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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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?
Robert Harvey
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