Sine waves, by a huge margin, are the most important waveform in electronics – we measure a circuit's frequency response with sine waves and represent all other signals through sine waves, with the help of the Fourier transform. However, Fourier transform on itself does not make sine waves special – after all, there are other ways to decompose a signal into a bunch of orthogonal functions (wavelet transform, for instance).
So there must be a fundamental physical reason for sine waves being so important. I would imagine that this reason is the fact that the electromagnetic wave equation (which can be readily derived from Maxwell's equations) is a second-order diff eq, so a sinusoid is its solution – that's why sine waves do not disperse in transmission lines and that's why a "frequency component" that has a particular propagation velocity in a medium, is a sinusoid.
Is the reasoning above correct? In a fantastical world where the equation of electric signal propagation was, for example, a third-order diff eq, would sine waves be as important as in our reality (I know it's a bit ridiculous to ask what would happen if the fundamental physical laws were different, but still)?
Best Answer
If you take the Maxwell equations, they can be combined to derive the wave equation for light. Since the orthogonal basis for the solution space of the wave equation is the complex exponentials, we can see that sine waves are intimately related to the Maxwell equations here. When wave propagation is defined by the Maxwell equations / the wave equation, they always must be able to expressed as linear combinations of complex exponential. This is really the reason why sine waves don't disperse. Any other wave forms contain superpositions of the sine waves that travel at different speeds, causing dispersion.
What's said here isn't much different from what you've already put together, but it is definitely a fundamental physical reason for sine wave popularity.
Regarding the fantastical third world: I think that sine waves would still be important because the solutions for linear third order differential equations are still exponentials. That said, it would also be important to consider (real) exponentials, which are sort of the other extremely fundamental function that is seen everywhere. Depending on the third order differential equation, there could also be exponential solutions, which could be a very important part of the fantastical world.