Cole equation of state

The Cole equation of state can be written, when atmospheric pressure is negligible, has the form

\[p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]\].

In it, \(\rho_0\) is a reference density around which the density varies \(\gamma\) is an exponent and \(B\) is a pressure parameter.

Usually, the equation is used to model a nearly incompressible system. In this case, the exponent is often set to a value of 7, and \(B\) is large, in the following sense. The fluctuations of the density are related to the speed of sound as

\[\frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,\]

where \(v\) is the largest velocity, and \(c\) is the speed of sound (the ratio \(v/c\) is Mach's number). The speed of sound can be seen to be

\[c^2 = \frac{\gamma B}{\rho_0}. \]

Therefore, if \(B=100 \rho_0 v^2 / \gamma\), the relative density fluctuations will be of about 0.01.