Hi Mike......

Great job starting this thread, and with the initial research.

Makes my brain hurt thinking about it. Reminds me of many years ago when I studied some of this.

The important thing that should be learned here by others (despite debate on the full details and physics) is that the prop does work, and that it is not just a flat piece of wood/plastic/other that pushes with the back side of the blade. And more importantly, the engine/prop/airframe work together as a system to create the performance of the "aircraft".

There are litterally thousdands of pages of actual prop test data in the old NACA and NASA archives. Most of it is boring, and good examples of what "not" to do. But at least someone tried it once Yet a good deal of it was application specific.... and of good value for others seeking to proppel similar applications/speeds/loads.

As for the pipe and a shaft run..... the engine will accelerate until it finds another "node" it is comfortable with. Just like throttling up or down... you can see where the harmonics are sometimes.

The ultimate fun thing is an in-flight adjustable pipe (been tried.... its not easy) ..... or taking time to design one that will multi-stage (the QM engines sorta do this) jumping from one harmonic to a higher frequency harmonic when the engine unloads.