So I believe the trail we're seeing in this old very popular picture,
is condensation caused by the propeller tip, compressing the
air, which since they're at sea, is probably pretty well laiden
with moisture, and then the flow expands around the tip of the
blade causing sudden expansion of the pressurized moist air
and causing it to condense into the white vapor we see. Also
the spiral is probably caused by the wind over the flight deck
in the interval of time the photog had the shutter open.
Anyway, to take a crack at the math:
propeller circumference (in ft) X rotations/sec = speed of sound at sea level (in ft/sec)
For Hamilton Standard F6F propeller: Diameter = 13ft 1in = 13.083 ft
Speed of sound at sea level = 1116.15 ft/sec
Plugging into formula above:
(pi x Diameter) ft X rot/sec = 1116.15 ft/sec
(3.14159 x 13.083 ft) X rot/sec = 1116.15 ft/sec
41.10 ft X rot/sec = 1116.15 ft/sec
rot/sec = 1116.15 ft/sec/41.10 ft
rot/sec = 27.16/sec
From: "Principles Of Aircraft Propulsion Machinery"
1949, Israel Katz
"... serious propeller losses are encountered as propeller-tip
speeds approach the sonic range. Reduction gears afford mechanical
means for accomodating high crackshaft speeds while limiting
linear propeller-tip speeds to about 1000 fps; reduction gears
currently (1949) have crankshaft/propeller speed ratios ranging
between 1.3/1 and 4/1. Since powerful engines require correspondingly
large propellers, there is a trend toward use of higher crankshaft
and lower propeller speeds in large engines; reduction ratios of
5/1 may well be employed in anticipated powerplants."
And a second source, the famous:
"Aircraft Propeller Design"
1930, Fred E. Weick, B.S.
(I might add, also mentioned by Kelly Johnson in his book)
"the aerodynamic force on an airfoil is a function of V/c, where
c is the velocity of sound in air (ie, what we call today
Mach Number), or the rate of travel of any compression wave.
While this compressibility function has been found to be unimportant
in connection with airplane wings, which even in the fastest
cases travel at velocities far below that of sound, the tip sections of
propellers on modern aircraft engines often approach and in some cases
exceed the velocity of sound. In these cases in which the velocity
approaches or exceeds that of sound, the effect of V/c (or Mach No)
is of vital importance, especially in regard to reduction of efficiency."
So even in 1930, their was an intentional effort to keep propeller tip
speeds from going supersonic.