Why does wake turbulence sink?
Lift is created by accelerating air downwards. The same goes for wake turbulence: The vortices form at the boundary between the downward-moving strip of air (wake) from surrounding air that tries to fill the void left by the wake.

Friction will eventually dissipate the kinetic energy of the wake, but in case of heavy aircraft the motion will continue for several minutes. By that time it will sink down hundreds of feet or hit the ground. This is driven by inertia; density differences do not exist unless the atmospheric lapse rate is above the adiabatic rate and the wake warms less quickly from increasing atmospheric pressure than the surrounding air.

In the end, the weight of any heavier-than-air vehicle is supported by an area of higher pressure on the ground. Some mechanism has to transfer the weight force from the flying aircraft to the ground, and that is the downward-moving wake. When it slowly dissipates, it will leave a slightly higher air pressure below it. When that happens at supersonic speed, the more focussed pressure change is high enough to be audible on the ground.