# Cole equation of state

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

## References

1. R.H. Cole, Underwater Explosions. Princeton University Press 1948
2. G.K. Batchelor, An introduction to ﬂuid mechanics. Cambridge University Press 1974