A square wave has an intense amount of harmonics, producing a bright sound. But a sine wave has none, and is a dull sound.
Why does the square wave create harmonics and not a sine wave?
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Sine waves (strictly speaking sine/cosine quadrature pairs) are the principal components of many signals, the main reason being that they are the Eigenfunctions of linear timeshift-invariant systems which many systems in reality are essentially or to first order approximation.
That means that a sine wave, filtered through such an LTI system, will result in another sine wave of equal frequency but possibly different phase and amplitude.
For that reason, sine wave audio signals suck for directional hearing since all you get to hear, regardless of attenuation or reverb, is another sine wave (though with possibly different amplitude and phase on either ear). You get much better locateability by superimposing sine waves at numerous frequencies with fixed phase relations. Square waves are such a superposition.
You can both represent sine waves as a superposition of square waves and vice versa, but it is only the sine waves that, for each particular frequency, are maintaining the same shape and frequency when passed through LTI systems.
So it makes more sense to represent signals as superpositions of a fundamental sine wave and harmonic multiples than in any other manner.
Now mechanical oscillators like strings and air columns (in flutes) have modes they resonate with, basically Eigenstates of the oscillator that will recur periodically, a certain configuration of mechanical energy. More often than not those modes result in first order approximation in proper harmonics, oscillations with an integral multiple frequency of that of the fundamental oscillation. But for things like thick strings or sounding bars, the first order approximation is not good enough and the modes are noticeably detuned from proper harmonics. This is typically experienced with the thick strings from a piano, an effect known as "disharmonicity" and somewhat accounted for by "stretch tuning".