This morning, just after 7AM in Austin, I was buzzed by a plane. It sounded like it was very low. The pitch change from the doppler effect was very fast, and the sound volume change was also very fast. I listened to the radio for a while, half way expecting to hear a report of a crashed plane.

I didn’t see it, but this got me to wondering about how I could tell how close it was given only a sound recording. I figured:

Let:

S = speed of sound

I = speed of plane toward observer (-I is then speed of plane away after it passes)

P0 = the pitch of the propeller’s sound in Hz in the plane’s reference frame

PI = the pitch of the propeller in Hz from the listener’s reference frame.

I think PI = S * PO / (S - I)

If R = the ratio of the incoming pitch to the departing pitch (both of which would be asymptotes assuming no collision), then

R = (S + I) / (S - I)

Solving for I we get:

I = (R - 1) * S / (R + 1)

The volume of the sound is inversely proportional to the square of the distance (sound pressure is inversely proportional to distance). Graphing the volume over time, the sound of a passing plane looks like a bell, with the peak being where (but not when, because the sound takes a while to travel to the observer) the plane is at closest proximity.

I think what needs to be done is to compare how much time elapsed from where this graph changes from 1/2 the peak height to the peak height. That establishes a time frame and a distance ratio. We then compare this with the known speed of the plane in the first step to get the distance, but I haven’t worked that part out, yet. I’m not sure what the proper units would be on the sound recordings volume. Would that correspond to sound pressure or sound intensity?