Except sinusoids are special in that they are natural solutions to the Helmholtz wave equation. There's other problems too like square waves having infinite energy. This article might make sense to a mathematician or computer scientist but neglects the underlying physics of sound and waves.
Excellent point, lots and lots and lots and lots of physical objects are harmonic oscillators. That does have pretty fundamental grounding in physics.
I can think of lots of other places I'd use fourier analysis (at least qualitatively as with doing diffusion modeling in my head) but you're right that sinusoids are more physically "real" whereas being possible to represent in any basis set is more "valid" if that makes any sense.
Not quite sure what the right word is on this one, but I agree "real" kind of suggests real oscillators underlying the phenomena. Square waves are less physical because of discontinuities in both the signal and derivative; nature really doesn't care for discontinuities.
Frequency domain also makes the math really easy for linear, time-invariant operations, which (approximately) describe a lot of systems that exist in nature.
The Gibbs phenomenon, for example, falls out naturally from the IFT of a frequency response where all the frequencies above some cutoff are zero.
I'm curious how the square wave frequency domain would describe the Gibbs phenomenon -- I think you'd have harmonics of the fundamental square frequency showing up as if the system were nonlinear.
