The gray strip is a one-lane highway. The vehicles marked $A$, $B$, $C$, $D$, move along it at various velocities. The space-time diagram at the top (the $(x,t)$~plane) conveys quite well the quality and quantity of motion even in the absence of animation.
This animation illustrates the solution of the initial value problem \begin{align} &u_t + c u_x = 0, \\ &u(x,0) = f(x), \end{align} with $c=1$ and $u_0$ as \begin{equation} f(x) = \begin{cases} \cos^2 \Bigl( \frac12\pi x \Bigr) & \text{if }|x|<1, \\ 0 & \text{otherwise} \end{cases} \end{equation} The solution holds for $-\infty \lt x \lt \infty$ and $0 \lt t \lt \infty$, although the animation window encompasses only $0 \lt x \lt 8$ and $0 \lt t \lt 10$.
This animation illustrates the solution of the initial value problem \begin{align} &(1+t)u_t + xt u_x = u, \\ &u(x,0) = 1-x. \end{align} The solution holds for $-\infty \lt x \lt \infty$ and $0 \lt t \lt \infty$, although the animation window encompasses only $-4 \lt x \lt 4$ and $0 \lt t \lt 6$.