On the emergence of the Green's function in the correlations of a diffuse field

TLDR: It is shown that a diffuse field is not devoid of phase information, but has a correlation function equal to the Green's function, and the cross-correlation between diffuse signals in two transducers is very nearly equal to an impulse applied to the other.

Abstract: A diffuse acoustic field is shown to have correlations equal to the Green’s function of the body. Simple plausibility arguments for this assertion are followed by a more detailed proof. A careful version of the statement is found to include caveats in regard to how diffuse the field truly is, the spectrum of the diffuse field, and the phase of the receivers. Ultrasonic laboratory tests confirm the assertion. The main features of the direct signal between two transducers are indeed recovered by cross correlating their responses to a diffuse field generated by a third transducer. The quality of the recovery improves with increased averaging and the use of multiple sources. Applications are discussed.