The rotating detonation engine is increasingly favored as a viable pressure gain combustion technology for both propulsion and power generation applications. Practical designs involve the discrete injection of fuel and air, which then partially mix to produce the reactive mixture that is processed by a continuously moving detonation wave within the detonation chamber. One of the fundamental challenges in the successful operation of rotating detonation engines is the design of a robust fuel-oxidizer injection system. To minimize pressure losses while ensuring efficient mixing that promotes stable detonation, the injection has to satisfy many different constraints. In this work, the impact of discrete injection on wave propagation is studied in an effort to understand the impact of fuel stratification on detonation stability. To this end, the structure of the detonation wave within a linearized rotating detonation engine with fully premixed hydrogen-air and hydrogen-oxygen injection at various equivalence ratios is analyzed. The direct numerical simulation of the linearized model detonation engine is compared with experimental results, and wave behavior is correlated to the wave structure and flow turbulence. The flow properties and chemical composition of the gases across the detonation wave are studied to examine the characteristics of the reaction zone.