Oh-oh, the uber-Geek engineer inside me just kicked in. (Sorry, it's genetic - my Dad was the same way, as is my Son)
The speed of sound at the sea level standard atmosphere is generally listed as 761 mph.
761 mph = 4,018,080 ft/hr (761 x 5280 ft per mile) = 48,216,960 in/hr (4,018,080 x 12 in per ft) = 803,616 in/min (48,216,960 / 60 min per hr).
Propeller tip speed (inches/minute) = Pi * propeller diameter (inches) * RPM or
RPM = Propeller tip speed / (Pi * diameter) or
diameter = Propeller tip speed / (Pi * RPM)
So, for the original question "what prop diameter is needed to keep from going supersonic at 3000 RPM", we get:
diameter = 803,616 (speed of sound in inches per minute) / (3.1416 x 3000 RPM) = 85 inches. At this RPM, the tip would *just* be at Mach 1.
For a 78 in diameter prop:
RPM = 803,616 / (3.1416 x 78) = 3279 RPM. Again, at this RPM, the prop tip would "just" be at Mach 1.
This is pure theoretical rotational math and does not take into effect any local airflow effects across the propeller.
Also, at the end of WWII, the faster fighters discovered compressibility effects as the airfoil approached the speed of sound which created much more drag than expected and therefore required much more power (or rate of descent) to maintain the speed.
I suspect that this may have been what Johathan observed in his case, even though, mathematically, his prop tips seem to be only about 90% of the speed of sound.
Dave York
DISCLAIMER: Any of my math may be wrong and should be considered suspect until verified. Do not use for navigational purposes.
Trivia side note: A side effect of the compressibility factor noted by the WWII planes was that there was often a control reversal or total loss of control as the planes approached the speed of sound. In Microsoft's Combat Flight Simulator 3, the P-38 aircraft will show this behavior, just like the original plane. If you dive on your opponent and allow your speed to exceed about 500 MPH, pitch control gradually goes away and leaves you in a terminal velocity, vertical dive with no way to pull out. This dive continues until you finally create a smoking crater in the ground. I got nailed several times before I realized what was actually happening.