For a transverse wave like a wave on a string, when the wave is traveling in the x-direction the pieces of string oscillate back and forth in the y-direction. For a longitudinal wave like a sound wave the oscillations are parallel to the direction the wave travels.

For a wave traveling in the +x direction, for instance, the oscillation of the particles in the medium in the x-direction can be described by:

s(x,t) = s_m sin(ωt -kx)

The oscillations of the particles produces small changes in pressure in the medium. A higher density of particles corresponds to a higher pressure, and a lower density corresponds to a lower pressure. Examining the simulation closely, it can be seen that when a particle has a displacement of zero, the pressure is at a maximum or minimum. High pressure at a point occurs when neighboring particles come toward the point; low pressure when the neighboring particles move away.

If the particles at a given point fluctuate following a sine, the pressure there fluctuates like a cosine, 90 degrees out of phase with the displacement.

Δp = Δp_m cos(ωt - kx)

The relationship between the maximum pressure change Δp_m and the maximum displacement amplitude of the particles s_m is:

Δp_m = (v ρ ω)s_m

This is derived in the book.