We read time and time again, how more efficient slower airfoils such as helicopter rotors, props, and sailplane wings are compared with rapidly rotating fan blades or faster (still subsonic wings). We also read where some people consider compression a negligible factor below a certain speed, even to the point of considering airflow < 200 mph as "incompressible"!
Yet looking at this commonplace and ubiquitous phenomena (indeed, while your prop is putting out 600 lbs of thrust, your wing is creating 4x more lifting force), is there a mathematical explanation for this involving the ideal gas law: PV=nRT?
Can we visualize creating and maintaining a compressed "bow wave" ahead of a moving surface involves substantial amounts of energy even at lower speeds?
I have yet to fully understand why "it is better to move a lot of air a little than a little air a lot".