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I recently did an experiment where I tried to measure the properties of an acoustic wave by using a microphone and speaker in a tube. I was wondering how would changing the waveform to a square change its properties? I feel like the microphone wouldn't be able to pick it up because in a sine acoustic wave you can ignore backward moving wave but in a square wave you can't.

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You can hear a square wave. It just has a different timbre (more “buzzy”) than a sine wave. This is because it has many higher-frequency harmonics, while a sine wave has none.

You can decompose any periodic wave into a sum of sinusoidal waves with various frequencies which are integer multiples of a fundamental frequency. This is called Fourier analysis. The higher harmonics go a long way toward explaining why different musical instruments sound so different even when playing the same pitch.

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