Cole equation of state

The Cole equation of state is the adiabatic version of the stiffened equation of state. (See Derivation, below.) It 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:

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

It is quite common that the name "Tait equation of state" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39).

Derivation

Let us write the stiffened EOS as

$p+p^{*}=(\gamma -1)e\rho =(\gamma -1)E/V,$

where E is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the first law reads

$dW=-pdV=dE$

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