It doesn't have to say the area of effect is stationary. It says the 20-radius zone is centered on a point you choose within range. A point in space you choose. If the point was then allowed to move it would have to then say in the description "If this point is a person or object capable of moving, the point and the zone of silence moves with it."

"It doesn't say it can't move" isn't a legal justification for thinking it does, because there are millions of other things it doesn't say it can't do.