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Yes, sound is carried in waves of matter. If sound moved at light speed, the.matter must also. Interesting thought of the day, your heartbeat would explode your chest.
Sound waves are pressure waves, so it isn't matter that moves in only one direction. It's matter that moves back and forth. Therefore I assume the wave can propagate faster than the matter moves.
Fun fact:
When two objects collide, the maximum internal propagation speed of the shock wave is limited by the local speed of sound in the material itself, which is typically 2000-6000 meters per second (m/s).
In low earth orbit, satellites move faster than that, around 7000-7700 m/s. So when they collide (at up to twice that amount in closing velocity), the structure of the satellites is moving faster than the resulting internal shock wave. In one case, there was still a signal sent from a satellite a split second AFTER it had smashed through another satellite, because the resulting shock wave hadn't finished destroying it yet.
Source: I'm a space systems engineer, and one of my colleagues wrote his PhD thesis on this topic.
Holy shit this makes so much sense!