Aspects of the invention are directed to a system for improving the damping performance of an acoustic resonator. According to aspects of the invention, a resonator, such as a Helmholtz resonator, can be attached to a surface of a combustor component in a turbine engine. The combustor component includes a region in which a plurality of passages extend through the thickness of the component. The resonator is attached to the component so as to enclose at least some of the passages. The passages are in fluid communication with a cavity defined between the component surface and the resonator. Resonator performance is a function of the length of the passages in the component. According to aspects of the invention, resonator performance can be improved by reducing the length of the passages in the component by reducing the thickness of the component in a region that includes the plurality of passages.