The sampling frequency determines the limit of audio frequencies that can be reproduced digitally. One of the most important rules of sampling is called the Nyquist Theorem, which states that the highest frequency which can be accurately represented is less than one-half of the sampling rate. So, if we want a full 20 kHz audio bandwidth, we must sample at least twice that fast, i.e. over 40 kHz. If we don't, bad things happen. Here's our example sine wave again:
The dashed vertical lines are sample intervals, and the blue dots are the crossing points - the actual samples taken by the conversion process. The sampling rate here is below the Nyquist frequency, so when we reconstruct the waveform we see the problem quite readily:
It's not pretty, and it's nothing like the input. This undesirable product is called aliasing and shows up as unmusical tones in incorrectly digitized material. Because of this, A/D converters must use lowpass filtering to remove all signals above the Nyquist frequency. Of course, it also means that in order to get high-fidelity sound, we have to take a lot of snapshots.
