\(a \equiv \text{speed of sound}\)
\(M \equiv \text{Mach number}\)
\(p \equiv \text{pressure}\)
\(T \equiv \text{temperature}\)
\(u_2 \equiv \text{particle velocity}\)
\(W \equiv \text{speed of wave}\)
\(\gamma \equiv \text{specific heat ratio}\)
\(\rho \equiv \text{density}\)
$$M \equiv \frac{W}{a}$$
Pressure Ratio:
$$\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}\left(M^2-1\right)$$
Temperature Ratio:
$$\frac{T_2}{T_1} = \frac{p_2}{p_1}\left(\frac{\frac{\gamma+1}{\gamma-1}+\frac{p_2}{p_1}}{1+\frac{\gamma+1}{\gamma-1}\left[\frac{p_2}{p_1}\right]}\right)$$
Density Ratio:
$$\frac{\rho_2}{\rho_1} = \frac{1+\frac{\gamma+1}{\gamma-1}\left(\frac{p_2}{p_1}\right)}{\frac{\gamma+1}{\gamma-1}+\frac{p_2}{p_1}}$$
Particle Velocity:
$$u_2 = \frac{a_1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\frac{2\gamma}{\gamma+1}}{\frac{p_2}{p_1}+\frac{\gamma-1}{\gamma+1}}\right)^{1/2}$$
Pressure Ratio:
$$\frac{p_5}{p_2} = \frac{(3\gamma-1)\frac{p_2}{p_1}-(\gamma-1)}{(\gamma-1)\frac{p_2}{p_1}+(\gamma+1)}$$
Temperature Ratio:
$$\frac{T_5}{T_2} = \frac{p_5}{p_2}\left(\frac{\frac{\gamma+1}{\gamma-1}+\frac{p_5}{p_2}}{1+\frac{\gamma+1}{\gamma-1}\left[\frac{p_5}{p_2}\right]}\right)$$
Density Ratio:
$$\frac{\rho_5}{\rho_2} = \frac{1+\frac{\gamma+1}{\gamma-1}\left(\frac{p_5}{p_2}\right)}{\frac{\gamma+1}{\gamma-1}+\frac{p_5}{p_2}}$$
Reflected Wave Speed and Mach Number:
$$W_R = \frac{\rho_2 u_2}{\rho_5-\rho_2}$$
$$M_R = \frac{W_R+u_2}{a_2}$$
$$\frac{p_4}{p_1} = \frac{p_2}{p_1}\left(1-\frac{(\gamma_4-1)(a_1/a_4)(p_2/p_1-1)}{\sqrt{2\gamma_1\left[2\gamma_1+(\gamma_1+1)(p_2/p_1-1)\right]}}\right)^{-2\gamma_4/(\gamma_4-1)}$$