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Big turbine engines produce very loud, ear damaging noise. Little turbine engines produce very high pitched, uncomfortable noises. Wouldn't it be great if this noise could be eliminated?
The noise is generated primarily by the turbine compressors and turbine expanders -- very little of it is due
to other causes (eg turbulence). On each blade of a turbine compressor, there's high pressure air being pushed in front of the leading edge, and low pressure air on the trailing edge. Similarly, on each blade of a turbine expander, there's high pressure air pushing on the trailing edge and low pressure air in front of the leading edge.
As these circularly arranged high and low pressure regions move off the turbine blades and strike the stator blades, they change from circular high and low pressures (which might cancel each other out, if they were sent down a long enough tube) to a more conventional sound wave (flat waves of sound). As the air moves through the next successive turbine rotor, more sound is generated.
What if the spacing, and relative angles, of the turbine air compressors and expanders were selected to create sound waves 180 degrees out of phase with the sounds made by the preceding and following rotors? In other words, what if each rotor's sound canceled out some of the sounds of the other rotors?
Of course, a turbine engine would probably only be acoustically tunable for a certain range of speeds, so we can't use this to make a completely silent airplane engine, but there are many fixed speed engine tasks that this would be wonderful for.
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||Very close to the blades the solution to the flow-equations might be clear, but further down the road, it is not possible anymore to predict, with any exactness what will be going on at a particular point in time.
||So the design would have to be such that pressure differentials actively create their counterparts, which is only marginally baked in digital active sound suppression, but you are proposing an analog version. Might take a tad to compute. (And who has a tad?)
||I can see it being doable for the compressor stage, but after combustion not so sure.