# Cole equation of state

The Cole equation of state [1][2] 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 of about 0.01.

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

$p = B \gamma \left[ \frac{\rho-\rho_0}{\rho_0} \right]$

## References

1. R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227
2. G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962