A variational principle for the scattered wave

A Schwinger-type variational principle is presented for the scattered field in the case of scalar wave scattering with an arbitrary field incident on an object of arbitrary shape with homogeneous Dirichlet boundary conditions. The result is variationally invariant at field points ranging from the surface of the scatterer to the farfield and is an important extension of the usual Schwinger variational principle for the scattering amplitude, which is a farfield quantity. Also, a generic procedure, physically motivated by the general principles of boundary conditions and shadowing, is presented for constructing simple trial functions to approximate the fields. The variational principle and the trial function design are tested for the special case of a spherical scatterer and accurate answers are found over the entire frequency range.