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Comment by hrydgard | Hacker News Reader

There's an infinite number of different sets of orthogonal bases that you can construct any signal from. Sine waves (with phase) are a quite natural and convenient choice though, for mathematical reasons (for example, the efficiency of the Fourier transform).

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Sure, but there may be a reason why non-sinusoidal sound waves cannot exist in practice. For example, if I try to generate a square wave by applying +5V and -5V to a speaker, the diaphragm does not move instantaneously so the resulting pressure wave is not discrete. However, a triangle waveform may be physically realizable. But I'm not sure about that either, because a speaker is ultimately an inductor and hence the current flowing through it is non-linear.

I did a deep dive into this, but this just seems to be the case about sinusoids. technically any well behaved periodic functions can be the basis function for all fourier-related stuff (there actually is a FT where a square wave is the basis function, but I can't seem to find it) but there seems to be something special about sin(x) and cos(x) that makes it more convenient to analyze our physical world. They seem to be the result (and the only result) for some class of very simple linear differential equations, which because of its simplicity, happens very often in nature. That's why FT uses sin(x) and cos(x).

>there actually is a FT where a square wave is the basis function, but I can't seem to find it

Haar transform? Technically that's a wavelet transform, but that's a distinction without tremendous difference.

i meant the walsh transform, but that's also neat

Any linear time-invariant system bounded in reality has a frequency response that will drop off in magnitude before reaching infinity. An ideal triangular wave is an infinite sum of sine harmonics because of the discontinuity in its derivative. More generally, the order of the derivative in which the discontinuity happens relates to decay rate of the harmonics. A square wave decays as sinc(f) whereas a triangular wave decays as sinc^2(f).

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