I hope it helps how I understand the relation of pressure and displacement in an acoustic wave.
An instantaneous acoustic pressure does not generate an instantaneous particle displacement. The process is delayed as pressure accelerates the particles until no pressure remains, there you have the max. particle velocity but the displacement continues.
For sinusoidal displacement of particles in acoustic sound waves,
the formulas have been derived long ago from Fluid Dynamics Law.
We can apply Fluid Dynamics as long as
the acoustic pressure is small compared to the steady pressure (e.g. ambient air) and the particle speed remains much below the speed of sound in the medium.
The particle dispĺacement is linked to a speed and time by a frequency.
The speed of flow by Fluid Dynamics has an equivalent pressure at rest.
Without expansion the equivalent resting pressure of the actual speed plus the actual pressure remain constant.
At max. pressure there is no particle speed and no more particle displacement hence this is also the moment of max. particle displacement in one direction.
The displacement is in phase with the pressure.