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I'm trying to go from, this equation:

$\frac{\gamma}{\gamma -1} \left( \frac{P}{ \rho} - \frac{P_0}{ \rho_0} \right) - \frac{1}{2} (P-P_0) \left( \frac{1}{\rho} + \frac{1}{\rho_0} \right)=0$

to this one:

$\frac{2 \gamma}{\gamma -1} \left( \frac{P}{P_0} - \frac{\rho}{\rho_0} \right) = \left( \frac{P}{P_0} - 1\right) \left( \frac{\rho}{\rho_0} + 1 \right) $

from which one can then obtain the Rankine-Hugoniot equations (for physics), however, the intermediate steps are proving to be quite vexing. Any help?

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To simplify, let $N=\frac{2\gamma}{\gamma-1}$. $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=(P-P_{0})(\frac{1}{p}+\frac{1}{p_{0}})$$ $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=(P-P_{0})(\frac{\rho+\rho_{0}}{\rho\rho_{0}})$$ $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}}{\rho\rho_{0}}$$ $$N(\frac{P\rho\rho_{0}}{\rho}+\frac{P_{0}\rho\rho_{0}}{\rho_{0}})=P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}$$ $$N(\frac{PP_{0}\rho\rho_{0}}{P_{0}\rho}+\frac{PP_{0}\rho\rho_{0}}{P\rho_{0}})=P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}}{P_{0}\rho_{0}}$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho}{P_{0}\rho_{0}}-1$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=(\frac{P}{P_{0}}-1)(\frac{p}{p_{0}}+1)$$ The equations don't look identical. Did you mix up the positive and negatives? (Or point out any errors I made)

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  • $\begingroup$ There was in fact a sign mistake in the parenthesis of the 1st equation, I fixed it now. Thank you! $\endgroup$ Commented May 16 at 18:18

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