Eq. 5 fails as an equation of motion only on its left-hand side where the pressure P appears rather than an explicit expression in terms of the degrees of freedom and their derivatives. To complete the derivation of the equation of motion, we must therefore determine the pressure in terms of the degrees of freedom. Such an explicit equation relating the driving forces in a system to the degrees of freedom is known as a constitutive relation.
To derive the constitutive relation for a gas (or fluid or solid), we
begin by noting that as we increase the volume of the gas, we expect
the pressure to decrease. For a small change in volume 
, we
expect the change in pressure 
to also be small and in
proportion to the change in volume, 
.
Moreover, for a given change in volume , we expect the
change in pressure to be quite small if the initial volume 
of
gas is large and thus 
. Thus, we expect
where B, the bulk modulus, is a constant characteristic of the
particular material making up the system under study. (Note that
throughout these notes, we always define our signs so that the change
in quantity Q as 
where 
is the initial value
of the quantity.)
To express the pressure in terms of the solution using (8), we begin by evaluating the relevant quantities for the chunk in Figure 3 directly in terms of the solution,
Substituting these results into (8), we find the final constitutive relation relating the pressure P to the solution s,
where we have taken the limit of a very thin chunk, 
in order to get the pressure at precisely the point
x. Note again the similarity to the string. Apart from the
constant background pressure 
, which cancels out in most physical
effects, the driving force is in direct proportion to the first
spatial derivative of the solution through a constant characterizing
the strength of restoring forces in the system, B for the gas (or
liquid or solid) and 
for the string.
