Analytical analysis of slow and fast pressure waves in a two-dimensional cellular solid with fluid-filled cells.

Wave propagation in cellular and porous media is widely studied due to its abundance in nature and industrial applications. Biot's theory for open-cell media predicts the existence of two simultaneous pressure waves, distinguished by its velocity. A fast wave travels through the solid matrix, whereas a much slower wave is carried by fluid channels. In closed-cell materials, the slow wave disappears due to a lack of a continuous fluid path. However, recent finite element (FE) simulations done by the authors of this paper also predict the presence of slow pressure waves in saturated closed-cell materials. The nature of the slow wave is not clear. In this paper, an equivalent unit cell of a medium with square cells is proposed to permit an analytical description of the dynamics of such a material. A simplified FE model suggests that the fluid-structure interaction can be fully captured using a wavenumber-dependent spring support of the vibrating cell walls. Using this approach, the pressure wave behavior can be calculated with high accuracy, but with less numerical effort. Finally, Rayleigh's energy method is used to investigate the coexistence of two waves with different velocities.