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I understand that playing a square wave from speakers cannot produce a PERFECTLY sharp division between compression and rarefaction. But it's sharp enough to sound distinctly different from a sine wave. As it travels, does the wave "smoosh out" even further as it travels, changing from a square wave into something closer to a sine wave?

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The rectangular wave has a frequency representation (sinc function if I am not mistaken), and if there is no dispersion, these frequency components all attenuate similarly, and so the original shape, too, is unchanged. Now, obviously in some cases you do notice that some frequencies behave differently from others (you will find, by playing a sine-wave in your room, that some frequencies resonate much more strongly than others), and as such, in general, the square wave will be distorted. Not a smooshing, though: When I listen to music from far away, it does not sound like a single sine wave –  alarge Aug 7 at 21:46
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Try plotting it: a square wave is formed by the infinite sum of waves with integral multiples of the fundamental frequency. The more frequencies you add, the more the sine shape resembles a square. Since air is dispersive, the square will be deformed, especially for the higher frequencies. Higher frequencies affect the edges of the waveform, therefore the edges will soften as you lose supersonic frequencies. –  auxsvr Aug 7 at 22:15

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Yes. Higher frequencies are attenuated more over distance than lower frequencies are, which has a rounding effect on the square wave as the upper harmonics are reduced.

Reference
Do low frequency sounds really carry longer distances?

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Correct. As well as attenuation, there is also a certain amount of dispersion. –  Floris Aug 7 at 22:34

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