You can't hear a meteor as it burns up in the atmosphere, but you can bounce radar off of it and convert that signal into sound. The U.S. Air Force Space Surveillance Radar in Texas tried this during the Perseid meteor shower, and the resulting recording sounds appropriately alien.
What you're about to hear in the following clip are two distinct noises caused by the brief but spectacular demise of a small space rock as it incinerates itself in our upper atmosphere. The first whooshing noise comes from the radar reflecting off of the passing meteor itself as it turns to vapor. The eerie high-pitched moan is caused by the trail of ionized gas that's all that remains of the meteor, which gradually fades back in to the atmosphere:

The picture above, incidentally, comes from astronaut Ron Garan, who go to spend two missions on board the ISS, lucky sod. Through some superhuman feat of premonition (or a long exposure) he managed to capture a shooting star from above, during the Perseid meteor shower back in August. Wild, wild stuff.

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