Why does waving a stick on fire sound like waving a flag?

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I lit a candle and when I waved the long match to kill the flame, it sounded like waving a tiny flag.

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When you wave a flag, the sound you’re hearing is displaced air. You’re pushing the air molecules out of the way.

Waving a small hot thing around does the same thing. Displaces the air molecules, because heat makes them move faster. You’ve also got the added bonus of the fire sucking oxygen from the surroundings, which also causes molecules to move, rushing in to replace the ones consumed by the fire.