Editing Cole equation of state
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The '''Cole equation of state''' | The '''Cole equation of state''' | ||
<ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref> | <ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref> | ||
is the adiabatic version of the [[stiffened equation of state]] | is the adiabatic version of the [[stiffened equation of state]]. (See ''Derivation'', below.) | ||
It has the form | It has the form | ||
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:<math> dW= -p dV = dE .</math> | :<math> dW= -p dV = dE .</math> | ||
Taking differences on | Taking differences on theEOS, | ||
:<math> dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , </math> | :<math> dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , </math> | ||
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which can be written as | which can be written as | ||
:<math> p = | :<math> p = ( p_0 + p^* / \gamma) (V_0/V)^\gamma - p^* / \gamma . </math> | ||
Going back to densities, instead of volumes, | Going back to densities, instead of volumes, | ||
:<math> p = | :<math> p = ( p_0 + p^* / \gamma) (\rho/\rho_0)^\gamma - p^* / \gamma . </math> | ||
Now, the speed of sound is given by | Now, the speed of sound is given by | ||
:<math> c^2=\frac{dp}{d\rho} | :<math> c^2=\frac{dp}{d\rho} , </math> | ||
with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain | with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain | ||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: equations of state]] | [[category: equations of state]] |