Sound waves and sound pressure

[Booney]

Homework Statement

Consider a sound wave in air of density 1.2 kg/m³. The displacement wave has the form s(x,t)=s_maxcos(kx-$$\omega$$t) where k=8.79rad/m, $$\omega$$=3021.6 rad/s and s_max=2.51 x 10^-7m.
Calculate the sound pressure $$\Delta$$P(x,t) of this wave at x=0.282m and t=0.00137s. Answer in units of Pa.

Homework Equations

$$\Delta$$P_max=$$\rho$$v$$\omega$$s_max
$$\Delta$$P=$$\Delta$$P_maxsin(kx-$$\omega$$t)

The Attempt at a Solution

I have a feeling I'm not using the correct equations because the equations I've found include a velocity, which isn't given, and don't include the displacement function. How do I relate the displacement function to the pressure equations?

 

[Xerxes1986]

The only way I can think to solve this is by using the kinetic theory of an ideal gas to derive pressure based on average velocity. You can review this derivation at http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kinthe.html#c3". The important equation is:

$$\frac{1}{3} \frac{N}{V} m \bar{v}^2$$

You can calculate velocity from your displacement equation.