Rankine-Hugoniot Relation

Passing through a shock front moving with a speed $V_s$, the physical variables $\rho$, $p$, and $u$ change abruptly. Since the basic equations of hydrodynamics is unchanged after chosing a system moving $V_s$, the continuity equation

$$\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}=0$$ (A.26)

gives an equation for a steady state as

$$\frac{\partial \rho u}{\partial x}=0.$$ (A.27)

Considering a region containing the shock front at $x_s$ which extends from $x=x_s-\Delta x$ to $x=x_s+\Delta x$ and integrating the above equation, we get

$$\int_{x_s-\Delta x}^{x_s+\Delta x}\frac{d \rho u}{d x}dx= \left[\rho u\right]_{x_s-\Delta x}^{x_s+\Delta x}.$$ (A.28)

Thus, we obtain the jump condition conserning the mass coservation as

$$\rho_1 u_1=\rho_2 u_2,$$ (A.29)

where the quatities with suffix 1 are for preshock and those with suffix 2 are for postshock.

Equation of motion for steady state

$$\rho u\frac{\partial u}{\partial x} =-\frac{\partial p}{\partial x},$$ (A.30)

gives

$$p_1+\rho_1 u_1^2=p_2+\rho_2 u_2^2,$$ (A.31)

where we used equation(A.29).