Auxiliary superfield method for statistical predictions of complex, structural acoustics systems: Saddle point approximation for the mean field

The auxiliary superfield approach is proposed as a method to obtain statistical predictions of the acoustic response of complex elastic structures. The potential advantage of the method is the full retention of interference and resonance effects associated with the degrees of freedom being averaged over. It is not known whether this approach leads to tractable problems for structural acoustics systems, however. We have applied the method to the idealized case of an infinite, thin plate with attached oscillators to obtain the mean Green's function. The oscillators are assumed to have an uncorrelated, Gaussian distributed mass and stiffness distribution representing a simple form of complex internal structure. Using the auxiliary superfield approach, the mean Green's functions are expressed exactly as a functional integral. For relatively small disorder, the integral may be estimated by a saddle point approximation which leads to coupled integral equations for effective mass and stiffness matrices that can be solved numerically for a given spatial distribution of the disorder. With the solutions for these matrices, one obtains a self-consistently determined, generalized fuzzy structure model. We give analytical solutions for the simple case of a uniform spatial distribution. The results are promising for the application of the method to more challenging geometries.