Yes, more power is being sent from the amplifier to the speakers.
The sound is made by wiggling the speaker cones using an electromagnet. The farther the speaker cone moves, the louder the sound (this is different than how *fast* the cone moves back and forth, that’s what controls pitch). Moving the cone back and forth farther while maintaining frequency requires pushing & pulling it harder, so the electromagnet has to work harder, so it needs more power. The amplifier does this by increasing the voltage of the signal going to the speakers. When you turn up the volume, you’re turning up the gain (increase) of the amplifier.
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