Number of propeller blades

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You remember correctly. It started with a 2-blade and went to a 3-blade.

IIRC, the first AUstralian restoration (or was it a reproduction?) also had a 2-blade.

But high horsepower 2-bladed props were deadly. The blades would bend forward at high power and, sooner or later, would shed a blade due to fatigue. That's what killed Jim Wright in his replica Hughes H-1 racer ... he shed a blade over very inhospitable terrain.

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Shortround6:
Is it the total surface area that matters? Like, can a three-blade prop with a wide chord (like the TA 152H) and a four-blade with a slimmer chord (like the Mustang) provide the same performance?

Just like wing area, you need enough blade area to prevent stall. The propeller with the thinner chord, assuming the same diameter, will have a higher aspect ratio on the blades, and lower induced losses as a result, so, no, it can't provide the same performance, all else being equal. There are many potential reasons why the Germans preferred three blades, not excluding prejudice on the part of VDM or the Luftwaffe. Without actually looking through the design documentation (alas, I don't read German), there is no way to know the reasoning. My first suspicion is that the primary reason was that the three-bladed propellers avoided some resonances in the engine operating range that the four-bladed propeller couldn't. A second is that the German propeller makers could not reliably produce higher-aspect ratio blades for their fighter aircraft, so stuck with three, vs four, blades.
 
About tip speed –would sweeping the prop at an angle greater than the supersonic shock wave angle allow sonic tip speeds? Some turboprops look like they're doing this.
 
Read in somewhere that the Germans were researching this by the war's end.
 
About tip speed –would sweeping the prop at an angle greater than the supersonic shock wave angle allow sonic tip speeds? Some turboprops look like they're doing this.

They are. For some serious sweep, take a look at propfans. There, the idea is because the vector sum of aircraft speed and tip speed is great enough so that transonic effects are significant over a large chunk of the blade.
 
Hi Tante Ju,

It takes some period of time to disable the firing mechanism in a machine gun and some period of time to enable it as well. The 3-blade prop must interrupt the MG only 3 times per revolution while the 4-blade prop must interrupt 4 times per revolution. So there is more firing time for a 3-bladed prop than for a 4-bladed prop.
 

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