Why does a low pitch propeller have better acceleration/take-off and climb characteristics than a high pitch propeller?

Why does a propeller with a low pitch have better acceleration/take-off and climb characteristics vs. a high pitch propeller?

And conversely, why is a propeller with a high pitch not as good during a climb, but is good for a cruise phase?

I am looking for a „physics and maths” explanation, I like the bicycle and car gears analogy and they are really intuitive, but they do not explain why does it works like that.

• One way to look at it if you ride a bicycle: how hard is it to get started in high gear, or to climb a hill in high gear? How is cruising on a flat stretch in a low gear vs. a high gear? It is very similar for propeller pitch in airplanes. A single speed bike is much like an airplane with a fixed-pitch propeller -- a compromise. Commented Jun 30, 2021 at 16:39
• So, in reality, you want the prop to take the same "bite" of air (at its optimal AOA), based on relative wind created by forward airspeed (higher in cruise) and rpm (lower in cruise). Commented Jul 2, 2021 at 10:30

Power

A lower pitch allows the blade to spin faster ($$\omega$$) for the same torque ($$T$$). The relationship between power ($$P$$) and speed and torque is $$P = T \omega$$, so this means that for the same $$T$$ we get more power out of the prop when it spins at a faster $$\omega$$. More power means a higher climb rate.

Stall

At low airspeed, significant sections of a high-pitch propeller will be stalled. This translates into lost thrust, and explains why a plane with a climb prop has a shorter ground roll than the equivalent with a cruise prop.

Lower tip speed

Efficiency goes down at high tip speeds, and basically falls off a cliff when the tip speeds reach the speed of sound. So a higher pitch allows for creating the same thrust at a lower tip speed, and thus gives higher efficiency.

Engine RPM

A secondary effect is that a prop which is spinning more slowly typically leads to a more efficient engine, and reduces wear. So while this isn't directly related to your question about the prop being more efficient, it is a very important effect in the overall system efficiency.