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A collection of works is presented in which the author explores mostly via computational modeling the solution of the PDE s governing wave motion in viscoelastic solids and in multi-phase porous materials. The aim: quantifying the dissipative effects on amplitudes, particle polarization, Q attenuation, dispersion, velocities, etc. Maxwell models are considered in the first section of the book. In the second the averaged motions of the solid frame and viscous fluids of porous materials are quantified using Biot, de la Cruz and Spanos and Sahay theories. A variety of finite difference and pseudospectral algorithms are used to discretizing the PDE s. Inquisitiveness bore fruits; the new findings include: 1-Biot fluid flows display vortexes; 2-Wavefronts can be traced as transient porosity oscillations; 3-Medium displays porosity relaxation; 4-Slow P-waves propagate tens of meters in finite viscosity materials; 5-Slow S-waves can propagate in hydrocarbon bearing rocks; 6-Waves can beelliptically polarized in a porous uniform medium; etc. Since this work deals primarily with waves the book can be useful in exploration and earthquake seismology, geotechnical, medical imaging, etc.