The phenomenon of wave dispersion in dry sand is studied both by purely analytical studies and by analytical-numerical experiments on the basis of gradient elastic and viscoelastic material models. These material models are employed in order to simulate the microstructural characteristics of dry sand. The analytical studies treat the material body as a one-dimensional (for the viscoelastic case) and three-dimensional (for the gradient elastic case) and for both material models provide explicit expressions for the velocity of propagation of harmonic compressional (P) and shear (S) waves. These velocities are found to be functions of frequency, i.e., dispersive. The analytical-numerical studies treat the material as a one-dimensional one and try to simulate P and S wave propagation along the axial direction of cylindrical dry sand specimens. Thus, a sinusoidal pulse with a specific frequency is applied at one end of the specimen and the response is determined at some other point by solving a transient dynamic boundary value problem with the aid of a numerical Laplace transform. This analyticalnumerical experiment is repeated for various frequencies. Thus, one determines the velocities of P and S waves as functions of frequency, thereby proving again that wave propagation in dry sand is dispersive.