Low frequency propagation through random polydisperse assemblies of cylindrical or spherical poroelastic obstacles

The effective wavenumbers, moduli, and mass densities are found for polydisperse assemblies of poroelastic obstacles (considering fluid flow and solid deformation in the porous medium). The obstacles are infinite length cylinders and spheres. To achieve this, recent formulas for the effective wavenumbers, given by Linton and Martin [SIAM J. Appl. Math. 66(5), 1649-1668 (2006)] and Norris and Conoir [J. Acoust. Soc. Am. 129(1), 104-113 (2011)] in the dilute monodisperse case (obstacles of identical sizes in a fluid matrix), have been modified. Given the uncertainty in predicting the distribution in size of the obstacles, three quite different probability density functions are studied and compared: uniform, Schulz, and lognormal. Specifically, the Rayleigh approximation (low frequency regime) is considered, in which the wavelengths can be assumed very large compared to the size of the obstacles. Within this limit, simplified formulas are provided for the concentrations depending on the parameter characterizing the size dispersion.