"Bill Daniels" wrote in message news:5_- It seems to me that the gear analogy is spot on. A variable pitch
prop has EXACTLY the same function as the gearbox on a car.

Not quite. Gears don't have preferred operating conditions, props do.

The engine has its preferred RPM and torque for optimum efficiency and the
prop blades have their optimum angle of attack. If the engine/prop
combination results in the prop operating at a higher (or lower) angle of
attack than optimum to absorb the torque of the engine (Prop governor
increases pitch to hold RPM setting.) then the combination operates below
optimum conditions.

Under some conditions, it would make sense to introduce a third variable
i.e. a gearbox between the engine and prop, to allow both the engine and
prop to operate at peak efficiency. This was the reason that two-speed
grearsets were installed in the nose case of some large radials. This, in
turn, allowed the propeller designer to optimize his prop blades for a
single AOA, thus gaining still more efficiency.

The problem, simply stated was this: How does a heavily loaded, long-range
bomber haul itself off a short runway and climb to cruise altitude and then
shift to highly efficient, long-range cruise. The answer was just emerging
from the labs as the world shifted to turbines. The flight engineer would
shift his engines into a "hole gear" by selecting a cam profile and engine
timing optimized for the low gear that would let the engines scream at high
RPM and pump massive HP into props set for maximum acceleration and climb.
Once in cruise, the engineer would shift his engines back to low RPM, high
efficiency settings.

First time I've ever heard of gear-shifted props in certified
engines. Which engines were these? I know that many radials (and other
engine layouts) used reduction gearing in the case nose to allow the
engine to run faster and produce more HP while keeping the prop within
safe limits, and that there were two-speed geared superchargers on
many of these engines, but two-speed props?
Jim Bede used a snowmobile-type propshaft drive in the early
BD-5s but abandoned it as unworkable. It still required a relatively
tiny prop to keep the tip speeds subsonic.
As far as the propeller pitch angles go, the constant speed prop
improves takeoff performance by more than just letting engine RPM
reach redline to produce max HP. It reduces the angle of attack so
that more of the prop is unstalled and producing thrust in the static
condition, improving acceleration and shortening takeoff distance. The
inboard sections of a fixed-pitch prop blade have a large angle so
that they still produce thrust in faster forward flight even though
they don't travel the circumferential distance that blade areas near
the tips do, but the large angle means a stalled blade, or at least a
really turbulent flow, at low forward speeds. A gear-shifted
fixed-pitch prop will still have those problems.

Dan