Pressure wave equation and displacement vs. pressure amplitude + example.
Starting with an animation of a longitudinal wave, we define the position wave function for a longitudinal wave y(x,t)=Acos(kx-wt) and clarify what this means in the context of the moving longitudinal wave: horizontal displacements y(x,t) which vary as a function of equilibrium position x and time.
Next we get into the derivation of the pressure wave equation for a longitudinal wave by looking at a column of air marked with a left position of x and right position of x+delta-x. The left and right ends of this column have displacements given by y(x,t) and y(x+delta-x,t) respectively.
This region of air is compressed, and that means the pressure must have increased relative to atmospheric pressure. So we write down the pressure change in terms of bulk modulus and volume change: delta-p = B delta-V/V. Writing down the volume change in terms of the displacements, we find an x partial derivative of y(x,t) which allows us to write down the pressure wave equation as p(x,t)=BAksin(kx-wt).
This means the pressure amplitude of the sound wave relates to the displacement amplitude as p=BAk, and that allows us to solve our example, finding the displacement amplitude of a pure sinusoidal sound wave given the frequency, speed of sound and pressure amplitude. We find for a roughly 65dB sound wave that the displacement amplitude is about 64nm, so very small!
