A combustor liner has a plurality of resonators formed thereon. Each resonator has a radially outer wall and at least one side wall. The outer wall can be free of holes. Each resonator has an inner cavity defined between the outer wall, the at least one side wall and the outer peripheral surface of the liner. The at least one side wall of each resonator surrounds a subset of a plurality of holes that extend substantially radially through the liner. A plurality of cooling passages extends generally longitudinally within the liner. Each cooling passage has an inlet in fluid communication with the exterior of the liner and an outlet in fluid communication with the inner cavity of a respective one of the resonators. A coolant, such as compressor air, can enter and flow along the cooling passages to thereby cool the liner and purge the inner cavity of the resonator.