Would you, as a stationary observer, hear a sound in reverse when the source of the sound travels with twice the speed of sound? Of course, he wouldn't hear anything at all before the airplane passed by and the sound reaches him, but what about after? Does it reverse due to the Doppler effect?
Acoustics – Understanding Sounds of Supersonic Objects
acousticsdoppler effectspeed
Related Solutions
why do textbooks never mention this?
Because in order to travel at supersonic speeds, human beings must be enclosed in a rigid metal tube of some sort. Also, these metal tubes they ride in at those speeds generally tend to be insulated against noise from the outside.
As for trying to place some sort of microphone outside said metal tube, the propulsion system would risk drowning out any atmospherically transmitted noise (i.e. noise can be transmitted through the body of the structure).
Now, if you run the math, it still doesn't work quite like hearing it backwards (see the comment by eudoxos). Although you would encounter the soundwaves in "reverse" order, the shockwaves around you would disrupt anything around you as to make the notion of noise from them irrelevant.
(I'm considering the speakers are emitting some kind of music or something nonperiodic, the situation gets a bit boring if you consider a uniform source)
It basically means Alice hears nothing. Atleast, not until Bob crosses (at which time your equation is no longer valid, the $-$ in the denominator becomes a $+$). She hears a sonic boom as Bob crosses her, and then hears two sounds at once. The first sound is whatever is being played by Bob after he crosses her, at a frequency $\frac{f}{3}$. The second, more interesting sound, is that whatever sounds were emitted by Bob are heard backwards, at a frequency $f$ (This comes from the $-f$ you derived). So, if Bob was playing Mozart's Symphony 23, and switched to Coldplay's Yellow when he passed Alice, Alice hears: boom; Yellow at one-third the pitch and simultaneously Symphony 23 playing backwards. Would probably sound horrible ;-)
Why is this?
Remember, Bob's speed is greater than the speed of sound. So, wavefronts emitted by Bob now are much closer to Alice than the wavefronts emitted in the past :
Here, the moving dot is Bob, and assume Alice to be another dot to the right of Bob in his path.
The edge of the cone that you see being formed is the "sonic boom". It's a region of a rapid rise and fall of pressure (extremely high pressure). Right after it passes, you see two kinds of wavefronts passing Alice. The first is the "left sides of the circular wavefronts". These have been emitted after Bob passed, and are playhed normally, with a third of the frequency (Yellow in my example). The other kind is the right side of the circular wavefronts emitted before Bob passed Alice. As you can see, these are heard top-down, i.e., the ones emitted last are detected first.
For comparison, here is the same diagram if the relative speed was $<v_0$:
To summarize, the negative frequency just means that the sounds emitted at that time are heard "backwards" at a later time--"reflected" at the point in time when Bob crosses Alice.
BONUS: http://what-if.xkcd.com/37/
Best Answer
Yes, the sound can be reversed. Thanks to JiK, we have this animation (Python source code) of a supersonic jet moving forwards that can illuminate what is going on:
The red circle represents the first sound produced by the object, the blue circle the second sound produced by the object and the remaining (black) circles representing the sounds produced elsewhen along the simulation. Two things to look for here:
Obviously point #2 is key to OPs question.
However, sound intensity follows an inverse square law: $$ I\propto r^{-2} $$ which means that the larger circles are "quieter" than the smaller circles. Thus, the sounds that you would hear as backwards are muffled in comparison to the normally-directed sounds. So while there is reversed sounds, it would be very unlikely that you would actually distinguish them from the normal-directed sounds.
The image below is a still from a slightly modified version of JiK's code (added an
elif (t_i > 7)condition). This shows a bit more clearly the details of the above paragraph, that the normal sounds (in orange and received in the normal direction) would be louder than the reversed sounds (because smaller is louder).