Abstract:
The flexural response of an infinite homogeneous plate attached to a random uniform distribution of single degree of freedom undamped oscillators constituting a ``fuzzy substructure'' is studied theoretically using methods from diagrammatic multiple scattering theory. A Dyson equation is posed for the ensemble average Green's function, and a Bethe--Salpeter equation is posed for the average of the square of the Green's function. The Dyson equation is solved within three different approximations: the ``first-order smoothing approximation'' (equivalent to that of Pierce, Sparrow, and Russell) the ``average T-matrix (or Foldy) approximation,'' and the self-consistent ``coherent potential approximation.'' The mean Green's function is found to be equal to the Greens' function proposed by Pierce, Sparrow, and Russell, if the individual sprung masses are not large. The Bethe--Salpeter equation is solved within the latter approximation and in the limit that the length and time scales of the diffuse field are long compared to wavelengths and periods. In the further limit that these length and time scales are also long compared to mean-free paths and mean-free times, it is found that the Bethe-Salpeter equation reduces to a diffusion equation. [Work supported by ONR.]