The oblique detonation wave (ODW) triggered by a semi-infinite cowl or wedge has been widely studied, but the effects of finite-length cowls require further clarification to enhance their applicability. Based on two-dimensional inviscid Euler equations and a two-step induction–reaction model, two structures with smooth and abrupt transitions are simulated, and their interactions with the cowl-induced expansion wave are investigated. The expansion waves located downstream and far away from the initiation zone do not affect the macroscopic structures of the ODW. However, when the expansion waves move upstream, the structures evolve into a decoupled shock and a reactive front. Alongside findings from previous studies, these results indicate that the initiation mechanism, rather than the transition type, determines the near-quenching evolution of the structure. Moreover, given the same parameter settings for the chemical and inflow gas dynamics, these numerical results show that the abrupt transition evolves into a smooth transition under an expansion wave disturbance. This demonstrates that the local wave interaction plays a key role in the ODW structures.