By carefully tuning the thickness of a compliant thin film placed within an acoustic cavity, we achieve coherent control of the cavity's acoustic resonances, analogous to the operation of an optical etalon. This technique is demonstrated using a supported membrane oscillator in which multiple high-frequency harmonic resonances are simultaneously optoexcited by an ultrafast laser. Theoretical and computational methods are used to analyze the selective strengthening or suppression of these resonances by constructive or destructive interference.