Two facts allow us to dramatically simplify our analysis.
A plane wave is a wave in which the disturbance is constant in planes perpendicular to the direction of motion of the wave, which we shall always take to be along the x axis in these notes. (Figure 2 illustrates such a wave.) Any type of wave can be decomposed into a superposition, or sum, of plane waves. Hence, in these notes we consider the simpler case of plane waves without any loss of generality.
Polarization defines the direction of the motion associated with the disturbance relative to the direction of motion of the wave. In a string, for example, the direction of the motion associated with the wave is up-and-down along the y-axis, whereas the direction of motion of the wave is along the x-axis. These two directions are perpendicular, a special situation defined as transverse polarization. A pressure wave, on the other hand, requires expansion and contraction of the gas (or liquid or solid) to create changes in pressure. The gas compresses and expands only when the planes move back-and-forth along the x-axis. In this case, the direction of motion of the disturbance and of the propagation of the wave are parallel, a special situation defined as longitudinal polarization.
Figure 2: Plane wave with longitudinal polarization: direction of wave
propagation (x), set of chunks undergoing the same displacement s(x)
(plane), initial location of chunks (dashed plane).
Thus, to study sound, we only need consider longitudinal plane waves.
This means, as in Figure 2, that the displacement
vector 
describes motion directly along the x axis.
Thus, 
only has a single component along the the x axis and
can be described by a single scalar value s. Moreover, because the
wave is plane, 
is the same for all points which
shares the same value of x, regardless of the values of y and z.
Thus, a single scalar (non-vector) function s(x) of the scalar x
suffices to specify the degrees of freedom.