Construct Contour Plot in Mathematica for Quasi-Linear 1-D Wave Eq

[Dustinsfl]

How can I construct a contour plot in Mathematica for
Consider the quasi-linear 1-D wave equation
$$
\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0
$$
with the piecewise constant initial conditions
When $x_0 = \pm 1$

$$
\rho(x,0) = \begin{cases}
4, & x < -x_0\\
3, & -x_0 < x < x_0\\
1, & x > x_0\\
\end{cases}
$$

 

[Dustinsfl]

How can I construct a contour plot in Mathematica for
Consider the quasi-linear 1-D wave equation
$$
\frac{\partial\rho}{\partial t} + 2\rho\frac{\partial\rho}{\partial x} = 0
$$
with the piecewise constant initial conditions
When $x_0 = \pm 1$

$$
\rho(x,0) = \begin{cases}
4, & x < -x_0\\
3, & -x_0 < x < x_0\\
1, & x > x_0\\
\end{cases}
$$

Would this be the correct contour plot?
View attachment 339