No. Nature doesn't use maths. Humans use maths to predict what nature does. Maths is just a collection of patterns that humans have noticed about the world and about patterns in patterns. And we give these patterns symbols and we write them down, and we call that maths.

But nothing, ever, in nature is due to our human definitions.

I completely agree, and I wasn't arguing otherwise. But modes are a human definition. What we do in physics is to look at the string and try to model it by mathematics. Of all the waves that can be described on the string by our model, some have a special property, and we call those modes. That property is that the waves form a

**complete, normal set**. That's math-speak to say that we can mathematically describe, with our model, any wave that exists on the string by adding up a bunch of these modes, and also that any one mode cannot be written in terms of any other modes.

The real power is what I've mentioned twice above: that you can mathematically describe a wave on a string in terms of just a list of the coefficients of these modes, i.e. a list of numbers rather than a very complicated equation.

The fact that the fundamental must be a sine wave is because of the relationship between the energy modes and the x/y dimensions of something spinning around a circle. It's the same pattern. That's why it's sinusoidal. There's probably other ways to see it as well.

I'm not sure what you're getting at here, but the fundamental mode is a sine wave because all modes are sine waves for the string. There aren't really other ways of getting at this, since modes are a mathematical definition within our model, and you only get them by enforcing completeness and normality.

Your argument that you just have to solve the wave equation doesn't really solve anything, since ANY waveform can travel obeying the wave equation. Square, triangular, anything. You need to show why sinusoids are so common.

Actually it does solve everything. Modes are a particular wave within our mathematical model, so it makes sense we have to solve the equation on which we base our model. As I mentioned, these more complex waves don't meet the requirements of what we call modes, but within our model, we can mathematically express them as a sum of modes.