In this study the characteristics method is applied to wave propagation corresponding to earthquake intensities in models of unsaturated and saturated soils. Shear waves developed in linear, or in strain-softening materials, and pressure waves in viscoelastic materials are considered. One-dimensional transmissions in unsaturated layered systems are treated first, then the method is applied to two-dimensional simulations by the lattice work approach. The latter method may also be extended to treat three-dimensional problems. Pore pressure transients are investigated in a one-dimensional approach based on Darcy's law. By prescribing reductions of void ratios as a function of time, zones of potential liquefaction may be identified and their movement traced. By use of a lattice work representation, transient pore pressure within the upstream portion of an earth dam are investigated.