(1) - This current thread was triggered by a remark from David Pike in a previous thread " as follows: "... upon first hearing an approaching high-flying aircraft is that it’s no good looking in the direction of the sound to see the aircraft. You need to look about five miles ahead of the sound".
From this initial remark, 2 comments were subsequently added:
(2) : Here, I replied : " ...when I start hearing a subsonic civilian aircraft at its cruising altitude, it is almost approaching its closest point from me, i.e. it is almost already reaching its highest elevation above the horizon for me."
(3) : Shortly thereafter Frank reminded us of an interesting rule of thumb : "... the angle between the sound of the aircraft and the visual aircraft is simple for an interesting practical case: angle = 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".
This problem is easily solved in the (generally slant/oblique) plane containing both the Observer and the A/C track where only 3 parameters fully define the situation:
- Maximum hearing distance (here assumed to range somewhere between 10 and 20 km). And:
- Closest Point of Approach (CPA) of A/C track to Observer. Such distance from Observer to A/C CPA is most generally a slant distance. And finally:
- A/C Mach Number, assumed to stay constant at all altitudes. This saves a bit more complex calculations.
The 2 examples given in this post are both underlined in yellow in the enclosed attachment which also includes a number of usual "average" Mach Values, and geometric configurations.
From this attachment, we can verify that all the statements listed here-above are generally correct.
As per (1) here-above, the distance between sound and aircraft is given by the distance "AB". It is equal to the "Maximum hearing range" multiplied by the Mach Number. In the examples given here it ranges from 7 to 16 km, but in "average situations" it lies between 8 and 12 Km, hence 5 miles is a decent order of magnitude to start up with.
As per (2) here-above where I was referring to a max hearing range of 20 km, then we can see that for the usual Long-haulers (Flying Mach Numbers between M0.80 and 0.85 and cruising between F350 and F450), then Angle β is close from 0°. This indicates that you start hearing the A/C when it is close from its CPA, i.e. close from being "abeam" of your position.
Finally, (3) is also fully verified in most cases since when the A/C is abeam of the Observer, i.e. close from its CPA, then as indicated in the attachment : "When A/C is close from abeam position - i.e. │β│˂12° - , then α ⩰ asin (Mach)".
I think it covers it all.
Best Regards to all,
Antoine M. "Kermit" Couëtte