Analysis of Shock Formation from Traffic Light Red Signal

[Dustinsfl]

Suppose traffic is moving uniformly with a constant density $\rho_0$ when a traffic light turns red.
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.

 

[Dustinsfl]

Suppose traffic is moving uniformly with a constant density $\rho_0$ when a traffic light turns red.
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.

I forgot to add that when $x<0$ and $x>0$ the density is $\rho_0$.
When $x=0$, the density is $\rho_{\text{max}}$.

 

[Dustinsfl]

View attachment 341

How can I convert this into boundary conditions.

I know that we must have $\rho_0$ for $x<0$ and $x>0$.

As before, we have that $t = r$.
So $\frac{d\rho}{dt} = 0\Rightarrow \rho = c$.
When $t = 0$ and $x = x_0$, we have $\rho(x_0,0) = c$.
Thus,
$$
\rho = \rho(x_0,0).
$$
Now, we have the ODE
$$
\frac{dx}{dt} = u_{\text{max}}\left(1 - \frac{2\rho(x_0,0)}{\rho_{\text{max}}}\right) \Rightarrow x = u_{\text{max}}\left(1 - \frac{2\rho(x_0,0)}{\rho_{\text{max}}}\right)t + x_0.
$$

 

[Sudharaka]

Suppose traffic is moving uniformly with a constant density $\rho_0$ when a traffic light turns red.
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.

I forgot to add that when $x<0$ and $x>0$ the density is $\rho_0$.
When $x=0$, the density is $\rho_{\text{max}}$.

Hi dwsmith, :)

What I don't understand in this problem is that in your first post you have mentioned that this model is valid only for \(t=0^+\) which I assume is, \(t>0\). Then in your second post the boundary conditions are given for \(t<0\) and \(t=0\). Am I missing something here?

Kind Regards,
Sudharaka.

 

[Dustinsfl]

Hi dwsmith, :)

What I don't understand in this problem is that in your first post you have mentioned that this model is valid only for \(t=0^+\) which I assume is, \(t>0\). Then in your second post the boundary conditions are given for \(t<0\) and \(t=0\). Am I missing something here?

Kind Regards,
Sudharaka.

Post 3 has the picture. Click and you will see the information.