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The extraction of the response of physical systems from field fluctuations is an area undergoing rapid growth. It is of relevance because it makes it possible to obtain the response of the system from passive field fluctuations instead of from an active point source. The impulse response is characterized by the Green's function G(t). The existing theory leads to the retrieval of G(t)- G(-t), the difference of Green's function and its time reversed counterpart. We show that the theory for Green's function extraction can be extended to many problems to yield both the difference and the sum of G(t) and G(-t). The importance of this derivation is that it extends the type of sources of field fluctuations that can be used for Green's function retrieval. The new formalism also opens up the possibility to extract the Green's function from field fluctuations in static systems. We show how this can be done for potential field problems and for direct current problems in conducting media, and illustrate this with a numerical example. For diffusive fields, the new formalism provides the possibility to extract the Green's function from field fluctuations excited either by injection sources or by current sources; previously this could be done only with injection sources. We show that with the new theory, the Green's function of acoustic waves can be retrieved from field fluctuations excited by body forces instead of from injection sources when the attenuation is constant. The Green's function extraction formulated here corresponds, for acoustic waves, to a Lagrangian formulation rather than the Hamiltonian (energy) principles that were used previously.

Lagrangian Green's function extraction, with applications to potential fields, diffusion and acoustic waves