Cole equation of state

The Cole equation of state 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 the adiabatic index, 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 about 0.01.

If the fluctuations in the density are indeed small, the equation of state may be approximated by the simpler:

$p=B\gamma \left[{\frac {\rho -\rho _{0}}{\rho _{0}}}\right]$