Wave equation first

This semester I kick off my general physics 2 course with waves. I really want the early focus to be on what waves are, and, more specifically, what the wave equation means. The reason I want to do this is because the wave equation is, for me, the biggest hallmark of a system being able to support waves. If you look at a system and find any parameter whose spatial curvature is proportional to its acceleration, you know it can support a wave and you can determine the speed of the wave from the proportionality constant.

When teaching about the speed of a wave on a string, which is often the first example that texts “cover,” I get frustrated with the various approaches to determining the speed of the wave, since often it’s a very contrived situation. Typically it’s considering a situation of moving one end up at a constant speed. That leads to a non-calculus calculation of the speed, but I’ve always thought it felt too contrived. I much prefer convincing the students that the wave equation is what to watch for and to simply look at the proportionality constant.

What I thought I’d do is start by asking my students about examples of waves. I don’t really want to get bogged down in SHM stuff, so really my hope is that we land at something like “disturbances that travel through the medium.” Then I want to see how far we can get talking about why the propagation happens. For a string, that’ll be talking about portions of the string where there’s a curvature (or, really, a change in slope). The idea would be to realize that a portion of the string that’s a constant slope will not have a force pulling the string up (or down) since the (constant) tension will balance out. However, if there’s a change in slope, then there will be a net force up (or down). So, a change in the slope will lead to an acceleration. The wave equation!

To figure out that the proportionality constant is related to the speed of the wave, I really want them to have a sense that any function f(x) simply turns in to f(x-vt). I’ll try to get to that by getting the students to comment on the similarities and differences of the motion of different portions of a string that has a disturbance traveling down (for now without friction). I’ll be hoping for thoughts like “they all do the same thing, only delayed” and “they’re delayed by how long it takes the disturbance to get there.”

If I can do both of those last paragraphs, then getting that $v=\sqrt{T/\mu}$ should be relatively straightforward.

So what about using this approach with other systems? Sound in air, for example? Well, what causes sound. Again you have to have a change in the pressure distribution or nothing happens. Again you’d have a second order spatial change proportional to an acceleration (of the air particles, say). I think it could work, but it’s not as clear cut as the string example.

Where this approach will come in handy is when we get to Maxwell’s equations and light. You futz with Maxwell’s equations and you end at a proportionality between the second spatial derivative of E (or B) and the second temporal derivative of E (or B) with the permeability and permitivity of free space sitting there as the proportionality constant.

• Why do you put the word “cover” in quotes?
• I really like the “one end moving up at a constant speed” approach. I like it better than this because . . .
• I, too, don’t really like the “one end moving up at a constant speed” approach. What I do instead (which is MUCH better than your plan) is . . .
• I don’t get all this. Just give them the equation and assign some homework.
• With friction the notion that “everyone does the same thing, only delayed” doesn’t work. How do you deal with that?
• In multiple dimensions the notion that “everyone does the same thing, only delayed” doesn’t work. How do you deal with that?
• I like this approach a lot. Please let us know how it goes!
• With sound in air, here’s a good way to get students to think about a second spatial derivative needed . . .
• For light, it’s not an acceleration. Will your students be ok with that?