Isotharmal shock

In the case of the gas is isothermal $p=c_{is}^2\rho$, equation (A.31) becomes

$$\rho_1\left(c_{is}^2+u_1^2\right)=\rho_2\left(c_{is}^2+u_2^2\right).$$ (A.32)

Eliminating $\rho$ from equations (A.29) and (A.32), we obtain

$$\left(u_1u_2-c_s^2\right)\left(u_1-u_2\right)=0,$$ (A.33)

which means

$$u_1u_2=c_s^2.$$ (A.34)

From equation (A.29),

$$\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{u_1^2}{c_s^2}.$$ (A.35)

This indicates the postshock velocity $u_1 \gg c_s$ the ratio of the postshock density to the preshock density becomes large.