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So a coarse prop, although it works well at low speeds, won't work so well at high speeds. These props are known as "climb props" because they help you climb faster, versus "cruise props" which are finer and help you cruise faster at the expense of climb performance.
I think you need to re-write this statement of yours. You have it backwards.
At low speeds eg take-off & climb a relatively fine pitch prop is needed. This is a
climb prop. At cruise & high speeds a relatively coarse pitch is more efficient. This is a
cruise prop.
As you quite correctly mentioned elsewhere, as speed increases an increasingly coarser pitch blade angle is needed to maintain the most efficient AofA.
If the blade angle of a fixed pitch prop is set so that it is most efficient at take off (a fine pitch prop aka. a climb prop) then it will become increasingly less efficient as the a/c accelerates in the cruise, resulting in reduced cruise performance.
If the blade angle of a fixed pitch prop is set so that it is most efficient at cruise (a coarse pitch prop aka. a cruise prop) then it will start at a low efficiency during takeoff becomeing increasingly more efficient as the a/c accelerates in the cruise. This results increased cruise performance at the expense of take-off performance.
Have a look at the Schneider Trophy racers of the '20s & '30s. Extremely coarse blade angles optimised for their eventual cruise speed (ie fast) but with the penalty of abysmal take-off performance.
In the imaginary ideal world, after take-off you would unbolt the very fine pitch take off prop after getting airborne, put on a slightly coarser climb prop for the climb, exchange that one for a coarser pitch prop for the cruise then back to the very fine pitch prop for the landing (in case maximum power & efficiency is suddenly needed during the flare).
Obviously this is impractical, hence the arrival of variable pitch props, commonly in the form of two speed props. Flick a switch & the blade angle changes from fine to coarse or vica versa. The blade is mostly efficient somewhere in the take-off & climb range of speeds, and also mostly efficient somewhere in the cruise range of speeds.
The next development was for a continously variable pitch prop. This replaced the two position type with a blade mechanism that allowed a blade setting anywhere between its finest & coarsest positions.
Great, but every speed or RPM change meant that the efficiency changed a bit unless the blade angle was reset.
Add a Constant Speed Unit to this VPP & it will automatically adjust the blade angle to maintain a designated RPM, relieving the crew of this task.