# Will you be able to hear a supersonic aircraft flying away from you?

I'm wondering if you could be able to hear the sound emitted by an aircraft say traveling at Mach 2 away from you.

I wondered if it might shift the emitted frequencies beyond the audible range. Would you be able to hear it?

• XKCD's "What If?" has a nice article about a supersonic Mach 2 sound source: what-if.xkcd.com/37 – Sanchises Nov 11 '19 at 10:14
• Ideally, avoid asking a question where the title asks it one way (will you be able to hear it) and the text asks it another (would it shift...beyond the audible range) as answers starting with (say) "Yes" are ambiguous. Your only answer so far starts with "yes," so I'd adjust the text to fit the title. (I've suggested an edit that does that.) – T.J. Crowder Nov 11 '19 at 18:33
• A tangential question I once heard was, when you're flying in a supersonic plane do you hear the plane? The answer was yes, in a way. You'll hear the wind noise, the vibration of the engines and other materials, etc. – CramerTV Nov 12 '19 at 0:08

Yes, actually you can only hear a supersonic aircraft after it has passed over you and is now flying away from you since it is moving faster than the sound moving towards you. The sound waves will still propagate in all directions and will eventually reach you:

The frequency will be shifted according to the Doppler formula:

$$f = \frac{c \pm v_r}{c \pm v_s} f_0$$

Here, $$v_r$$ is the the speed of the receiver ($$v_r = 0$$ for you standing still) and $$v_s$$ is the speed of the sender. It is added when the sender is moving away from you. In your case of $$v_s = 2c$$ you would get

$$f = \frac{c}{c + 2 c} f_0 = \frac{1}{3} f_0$$

The audible range for a typical human is about 20Hz to 20kHz. So you would still be able to hear any $$f_0$$ greater than about 60Hz after the Doppler shift. The following image shows the frequencies emitted by a Boeing 747 as measured from the ground:

There are a lot of contributions from the aircraft above 60Hz, which should make it clearly audible. Of course an aircraft flying at Mach 2 will be different than a 747, but the general spectrum should not change too much.

Note: the formula above does not work while the aircraft is moving towards you because the resulting frequency would become negative. This is because sound emitted at a later time reaches you before the sound emitted at an earlier time because the aircraft has moved more towards you than the sound:

• I think it's awesome that the formular gets negative frequencies, and that acutally means something (sound is inverse) – Christian Nov 11 '19 at 15:31
• @Christian Negative frequencies aren't really a thing, but one can interpret it that way :) – Bianfable Nov 11 '19 at 16:14
• @Christian: A better interpretation is that the result/value of the function isn't physically meaningful over that part of its domain, and therefore you need to special-case it. Sometimes physics does work like that, e.g. optics where a negative image distance can be physically meaningful: a lens or mirror forming a virtual (behind the mirror) vs. real (projected onto a screen) image. – Peter Cordes Nov 11 '19 at 20:58
• But here notice that the formula has a singularity (f * c / 0) at v_s = -c: the formula breaks down and we need to consider what happens to real non-ideal fluids as sound and shock waves "pile up" around an object moving exactly at the speed of sound. You don't just get all the sound energy building up forever delivered in one huge burst. For an object moving faster than sound, it might actually sort of invert the wave-form, if you ignore all the added noise from moving faster than the speed of sound in real air. (shock waves are loud) – Peter Cordes Nov 11 '19 at 20:58
• The speed of sound at 20 degrees temperature through the air medium is 0.343 km/s. At Mach 2 the supersonic fighter jet will travel faster than the speed of sound. Though the jet is traveling at such a speed and if you are on the way of the direction of sound then eventually after a couple of seconds later you will hear the sound of the jet that just passed you. – mvr950 Nov 11 '19 at 21:40