Math 142 HW 4: Traffic: Traveling Waves and Characteristics
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Math 142 HW 4: Traffic: Traveling Waves and Characteristics
• 67) 1, 2, 3
• 71) 1, 9
• 73) 1, 2, 3, 5, 6
Hint for 2, 3: See Eq. 73.8, from the derivative of Eq. 73.7. Note that the discussion below 73.8 already gives the answer to Question 3. For Question 2, take the derivative and use x0 = 
.
• A) For the following three questions that you did from 73:
i) find the position of the car x(t) using ρ(x, t) that you already found, and the given initial conditions. That is, solve 
= u(ρ(x, t)). This will be a linear first order ODE, so use integrating factors.
ii) Draw the path of the car, along with the ρ=constant lines in the xt-plane, in the same manner as Figure 73-8.
73) # 1 with x(1) = a.
73) # 5 with x(1) = − 
.
73) # 6 with x(0) = −x0 .
• 74) 1, 2
• 77) 1, 2, 3, 5, 6, 7.
77.3: assume q(ρ) is differentiable to get the Taylor series.
What 77.5-77.7 is about:
Since the shock we derived in Section 77 is discontinuous, it seems to be a somewhat crude model of what actually happens. This is because we have only used the most basic idea of how things are moving on the freeway (conservation of cars) to derive all of our conclusions. On an actual freeway there would be many other things going on. But accounting for these things always comes with the cost of making the model more difficult to deal with.
To make things more realistic, we can include something about human behavior that we already looked at in 63.7 from the last homework. We can assume that cars will naturally slow down when there is increasing traffic visible up ahead, and similarly cars will speed up when the traffic ahead is decreasing. This turns out to add the new term ν
in 77.5c) with a spatial second-derivative, which is called a diffusion term. This will have the effect of smoothing out the overall distribution ρ(x, t), and our shock wave will also be represented by a smooth function. More specifically we get a smooth function f (x) and our shock wave will now be given by ρ(x, t) = f (x − ct). Note that in particular this means that the shock wave is a traveling wave! My goal by the end of the class is to revisit diffusion and traveling wave solutions like this for spatial population models in ecology.
77.7 will show that 63.7 leads to the model in 77.5, and then 77.6 will show how this gives a shock moving at the same wave velocity as our discontinuous shock, but the density ρ(x, t) is now a smooth function (f (x − Vt)). Detailed hints below.
– For 77.5b), we have that the conservation equation 
= q(a, t) − q(b, t) is the same as before, and correspondingly 
+ 
= 0 is still true. What is different is that q is no longer q = q(ρ) so we do not get 
+ 
= 0. You can still assume that q = uρ though, and go from there to get 77.5c).
77.6 shows how adding the diffusion term effects a shock. This one is pretty hard without hints:
– 77.6b) Integrate 77.6a) and you will get an equation that can be written νf′ = Q(f ),
where Q(f ) is a quadratic function (parabola). By ‘graphical methods’, he means the method of sketching the solutions to an autonomous ODE by using a sign chart and checking where the solutions have to be increasing/decreasing. This should be covered in 33B, but in case you haven’t seen it you can look at my ODE notes. His assumption that f → ρ 1 and f → ρ2 at 干∞ means that ρ1 and ρ2 have to be zeroes of Q(f ). Now do the sign chart and see if f has to be increasing or decreasing for there to be such a solution. (Check if the parabola points up or down.)
– 77.6c) See the first equation above for f′ with integration constant C. From your sketch in b) you should see that f′ (x) → 0 as x → 干∞ . Use this to get an expressions for C in
terms of both ρ1 = lim f (x) and ρ2 = lim f (x). Set them equal to solve for V.
x→−∞ x→∞
So 77.6bc) says that when a lower density region of traffic ρ1 (higher speed) runs into a higher density region of traffic ρ2 (lower speed), we now have a smooth, rather than discontinuous, shock that moves with the same speed as before. The specific curve f (x) can actually be solved for by separation of variables, but there are so many constants etc. that it is a little too tedious to do it explicitly here. It turns out to be a (shifted) hyperbolic tangent. We’ll do the explicit solution for a slightly different parabola Q(f ) in Section 39 when we get to population modeling.
– 77.7a) Let U (ρ) be a fixed function, that represents the ideal velocity distribution, and u be the actual velocity. You want to accelerate when u(x, t) < U (ρ(x, t)) and decelerate when u(x, t) > U (ρ(x, t)). From this you can derive a simple first order ODE for u(x(t), t) (equivalent to a first order PDE for u(x, t) by 6
– 77.7b) replace the acceleration in your ODE by something from 63.7, and see how this gives the equation of 77.5.
2023-07-27