Method of characteristics in gas dynamics

Remarks

Facts 1 and 2 follow from the equations: for a $C^1$ solution, $r_t + (u + c) r_x = 0$ and $s_t + (u - c) s_x = 0$. In practice one starts from the data on an initial line, follows both families of curves, and updates $u$ and $c$ at each crossing point.

Example: a gas at rest with constant sound speed $c_0$ has $r = 2c_0/(\gamma - 1)$ and $s = -2c_0/(\gamma - 1)$ everywhere. The characteristics are the straight lines $x = x_0 + c_0 t$ and $x = x_0 - c_0 t$, and indeed $u = (r + s)/2 = 0$ and $(\gamma - 1)(r - s)/4 = c_0$. If $s$ is constant in a whole region, the flow is a simple wave and all $C_+$ characteristics there are straight lines.

Non-example: across a shock wave the solution is not $C^1$, so the method does not apply there. The invariants jump across the shock, and the entropy changes, so the isentropic relation $p = K\rho^\gamma$ fails behind a strong shock. For non-isentropic flow a third family of characteristics, the particle paths $dx/dt = u$, carries the entropy.

In steady two-dimensional supersonic flow the characteristics are the Mach lines. Along each family of Mach lines one of the two quantities $\theta + \nu(M)$ and $\theta - \nu(M)$ is constant, where $\theta$ is the flow angle and $\nu$ is the Prandtl–Meyer function.