Here’s why pipe organs seem to violate a rule of sound

[C C]

https://www.sciencenews.org/article/pipe...ration-air

EXCERPTS: In 1860, physicist Hermann von Helmholtz [...] devised an equation relating the wavelength of a pipe’s fundamental tone (the lowest frequency at which it resonates) to pipe length. Generally, the longer a pipe is, the lower its fundamental tone will be.

But the equation doesn’t work in practice. A pipe’s fundamental tone always sounds lower than the pipe’s length suggests it should according to Helmholtz’s formula. Fixing this problem requires adding an “end correction” to the equation [...] 0.6 times the radius of the pipe. Why this was, nobody could figure out.

A break in the case came in 2010 [... by studying...] how air moves through playing organ pipes using cigarette smoke. When an organ pipe sounds, a vortex indeed forms over the pipe’s rim, the team reported March 14...

[...] What’s more, this vortex is capped by a hemisphere of resonating air. This vibrating air cap ... effectively lengthens the organ pipe by the exact amount that must be tacked on to Helmholtz’s formula to explain the pipe’s fundamental tone... (MORE - missing details)

[Kornee]

The only 'mystery' is why it has taken so long to do a simple experimental probe - akin to the equivalent long done in the case of aerodynamics i.e. wind tunnel smoke trails.
Probably because e.g. airfoil design has far greater practical/commercial/military significance than solving an academic issue involving organ & allied musical instruments!
The simple 0.6xr empirical formula was perfectly adequate after all.