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Consider an acoustic wave in some medium, expressed as particle displacement:
$$s(t,x) = e^{j(-\omega t + kx)} $$
I understand that pressure must be at its maximum when the particle displacement is zero and vice versa. But there are many functions that satisfy that criterion.
How can the pressure be expressed in terms of particle displacement?
In a free plane sound wave the acoustic pressure $P \,(\textrm{N/m}^2)$ and acoustic particle velocity $U\,(\textrm{m/s})$ are in phase and proportional to each other. $Z=P/U$ is the acoustic impedance, which for air at STP is 428 Rayl.
The particle displacement $D$ for sine wave is $D = U/2\pi$, and it is 1/4 cycle out of phase, lagging pressure and particle velocity. $D$ is zero when $P$ is positive peak value and $U$ is positive peak value in the direction of propagation. The peak positive value of displacement $D$ occurs 1/4 cycle later.
For sine waves $P = U Z = 2 \pi D Z$.
According to Wikipedia's page on particle displacement, the maximum displacement in terms of acoustic pressure is given by $$\xi=\frac{p}{Z_0\omega}=\frac{p}{\sqrt{B\rho}\omega}$$ where $Z_0=\rho c$ and $B$ is the bulk modulus of the medium.
I'd guess that a derivation can probably be found in most textbooks on sound.
There's no such formula.
You can think of pressure as the (negative) spatial derivative of displacement. If the particles at $x = -\epsilon$ are displaced a little to the left and the particles at $x = +\epsilon$ are displaced a little to the right, then there's low pressure at $x = 0$ since less particles are there.
With that in mind, pressure/displacement are analogous to velocity/position in a standard wave. Now think about a typical water wave you see at the beach -- it doesn't make sense to say "if this part of the wave is 2 feet high, how fast is it moving?" It depends on the wave.
In the special case of a sinusoidal wave, the pressure and displacement are like $A \cos(kx - \omega t)$ and $B \sin(kx - \omega t)$, where $A/B$ is given by the formula DumpsterDoofus posted.
Your original equation is missing the amplitude in front of the exponential. It should be something like $ s(t,x)=s_0 e^{j (-\omega t +kx)} $ The pressure wave has an amplitude given by the formula above (answer by DumpsterDofus) and a phase shift of 90 degrees relative to the displacement. So you have all you need to write the equation for the pressure wave.
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.
