= Solution
The one-dimensional <Isentropic Euler equations> are
$$
\rho_t+u\rho_x+\rho u_x=0,
\qquad
u_t+uu_x+\frac{c^2}{\rho}\rho_x=0,
\qquad
c^2=\frac{dp}{d\rho}.
$$
Define the <Riemann invariants for one-dimensional isentropic flow>
$$
\boxed{R_\pm=u\pm\int_{\rho_0}^{\rho}\frac{c(s)}s\,ds}.
$$
Combining the two equations directly gives
$$
\boxed{\left[\partial_t+(u\pm c)\partial_x\right]R_\pm=0},
$$
so $R_\pm$ is constant on the corresponding <characteristic curve> $dx/dt=u\pm c$.
Initially both invariant disturbances are supported in $0\leq x\leq L$. Once the two characteristic families have separated, the left-moving packet is a <simple wave> with $R_+$ at its equilibrium value, while the right-moving packet is a <simple wave> with $R_-$ at equilibrium. Between and outside them, both invariants retain their equilibrium values. The schematic <spacetime diagram> below uses the equilibrium slopes $\pm c_0$ and shows separation after $t=L/(2c_0)$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4-simple-wave-separation.png]
{title=Characteristic curves and simple-wave separation}
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