Derivation

Expression for Stagnation Pressure $(P_s)$

$$\frac{P_{s}}{P_1}=\left[\left(1+\frac{k-1}{2} M_{1}^{2}\right)^{\frac{K}{K-1}}\right] ..(i)$$

Expression for Stagnation Density $\left(\rho_ {s} \right)$

For Adiabatic process

$$\begin{aligned} \left(\frac{\rho_{1}}{\rho_{s}}\right)=\left(\frac{p_{1}}{p_{s}}\right)^{\frac{1}{K}} \text{or}\left(\frac{\rho_{s}}{\rho_{1}}\right)=\left(\frac{p_{s}}{p_{1}}\right)^{\frac{1}{k}} \end{aligned}$$

(Taking reciprocal)

$$\quad \rho_{s}=\rho_{1}\left[\frac{p_{s}}{p_{1}}\right]^{\frac{1}{K}}$$

Substituting the value of $\left(\frac{p_{s}}{p_{1}}\right)$ from equation $(i)$,

$$\begin{aligned} \rho_{s} &=\rho_{1}\left[\left(1+\frac{K-1}{2} M_{1}^{2}\right)^{\frac{K}{K-1}}\right]^{\frac{1}{K}} \\ \rho_{s} &=\rho_{1}\left[1+\frac{K-1}{2} M_{1}^{2}\right]^{\frac{1}{K-1}} ..(ii) \end{aligned}$$

Above …