::cs_5 = \sqrt{\gamma \frac{P_R}{\rho_R}} where \gamma is the adiabatic gamma. The first one is the position of the beginning of the rarefaction wave while the other is the velocity of the propagation of the shock. Defining: ::\Gamma = \frac{\gamma - 1}{\gamma + 1}, \beta = \frac{\gamma - 1}{2 \gamma} The states after the shock are connected by the Rankine Hugoniot shock jump conditions. ::\rho_4 = \rho_5 \frac{P_4 + \Gamma P_5}{P_5 + \Gamma P_4} But to calculate the density in Region 4 we need to know the pressure in that region. This is related by the contact discontinuity with the pressure in region 3 by ::P_4 = P_3 Unfortunately the pressure in region 3 can only be calculated iteratively, the right solution is found when u_3 equals u_4 ::u_4 = \left(P_3' - P_5\right)\sqrt{\frac{1-\Gamma}{\rho_R(P_3'+\Gamma P_5)}} ::u_3 =\left(P_1^\beta-P_3'^\beta\right) \sqrt{\frac{(1-\Gamma^2)P_1^{1/\gamma}}{\Gamma^2 \rho_L}} ::u_3 - u_4 = 0 This function can be evaluated to an arbitrary precision thus giving the pressure in the region 3 ::P_3 = \operatorname{calculate}(P_3,s,s,,) finally we can calculate ::u_3 = u_5 + \frac{(P_3 - P_5)}{\sqrt{\frac{\rho_5}{2}((\gamma+1)P_3 +(\gamma-1)P_5)}} ::u_4 = u_3 and \rho_3 follows from the adiabatic gas law ::\rho_3 = \rho_1 \left(\frac{P_3}{P_1}\right)^{1/\gamma} ## References - - ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Sod_shock_tube) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Sod_shock_tube?action=history). ::