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shock wave 2

dwsmith

Well-known member
Feb 1, 2012
1,673
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.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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 $\rho_{\text{max}}$.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
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.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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.