Well, that theroem kinda presupposes that the function is a periodic function (because if it isn't, it does not make sense to speak of it in terms of cycles-per-second). That means that your example with the second degree equation fails - the theorem says that the points are to be placed 1/2B seconds apart, whith means that they are 1) equally spaced, and 2) goes on to infinity. Now, most sounds does not go on to infinity (except Justin Bieber songs) so there are some edge conditions, but those are for all intents and purposes inconsequential. We can approximate "goes on to infinity" with "goes on until the sound ends".
But, yes, periodic. A periodic sound has some other properties, and one of those is that we can see it in two different ways. One is in the time domain, where we see it as basically a graph of air pressure over time - the classic "waveform display". The other way is to see it in the frequency domain: Every periodic sound can also be represented as a sum of the separate frequency components that the sound contains. That is, a graph of frequencies and their intensities. That is what you get in a sound spectrum.
(some graphs here:
http://en.wikipedia.org/wiki/Frequency_spectrum#Spectrum_analysis )
The only sound that has no more components than its base frequency is a pure sine wave. A pure sine contains only its base component and has no overtones, and that means that we can see that spectrum diagram as a chart of the sines that make up the sound. In fact, you can perfectly reconstruct a periodic sound by adding together sine waves of the frequencies and intensities given in such a diagram (you can also do it in a myriad other ways, for instance by adding together a special class of square waves called Walsh functions, but that means you'd have to do a lot of math to convert between the two).
Now, and this is important: We can shift between these. Every periodic sound can be fully represented by its time-airpressure graph, or by its frequency-intensity graph. We can go from one to another (with some limitations that I can detail later, if you like), and that means that when Nyquist above says that it contains no frequencies above B cps, he's really talking about it in the frequency domain. The actual waveform might look completely chaotic, but
if it is a periodic waveform, then it
is possible to analyze it in the frequency domain, and
if in that frequency analysis there are no components above B cps, then it
will be fully represented by giving its ordinates (that is, by sampling it) at a series of points spaced 1/2B seconds apart (that is, at the frequency double-B ). Once again, that means that the equally spaced points go on forever, or at least for as long as the sound goes on.
This is, then, dependent on a few things. The first is that the sound really must contain no frequency components above that limit. That is ascertained by an input filter that removes everything above the limit. It also is dependent on the playback mechanism not trying to play back any frequencies above that limit, either, and that is ascertained by an output filter that removes everything above that limit. That last bit is the magic thing. Intuitively, we think that if we have a perfect top-to-bottom swing between every sample, then our sine-at-the-limit will look like a square wave - Top-bottom-top-bottom with no slopes inbetween. Right. And that is correct, it does look like that, but a square wave
is not a sine (well, duh, that is the point) and that means that a square contains other frequency components. A square wave at the frequency B will contain strong components at 2B, 4B, 6B, 8B, and so on to infinity (or at least a long way). That means, for our sampled-sine-that-looks-like-a-square at 20kHz, that it will have components at 40kHz, 80kHz, 120kHz, and so on. But! Our output filter removes all our components above the playback frequency, which means that those 2B, 4B, 6B, 8B components disappear. Which means that the only component of that very square looking wave form that actually gets played back is the B component - Which is just one pure component, and therefore is a sine.
There are, of course, problems here, namely that if our sampling frequency is exactly double that of our sampled signal, that might mean that we capture not the top and the bottom samples (and thus get the correct intensity), but instead get the positive and negative zero crossing every time, resulting in the illusion that our signal has a zero intensity. Or, really, any other particular phase shift of the signal would get another incorrect intensity. That is a theoretical problem, though. The possibility that a natural sound will have that exact frequency and hold that unwaveringly is slim to say the least. But yes, that means that a signal on the exact half of the sampling frequency will sometimes not be reproduced at the correct intensity (unless I miss something important here - My mind is a bit slow after midnight).
Does this help? Or did I misunderstand the question?
