Everyone's heard of a sinewave and most people (even those who haven't done trigonometry since high school) are aware that it's considered a "pure tone" in audio engineering. But what is it exactly? Why, in short, is this shape of wave so special?
A lot of readers probably remember "sine = opposite over hypotenuse" from year 8 trigonometry, and try to relate it to right triangles. That's not necessarily the best way to look at it.
Here's how I see it. We start at (0,0) and facing North (0,1). The sine and cosine functions both take an angle (let's work in degrees, like 360 degrees) as an input — we then turn that many degrees clockwise. They then output the coordinates (X and Y) of the point one unit in that direction.
So if we feed the number of 90 degrees into these functions, we turn to face East and then go one unit in that direction. We're now standing at (1,0).
Now, think about what happens if you continually turn on the spot. The X and Y coordinates will smoothly oscillate as the point in front of you moves around where you're standing.
This oscillation is not just smooth in terms of its change over time. It's also smooth in terms of its acceleration. A graph of the acceleration of an object following a sinewave will also be a sinewave.
This is a critical mathematical property. The only other functions that share this property are ones that never change. In math speak, the derivative of the sine function is another sine function (with a slight lag).
This is why sinewaves are considered pure tones. Almost every other function's derivative is a different shape to it. This means that anything (like a speaker cone) that changes in position over time according to such a function will experience resonances of sinewaves at other frequencies.
We now know that anything that moves or changes size with momentum will have resonances that follow sine patterns. Since this covers almost everything that can generate or record sound, we also know that we can understand sound in terms of combinations of sine waves.
So that's what's so special about sinewaves in particular. Their origin is in geometry, but the way they behave over time is unique and special. The fact that their acceleration is exactly the same as their change in position over time is why they're the fundamental building block of audio.