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$$\begin{array}{cc} T_1=T_2\land T_2=T_3\implies T_1=T_3&\text{0th law}\\ \Delta e=0&\text{1st law}\\ \Delta s\ge 0&\text{2nd law}\\ S=k\log W&\text{3rd law}\\ de=Tds-pdv&\text{combined 1st/2nd law}\\ p=\rho RT&\text{ideal gas law} \end{array}$$ $$\begin{array}{cc} \text{1d flow}\\ \hline \rho_1u_1=\rho_2u_2\\ p_1+\rho_1u_1^2=p_2+\rho_2u_2^2\\ h_1+u_1^2/2=h_2+u_2^2/2\\ \end{array}$$ $$\begin{array}{c} \text{shock wave}\\ \hline M_2^2=\frac{M_1^2(\gamma-1)+2}{2\gamma M_1^2-(\gamma-1)}\\ \frac{T_2}{T_1}=\frac{h_2}{h_1}=\left(\frac{a_2}{a_1}\right)^2=1+\frac{2(\gamma-1)(M_1^2-1)(\gamma M_1^2+1)}{(\gamma+1)^2M_1^2}\\ \frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M_1^2-1)\\ \frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}\\ \frac{p_{02}}{p_{01}}=\left(\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}\right)^{\frac{\gamma}{\gamma-1}}\left(\frac{\gamma+1}{2\gamma M_1^2-(\gamma-1)}\right)^{\frac{1}{\gamma-1}}\\ \tan\delta=2\cot\beta\frac{M_1^2\sin^2\beta-1}{M_1^2(\gamma+\cos 2\beta)+2}\\ M_{1n}=M_1\sin\beta\\ M_{2n}=M_2\sin(\beta-\delta)\\ \end{array}$$