Cole equation of state

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

$p=B[{(\frac{\rho }{{\rho }_0})}^{\gamma }-1]$

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

$c^2=\frac{\gamma B}{{\rho }_0}.$

where $\gamma$ is the adiabatic index. Therefore, if $B=100{\rho }_0{v}^2/\gamma$, the relative density fluctuations will be of about 0.01.

If the fluctuations in the density are indeed small, the equation of state may be rewritten thus:

$p=B\gamma [\frac{\rho -{\rho }_0}{{\rho }_0}]$