Suppose that a solution to Burgers Equation exists as a density wave moving with

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Suppose that a solution to Burgers Equation exists as a density wave moving without change of shape: rho(x, t) = f(x - Vt ) at velocity V. What Ordinary Differenial Equation is satisfied by f ? Integrate this ODE once. By graphical techniques, show that a solution exists such that f rightarrow rho2 as x rightarrow + infinity and f rightarrow rho 1 as x rightarrow -infinity, only of rho 2 > rho 1. Give a rough sketch of this solution and an interpretation of the the result in terms of traffic. Show that the velocity of wave propagation, V, is the same as the shock velocity separating rho = rho 1 from rho = rho 2, occuring if nu = 0. For this problem we will begin with the conservation of cars of equation where rho := traffic density, u( rho ) := car velocity, and q( rho ) := rho u( rho ) = car flux, such that c( rho ) = q'( rho ). Only now we will enrich the model by allowing, instead of the simple u = u( rho ) model above, the new model where nu is some constant. What sign should nu have for the above expression to be reasonable based on what were trying to model (traffic)? What is the new conservation of cars equation? Assume U( rho ) = umax( 1 - rho /rho max). Derive Burgers Equation:

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