An acoustic-absorber system incorporates a plurality of acoustic-baffle assemblies in cooperation with an acoustic cavity. Each acoustic-baffle assembly incorporates a plurality of nested fluted cups, wherein at least one fluted cup of a first type having at least one orifice at at least one radially-inwardly-extending grooved portion is interleaved with at least one fluted cup of a second type having at least one orifice at at least one radially-outwardly-extending ridged portion. Externally-generated soundwaves entering either from the inside of an innermost fluted cup, or from the outside of an outermost fluted cup, travel through the at least one orifice thereof, and along a circuitous path along the gaps between adjacent cups and through at least one orifice of each of other fluted cups from one gap to another, until finally propagating into the acoustic cavity through at least one orifice of the outermost, or innermost, fluted cup for subsequent additional attenuation by destructive interference from phase cancellation within the cavity.