6 ms·
I think it’s better to know that sometimes we only know how to describe something by relating rates of change to other states. And that’s ok. Maybe it has a clo
by docfort 5y ago
I think it’s better to know that sometimes we only know how to describe something by relating rates of change to other states. And that’s ok. Maybe it has a closed form equation, or maybe can only be solved numerically. But if I see that a differential equation looks like a wave equation, then I get intuition that it’s describing waves. And why do the waves appear? Because the physical process the PDE describes has a speed limit on information passing from time into space!
Don’t like traffic waves? Well, why is there some limit on spatial information connected to temporal information? It’s because I cannot see through the cars in front of me. The “fog of war” creates the waves. The denser the fog (e.g. I’m surrounded by semitrucks), the greater the likelihood of waves developing.
This intuition is formed by being able to recognize the form of the PDE with general knowledge of the solutions, without needing to actually solve the PDE. Sure, additional insights are possible if you solve it, but knowing that traffic is like springs gives you leverage to use your ordinary intuition to understand unfamiliar things.
Point of fact, James Maxwell of E&M fame saw the wave equation and the separate electric and magnetic field PDEs and came up with a detailed spring model to give himself a more familiar analog to play with.
- dan-robertson 5y agoTo give Maxwell a little more credit (not that you aren’t), the wave equations and PDEs of today are much nicer thanks to modern knowledge and computational techniques. Maxwell didn’t have div, grad or curl and so he had dozens of equations to look at instead of just a few, and I think the terms and patterns weren’t as well known as they are today.
- aidenn0 5y agoDo you know if the surface/line integral forms taught to those without vector-calculus under their belts an actual stepping stone to the modern ones? The number of equations are the same, they are just a lot more hairy.
- dan-robertson 5y agoI don’t know enough about the history and I don’t know what forms you are talking about. When I learned ‘vector calculus’ at university, we were introduced to surface/line integrals and div/grad/curl at around the same time.
- bernulli 5y agoIt's really cool how a Mach number emerges from traffic flow, with speed of cars vs speed of information, completely with shock waves and everything!
- pfdietz 5y agoThe equation you want for this is the Burgers Equation. https://en.wikipedia.org/wiki/Burgers%27_equation https://en.wikipedia.org/wiki/Burgers%27_equation This equation was initially thought of as the appropriate continuous version of the discrete problem investigated by Fermi-Pasta-Ulam-Tsingou (on a very early digital computer shortly before Fermi's death), but then it was realized the KdV equation was better for that. https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_problem https://en.wikipedia.org/wiki/Fermi%E2%80%93Pasta%E2%80%93Ul... https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equation https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equa...
- FabHK 5y agoNice short course on edX, if you want to see it in action: Intro to Traffic Flow Modeling and Intelligent Transport Systems (Note that thanks to aggressive monetisation of MOOCs they shut off access to the course a few weeks after you enrol in it.) https://www.edx.org/course/intro-to-traffic-flow-modeling-and-intelligent-tra https://www.edx.org/course/intro-to-traffic-flow-modeling-an...
- agumonkey 5y agoemergent oscillations from partial information seems interesting, does this have a name ?
- docfort 5y agoThe fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initial conditions, and maybe a forcing function (energy or information pump). Change one of those things and you’ll get a different solution. You might be reminded of fractals, which are a special case of iterated function systems, roughly a discrete version of iterated function systems. Or maybe you might think of cellular automata, all based on local update rules and some initial conditions. The answer to your actual question is literally called the wave equation (first in the list on the linked webpage). The left side talks about some variable u and how it changes over time. The right side is how u changes over space. And the two sides are linked via a constant c. By observing the solution or by working out the units, we can understand c to be the phase velocity, or roughly, the wave speed. So the way that u evolves over space is limited by how it evolves over time (and vice versa)! u cannot react to all things in space instantaneously. Therefore, a wave emerges, carrying updates from one part of space to another.