Meaning of negative frequency of sound wave

Question

Suppose that Alice and Bob are both holding speakers emitting sound at a frequency $f$. Alice is stationary while Bob is moving towards Alice at twice the speed of sound.

In the case of Alice, if I forget for a minute that she is holding a speaker herself, and just think about the Doppler effect from Bob moving towards her, I get $$f_{\text{effective}}=f\frac{1}{1-\frac{2v_a}{v_a}}=-f$$

I do not know what "negative" frequencies mean in the context of sound waves, and so I do not really know how to describe what Alice will hear as Bob becomes very close to her.

I also do not know how the fact that she is emitting sound at a frequency $f$ will affect, if at all, what she hears.

And, from Bob's perspective, I get $f_{\text{effective}}=3f$, which is a bit more meaningful, but does not factor in the fact that Bob is emitting sound himself.

Any clarification is appreciated.

Answer 1

(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 Atleast, not until Bob crosses (at which time your equation is no longer valid, the $-$ in the denominator becomes a $+$).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 $<v_0$:

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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/