For a loose definition of "wave". All of the math behind information theory and sampling signals assumes waves are sinusoids. It also happens that waves in nature behave like (dampened) sinusoids. It's a completely natural way to model them mathematically when one has no a prior knowledge of the source, which is what the comment above you is pointing out.

To recognize and then reconstruct a sawtooth with no a priori knowledge, you need to sample much higher than the frequency of the sawtooth. You can compress said information quite well if you have not only wavelets, but sawtooths in your encoding. I am no audio expert but I don't think codecs exploit sawtooths (sawteeth?) for compression because they sound unnatural (because they are).

Note that even digitally you can't create a perfect sawtooth wave because there is a fundamental quantization of time in digital systems. It's a question of, again, how fast you can alter voltages, i.e. a frequency, so you end up generating a step-like function, inescapably. Yeah, sure, you can switch digital systems at MHz or GHz, but still.