I believe, you’ve got some big misunderstanding in that. The wave is the variation in pressure. It doesn’t exert pressure beyond that. There is general air pressure, but that is always there, whether you’ve got a wave going or not.

Maybe it helps you understand it logically to think of how a speaker membrane moves. It does this motion:

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So, when it bulges to the right, it pushes air molecules closer together, which locally increases pressure on the right and continues to travel to the right, because the molecules push each other.

Then the membrane bulges to the left, which locally reduces pressure on the right and continues to travel to the right, because the molecules make room for each other, so the general air pressure causes air molecules to fill that room back in.

Both of those travel at the same speed, so they reach your eardrum, which is just another membrane and therefore does the same motion that the speaker membrane did.

So, a speaker is like a Newton’s Cradle, it doesn’t actually cause the molecules on the way to traverse across the room, like a fan would.