a few things to wrap this up for me, and thank you for your civil answer to a not so civil comment.
1st: I should have made this a 'yes and' rather than a 'no but' comment. It's a well written article with nice illustrations, and you put in the time to do so and share it with the world. Thank you for that!
2nd: Obviously you're right that it's possible to calculate the distance to plane by observing when it passes a certain point and then measuring the time until the sound hits you from that very point.
3rd: My smartassery still stands that "How far behind a plane is its noise?" is a misleading way to frame it - the noise is not behind the plane at all, it's actually significantly ahead of it. But (!), again, I should have made that a 'yes, and' instead, to add a facet to your nice article.
Cheers, and hope to read from you again!
My point is that your one instant in time is completely arbitrary. You do not know where the position is, and you do not know when the sound was emanated. I.e., you cannot calculate anything.
Your math would work iff you observe a discrete event where you can tie sound and light - engine blow-up, for example. In all other cases, it means nothing.
Again - why don't you hear the sound of the aircraft taking off if speed of sound is the only effect?
What one would formally do is represent these as points of an imaginary unit sphere centered on the observer. There is an angular distance between those two points (one located using our hearing, another located using eyesight), which is what is observed as "plane not being where you hear it".
While one can't calculate much from these observations, it doesn't take much to get there: all you need is another similar observation (eg. at the time when the plane sounds as if it's where you first observed it) along with measuring the time between those measurements (say, counting seconds if you don't have a stopwatch, since we are very imprecise anyway). The time it took the airplane sound to "move" between these two points is how long it took for sound to reach you from the first observed plane position. And now you've got the distance from you (speed of sound multiplied by time) at that point.
With a few assumptions which are applicable to experiments like this (eg. constant airplane velocity, and climb rate being either small or constant), you can also establish where the plane was when the original sound was emitted, what's the speed it travels at, etc.
Error bars in all the measurements would be pretty huge if only observing with eyes and ears and by counting, but the fact that the effect is easily noticed even with all those "measuring errors" is fun to consider.
You can also move to much better instruments to measure all of these (including direction the sound is coming from).
One is our eyesight, which is pretty precise (error depending on the speed of light) — of course, I am not getting into all the potential ways our eyesight can fail us.
Another is our stereoscopic sound positioning based on having two ears, which is ultimately much less precise, but can still point in a general direction of where the sound you are hearing right now is coming from. If you've got both functioning ears, you can usually tell if someone is calling you from the front, either side or from the back. Some people are better at it than others, though.
Sound you are hearing right now (a discreet event) has been emitted at a particular point by an airplane a number of seconds/minutes away — this is what identifies the "discreet" point and requires no remarkable event like an explosion, and you can do the math based on that and the two "instruments" you'd use. I am not sure why would it matter that airplane produces the sound continuously for discussing this discreet moment in time when a particular sound was produced? (unless you are going for there being no vibration of air possible without time passing, which is needlessly picky).
I may have been assuming that it was clear in the article, but to help show the effect, the diagrams show only the sound emanating from the plane at one instant in time. In reality, the aircraft is continuously moving, and there's a continuously changing "where is the sound coming from" vector. The main idea is that this "where is the sound coming from vector", if you will, may be behind the plane's light vector (which we can just say is the plane's position) if you're some distance away, leading to this oddity.
> the sound may be able to reach you much much earlier than the plane
I completely agree! I didn't mean to say that a plane's sound is always behind it — it very much depends on the position of the observer. If a plane is flying in any other direction than perfectly perpendicular, the math and the effect will be different.
For "If the plane was moving very slowly, it wouldn’t outpace its sound by much.", I meant in the sense that we, as observers, are perceiving the sound. More that the plane's position vector wouldn't outpace the "where is the sound coming from vector" by as much (a smaller X in the diagram, if you will), leading to "where is the plane" being closer to "where is the plane's noise". Going faster than the speed of sound leads to all sorts of very interesting questions, but I don't believe it would affect this in the simple case we're looking at.