The compressibility effect on opposition drag control is studied via direct numerical simulation of turbulent channel flows at a bulk Reynolds number Reb=3000 for three different bulk Mach numbers: Mb=0.3, 0.8, and 1.5. For all Mb, the drag reduction (DR) has a similar trend as that of the strictly incompressible case; namely, DR first increases and then decreases with the sensing plane location yd+. With increasing Mb, DR slightly decreases at small yd+ but increases at large yd+. Consequently, yd+ for achieving maximum drag reduction (DRmax) shifts to larger values, namely, from yd+=12.5 for Mb=0.3 to 20 for Mb=1.5, consistent with the outward shift of the peaks of Reynolds stresses at higher Mb. By rescaling the sensing plane with semilocal units, a better collapse of DR is achieved among different Mb, particularly for small yd∗. The optimal sensing plane is found to be yd∗≈15 with DRmax≈23%. Interestingly, for large yd+ cases, a resonance buffer layer characterized by a streamwise periodic array of spanwise-coherent rollers is established, one of the main reasons for the deterioration of drag reduction performance. This layer of hydroacoustic instability resonance results from the intense interaction of wall-normal wave propagation with the background mean shear. Space-time correlation of wall-normal velocity reveals near-wall organized spanwise structures with a well-defined streamwise wavelength λx, decreasing with increasing Mb.