# Difference between revisions of "Cole equation of state"

Carl McBride (talk | contribs) m (Added a couple of internal links) |
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can be written, when atmospheric pressure is negligible, has the form | can be written, when atmospheric pressure is negligible, has the form | ||

− | :<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math> | + | :<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math> |

In it, <math>\rho_0</math> is a reference density around which the density varies | In it, <math>\rho_0</math> is a reference density around which the density varies | ||

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where <math>v</math> is the largest velocity, and <math>c</math> is the speed of | where <math>v</math> is the largest velocity, and <math>c</math> is the speed of | ||

− | sound (the ratio <math>v/c</math> is [[Mach's number]]). The speed of sound can | + | sound (the ratio <math>v/c</math> is [[Mach's number]]). The [[speed of sound]] can |

be seen to be | be seen to be | ||

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If the fluctuations in the density are indeed small, the | If the fluctuations in the density are indeed small, the | ||

− | + | [[Equations of state | equation of state]] may be rewritten thus: | |

:<math>p = B \gamma \left[ | :<math>p = B \gamma \left[ |

## Revision as of 14:37, 23 May 2012

The **Cole equation of state** ^{[1]}^{[2]}
can be written, when atmospheric pressure is negligible, has the form

In it, is a reference density around which the density varies is an exponent and 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 is large, in the following sense. The fluctuations of the density are related to the speed of sound as

where is the largest velocity, and is the speed of sound (the ratio is Mach's number). The speed of sound can be seen to be

Therefore, if , 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:

## References

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