NavList:

When you "first hear" an aircraft is a complex issue. I'll set that aside. But the angle between the sound of the aircraft and the visual aircraft is simple for an interesting practical case:
    angle = mach number.

That is, the angle as a pure ratio (or angle "in radians") is approximately equal to the speed of the aircraft relative to the speed of sound. It doesn't depend on distance. This is a good estimate for moderate mach numbers (very good up to 0.25, a good rough estimate up to 0.80). If we want the angle in degrees, just multiply by 57:
   angle[°] = 57° · (mach number).
This simple result works when the aircraft's path of flight is nearly perpendicular to the line of sight or equivalently when the aircraft is making its closest approach as it flies past the observer. You can draw triangles and try to come up with more intricate formulae, but then you end up dealing with more details of the flight path that are not usually available to a casual observer. For a quick BOTE (back of the envelope) estimate of angle from speed (or speed from angle), and given the generally poor ability to estimate the angular location of a sound, this is really all you need -- "the angle is the mach number". I have used this relationship as a regular "backyard" observing tool for decades.

Can you derive that result? Of course you can... :-)

Frank Reed