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Density wave equation

  1. Sep 14, 2012 #1
    Traffic is moving with a uniform density of [itex]\rho_0[/itex].
    $$
    \frac{\partial\rho}{\partial t} + c(\rho)\frac{\partial\rho}{\partial x} = \beta_0
    $$
    where
    $$
    c(\rho) = u_{\text{max}}\left(1 - \frac{2\rho}{\rho_{\text{max}}}\right).
    $$
    Show that the variation of the initial density distribution is given by
    $$
    \rho = \beta_0t + \rho(x_0,0)
    $$
    along a characteristic emanating from [itex]x = x_0[/itex] described by
    $$
    x = x_0 + u_{\text{max}}\left(1 - \frac{2\rho(x_0,0)}{\rho_{\text{max}}}\right)t - \beta_0\frac{u_{\text{max}}}{\rho_{\text{max}}}t.
    $$

    So we have [itex]\frac{dt}{ds} = 1[/itex], [itex]\frac{dx}{ds}=c(\rho)[/itex] and [itex]\frac{d\rho}{ds} = \beta_0[/itex].
    Then [itex]t(s) = s + c[/itex] where [itex]t=s[/itex] when [itex]t(0) = 0[/itex].
    Not sure how to handle the other two though.
     
  2. jcsd
  3. Sep 16, 2012 #2
    What I have done so far is:
    [itex]\frac{dt}{dr} = 1\Rightarrow t = r + c[/itex] but when [itex]t = 0[/itex], we have [itex]t = r[/itex].

    [itex]\frac{dx}{dr} = c(\rho)\Rightarrow x = tu_{\text{max}}\left(1-\frac{2\rho}{\rho_{\text{max}}}\right)+c[/itex] but when [itex]t=0[/itex], we have
    $$
    x = tu_{\text{max}}\left(1-\frac{2\rho}{\rho_{\text{max}}}\right) + x_0.
    $$

    [itex]\frac{d\rho}{dr} = \beta_0\Rightarrow \rho = t\beta_0 + c[/itex]

    How do I get to
    $$
    \rho(x,t) = t\beta_0 +\rho(x_0,0)
    $$
    and their characteristic?
     
    Last edited: Sep 16, 2012
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