisfying the inhomogeneous scalar wave equation. The discussion includes expansions of the type originally used by E. T. Whittaker involving only homogeneous plane waves, and of the type introduced by H. Weyl involving both homogeneous and inhomogeneous plane waves. Expressions for the plane-wave amplitudes for both types of representations are obtained in terms of the source function, and precise conditions under which each expansion is valid are given. It is shown that when both types of expansions are valid, the superposition of inhomogeneous plane waves in the Weyl-type representation is equal to the superposition of the homogeneous plane waves that propagate into a specific half-space in the Whittaker-type representation. It is shown also that in restricted space-time regions only a certain subset of the plane waves in the Whittaker-type expansion contribute to the field. This result leads to a simple expression for the field valid at large distances from the source.