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)?