The insight and engineering rule of thumb is that
we need to observe at least half of a sine wave to determine its amplitude and frequency.
Of course, as a rule of thumb it is a drastic cutoff, but it is quite intuitive and close to "real".
Now take the sum of two sinusoids with near frequencies
$$
\sin \left( {2\pi \left( {f - \Delta f/2} \right)t} \right) + \sin \left( {2\pi \left( {f + \Delta f/2} \right)t} \right) = 2\cos \left( {2\pi \,\Delta f/2\,t} \right)\sin \left( {2\pi \,f\,t} \right)
$$
we obtain a sine of frequency $f$ amplitude-modulated by a (co)sine of frequency $\Delta f/2$.
(a well known phenomenon of which I do not remember the name in english).
According to the given rule, we need to "see" at least half of the modulating wave to be able
and reconstruct the signal, and thus the difference in frequency. If the duration , i.e. "persistance" of the
signals, is much less we will only see one sinusoid with the average frequency and double amplitude.
Calling $T$ the duration of the observation window we shall have
$$
{1 \over 2}{1 \over {\Delta f/2}} < T\quad \Rightarrow \quad 1 < T\Delta f
$$
Passing to a more rigorous mathematical analysis, consider that
the Box function (a window of duration $T$)
$$
R(t,T) = \left\{ {\matrix{
1 & { - T/2 \le t \le T/2} \cr
0 & {otherwise} \cr
} } \right. = U(t + T/2) - U(t - T/2)
$$
where $U$ is step function, has a frequency spectrum (bilateral Fourier Transform) given by
$$
G(f,T) = T{\rm sinc}(f\,T) = {{{\rm sin}(\pi f\,T)} \over {\pi f}}
$$
A sinusoid of duration $T$ is the product of the Box function for
a sine function.
The spectrum will be the convolution of a Dirac at frequency $f$ (apart that at $-f$)
with the $G(f,T) =T{\rm sinc}(f\,T) $ function , i.e. the same centered at $f$.
Then, approximating the $G(f,T)$ to a box function with a cutoff at $f \pm 1/(2T)$
leads to that we can distinguish between frequencies with a separation
of at least $1/T$, i.e. to the formula given above.
Best Answer
Your concept of "wave" is a little vague to me. But:
A) A "wave with constant frequency but varying amplitude" is (informally) what one have in AM : amplitude modulation. It can be shown that if the amplitud varies "slowly" (as compared with the main period), the signal can be expressed as a (in general, complicated) combination of sinusoids of frequencies near the main frequency.
In the wikipedia article it's shown the simplest case, $ y(t)=[1 + M \cdot \cos(\omega_m t + \phi)]\cdot \sin(\omega_c t)$ Here we have a sinuosid of "central frequency" $\omega_c$ and its amplitude varies by a sinusoid of frequency $\omega_m$ ("modulation frequency"). It's easy to show (oops) that this signal can be expressed as the sum of three sinusoids of frequencies $\omega_c$, $\omega_c+\omega_m$ and $\omega_c -\omega_m$.
B) The sum of two sinusoids of same frequency and distinct amplitude (and perhaps phase) results in another sinusoid of the same frequency - this is easy, and it's fundamental property of sinusoids.