Effective mass density and stiffness derived from P-wave multiple scattering

A random distribution of cylindrical fibers or spherical particles immersed in a non-viscous fluid is considered. In this medium, propagation of P waves is subjected to multiple scattering. Recently, these effects were predicted analytically by Angel and Aristégui in the framework of Waterman and Truell. These new predictions concern the coherent waves that propagate inside and outside a layer of finite thickness, and are obtained without making any assumptions regarding the effective constitutive response of the medium. In this paper, it is shown that such assumptions should not be made, even though they may be approximately correct in some special cases. An analytical expression is obtained for the effective mass density, which is complex-valued and frequency-dependent. This expression coincides in the low-frequency and low scatterer-concentration limits with that corresponding to a weighted average of the masses of the two components. The effective stiffness is also obtained. It is complex-valued and frequency-dependent. It agrees with previous expressions in the respective limits of low scatterer-concentration and low frequency. Numerical results are presented for various values of frequency and scatterer concentration. The analytical expressions of this paper for mass density and stiffness are direct consequences of the approach proposed by Waterman and Truell.