Wave propagation is a fundamental property of all physical systems. The wave speed is directly related to the compressibility of the system and determines the rate at which local disturbances are propagated into the bulk of the material. The wave propagation characteristics of conventional forms of matter are well understood and well documented. In contrast, waves in granular materials are more complex due to the heterogeneous nature of these systems. The key element of the mechanics of a granular system is the force chain. It is along these preferentially stressed chains of particles that waves are transmitted. These nonlinear chains are heavily dependent on the geometry of the bed and are prone to rearrangement even by the slightest of forces. Results from both experiments and simulations on wave propagation in granular materials are presented in the current study. The experiments measure the pressures at two points within the granular bed that result from the motion of a piston at one end of the bed. The simulations are a two-dimensional version of the experiments and use a discrete, soft-particle method to detect the wave at both the output of the simulated bed and at any point within it. In addition to examining wave propagation in a granular bed at rest, simulations and experiments are also performed for a granular bed undergoing agitation perpendicular to the direction of the wave input. Imposed agitation increases the granular temperature of the bed and allows for the exploration of the effect of granular state changes on the wave propagation characteristics. Such information may provide a means to diagnose the state of a flowing granular material. Measurements of the wave speed and attenuation in the bed reveal the unique properties of waves in granular systems that result from the nonlinearity of the bed and the heterogeneity of the force chains. Sinusoidal waves demonstrate the nondispersive nature of a granular bed and show the transient effects of force chain rearrangement. Pulsed waves display a semi-permanent shape qualitatively similar to predictions from nonlinear wave theory. In an agitated granular bed, measurements of the wave characteristics were found to be possible even in the presence of significant agitation. The prevailing confining pressure, which changes throughout the agitation cycle, was determined to be the system parameter that correlates best with changes to the wave speed.