Slower props are usually larger diameter though aren't they? That to a large extent would offset the tipspeed gains by using a lower RPM. Or do they run slower props at a steeper root angle (more 'bite')?
E.g., turbo prop (source https://www.quora.com/How-fast-is-the-tip-of-a-turboprop-bla...)
I double checked this math and got 474mph.
Meanwhile a Piper Arrow (4-seater piston, pretty comparable numbers to most Lycoming/Continental singles) takes off at 2700 rpm and has a 74” propeller, for a tip speed of 594mph. You can use a 3-blade propeller to reduce the diameter and cut noise somewhat (among other benefits).
Both of these are just the tip speed relative to the plane, I suppose you’d need to factor the airspeed too and use Pythagorean theorem but I’m no aerospace engineer.
As you noted, this is possible because the constant-speed prop on the turboprop allows it to take larger bites of air.
The Arrow needs that rpm to produce full takeoff power. However after takeoff it can be adjusted to a lower speed with minimal loss in power (e.g., 2500 rpm greatly reduces noise while losing 7-8% power). To reduce RPM like that in a fixed pitch you’d also need to reduce manifold pressure, causing more power loss.
If it had even a modest reduction drive, you could run the engine at a reasonable power setting while maintaining a more efficient tip speed but the prop and engine engineers had to meet in the middle resulting in a highly compromised solution.
Sure there have been improvements in modern engines, but these primarily target efficiency across power outputs whereas (as sibling noted) planes tend to have stable power requirements.
Not sure what your point is about constant speed or gearing because I already said that.
Is it? I've always thought that while GA engines are in many ways stuck in the 1940'ies, they are actually quite efficient. Wikipedia seems to back that up at https://en.wikipedia.org/wiki/Brake-specific_fuel_consumptio... , with a couple of Lycoming engines producing BSFC numbers in the same ballpark as a modern car engine or Rotax.
Edit: A dyno graph from an late 1990's Corvette C5 engine (5.7L) shows about 170HP @ 2700 RPM. That's at the wheels, so probably close-ish to 195HP at the crank: https://cimg5.ibsrv.net/gimg/www.corvetteforum.com-vbulletin...