I was reading about how at how acoustic beats work.
If we combine two waves with frequencies $f_1$ and $f_2$ and unit amplitude, their combination is
$$
\begin{align}
A
&= \cos\left(2\pi f_1x\right) + \cos\left(2\pi f_2x\right) \\[10px]
&= 2\cos\left(2\pi \, \frac{f_1-f_2}{2}\,x\right)\cos\left(2\pi \, \frac{f_1+f_2}{2} \, x\right) \,.
\end{align}
$$
According to "Beat (acoustics)", Wikipedia:
Because the human ear is not sensitive to the phase of a sound, only its amplitude or intensity, only the magnitude of the envelope is heard.
So obviously the beat frequency is twice the envelope (since you're squaring it) and you get $$f_{\text{beat}} = f_1-f_2$$and not half that.
Now consider a regular cosine wave $A = \cos{\left(2\pi f_T\right)}$ with frequency $f_T$. Taking the magnitude (as Wikipedia says, i.e. by squaring $A$) gives you an audible frequency of $2f_T$… so do people hear frequencies as twice what they are in their amplitude wave?
EDIT:
The answer is we DO perceive twice the frequency — a sound wave that we define as having a frequency f will stimulate our ears with twice that frequency.
This frequency f is just a convenient name we give to the waves our machines make. This doesn't bother anyone as people can't hear in slow motion and i.e. count 200 'ticks' per second when playing a 100 Hz wave.
Best Answer
Humans hear the correct perceptive signal for a sound wave of that frequency.
We really can't say much more than that. The psychology of acoustics are very complicated and could fill volumes.
It's closer to say we have cells which act resonant at a specific frequency. Our brain identifies which cells are resonating at any point in time, and constructs the signal from that. Our brains receive information that cell A or cell B is signalling. The association between those neural signals and frequencies is a learned response that we pick up early on, as an infant or perhaps even in the womb.