The stiffness of soil is an important parameter that has implications on soil-structure interaction, on the response to earthquake motion and on the response of soils to dynamic loadings. Stiffness reduces and behaves plastically at medium to high strains; however, at small-strain the stiffness has been observed to be a constant value and elastic. Small-strain stiffness governs the soil-structure interaction during construction projects and site response during dynamic loading due to earthquakes and man-made operations. Quantifying stiffness, in particular shear stiffness, at small-strain is difficult due to the effect of sample fabric on the values measured and the resolution of the testing equipment that is available. Wave propagation has been used to measure the stiffness of samples by propagating waves in different directions and in different planes. This thesis aims to examine the propagation of stress waves through a granular medium. Samples were created using the numerical discrete element method (DEM) in two- and three-dimensions. Waves, created by a point source, were propagated through the samples and this propagation was measured using micromechanical data. The speed of the propagating wave was assessed using existing techniques and novel methods developed during the research. The effect of macro-scale parameters, such as sample boundary conditions, and the effect of micro-scale parameters, such as interparticle contact laws, on sample stiffness were examined. Randomly packed samples were created with a quantifiable fabric tensor, measured using the contact force network. Wave propagation in different directions was examined to quantify the effect of inherent anisotropy on the sample stiffness. Samples were confined at anisotropic confining pressures to isolate the effect of induced anisotropy on the sample stiffness. Wave propagation results were compared with the results of small amplitude stress probes for a number of simulations and with experimental work carried out in the University of Bristol.