Today I was helping some students of mine get ready for my final. These are science teachers who are in an alternative licensure program for physics. We were talking about the equations for the resonant frequencies in various systems and we developed some great insights about how to approach the equations involved.
One of the things we talked about were the similarities and differences between these two equations:
Here the left equation is for resonant systems where the two ends are the same (a string tied at both ends, or a tube open at both ends) and the right equation is for systems where the ends are different (a tube closed at only one end). We had derived both of these equations earlier in class and the students were asking why one has a 2 and one has a 4 in the denominator. As I was reminding them of the derivation, it hit me that they really both have a 4 in the denominator but we divide out a common factor of 2 in that case. In this post I’m arguing that it’s a mistake to do that, because, if you don’t, you get only one equation where the math leads to the physical principles.
Here’s what I mean. To derive either equation, I encourage my students to find patterns of nodes and antinodes that obey the appropriate rules. Really there’s only two rules: make sure the ends are correct (both nodes? both antinodes? one of each?), and nodes and antinodes should alternate.
Here’s what their whiteboards start to look like while studying a system with nodes on both ends:
N A N
N A N A N
N A N A N A N
After three they tend to get the pattern. Now we do some analysis. Prior to this point we’ve worked out that the distance between any two nodes and antinodes is a quarter of a wavelength. For the pattern above we find that there’s always an even number of those quarter wavelengths:
When they analyze the other type of system they find that you always have an odd number of quarter wavelengths, leading to the right equation up above.
Here’s my point, if you don’t cancel the common factor of 2 in the second step above, you end up with
which looks the same as the equation for the systems with different ends, only you use even numbers instead of odds. What my students and I discussed this morning was that if you leave it that way, you have one equation for both applications. The “4” that they asked about is immediately traced to the fact that there’s is a quarter wavelength between nodes and antinodes. You use evens for one and odds for the other because of the nature of the ends.
I like this way of thinking about it better than what I usually do (the first equation of this post). Later today we were discussing another situation and I found another time not to cancel something. This time we were studying the diffraction pattern of a single slit. We thought about ways that the infinite sources inside the slit could conspire to give a dark spot. We reasoned that if the top half cancelled the bottom half, we’d be in business. Of course, this is how a lot of texts explain the first dark spot in the single slit pattern.
When you get to the math of it, you find that the very top portion of the slit cancels with the portion just below the mid line. The analysis at that point is very similar to finding a dark spot in a double slit pattern () only you replace d with half of the spacing:
There’s nothing wrong with this result, but I commented to my students that many texts simplify it to . My point is that leaving in the factors of 2 lets you see the physics: it’s a/2 because the top half and the bottom half cancel each other. It’s /2 because you’re trying to get destructive interference.
What are some other examples of times with math simplification leads to physics obfuscation?