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At time $t = 0^+$, the initial density profile is then modeled according to the figure below.

The resulting wave motion of the disturbance is governed by

$$

\frac{\partial\rho}{\partial t} + c(\rho)\frac{\partial\rho}{\partial x} = 0

$$

where

$$

c(\rho) = u_{\text{max}}\left(1 - \frac{2\rho}{\rho_{\text{max}}}\right)

$$

Argue that two shocks emanate from the origin and obtain expressions for the shock speed.

From method of characteristics, we have $t=r$ and $x=tu_{\text{max}}\left(1-\frac{2\rho}{\rho_{\text{max}}}\right)+x_0$.

I don't know what to do now.