What follows is a series of simplified suppositions which may explain the derivation of this formula. I look forward to reading the comments of some of the accomplished physics and engineering types on this forum!
1) The velocity at which an object will aquaplane is dependant upon the area in contact with the water surface and the downward force applied upon the object. (area x pressure = force applied.) The vertical component of the hydrodynamic force vector must exceed this value in order to lift the object to the surface or support it there.
2) Pressure generated at the point of contact is proportional to the square of the velocity the object moves through or across the water.
3) Each pneumatic tire has a design footprint area and design load carrying capability. The design inflation pressure of the tire is likely of a certain ratio (perhaps even equal?) to the pressure across the footprint area and is sufficient to support the tire structure at that load under the environmental conditions the tire is designed to operate under.
If actual tire loading determines aquaplaning speed and inflation pressure is in proportion to design tire loading, then design inflation pressure would also have a relationship to aquaplaning speed! Using the tire inflation pressure is certainly a more convenient method for pilots to calculate aproximate aquaplaning speed than the alternative.
Well, that's one theory anyway. Let's hear yours. Fire away!
Best regards,
Westhawk