Topological limitations of effective-medium approximations in fluid-solid systems having two longitudinal-acoustic modes

We consider the acoustic properties of two-component composites. We show that in order for the system to have two distinct nondispersive longitudinal modes, one component must be a fluid, the other must be a solid, and each component must be part of an infinite, percolating cluster. Although the Biot theory is able to describe these modes, we show that effective-medium theories, within the single-site approximation, cannot predict them precisely because some or all of the constituents are taken as unconnected inclusions. In the special case in which the solid constituent occurs as isolated particles within the fluid (a suspension) there is only one longitudinal mode and the Biot theory and the average-$t$-matrix approximation are in essential agreement with each other and with experiments in predicting its speed.