## Friday, September 28, 2007

### The Superposition Principle

For readers that aren't familiar with this principle, I'll give a quick introduction. The idea behind it says that you can add things. A more formal definition says that when you add two linear functions, the result is still a linear function. Woo hoo.

This is probably easiest to see with a few examples. The most common one is for waves. If you take two waves, and they overlap, at some points, you'll get constructive interference and the total wave will be even bigger. In other places, you'll get destructive interference and there won't be much of anything. You can find a nifty little Java Applet here to play with that idea if it doesn't make sense.

Another example is a bit more buggy. Most of us are familiar with the annoying whine of cicadas that crop up every summer. However, not every species comes out every year. Various breeds' eggs stay buried for longer periods of time. Most notably, there's a 13 year species, and a 17 year version. A few summers back (I believe it was 2004), both happened to be out at the same time. If one species was out, you'd have their drone, but because both species were out at the same, the superposition principle was in effect and the noise was twice as loud. Annoying, but we don't have to worry about it for another 221 years from then, which will be the next time both species are out at the same time.

But we can apply this to lots of things with periodicity. In my case, homework, tests, and meetings often come with various fixed periods. Weekly, I have ~3 homework assignments. Every ~3.5 weeks, I have another class that gives an assignment. Tests come every ~6 weeks with a bit more spread. Undergraduate committee meetings about every ~4 weeks. Lab reports every 3 weeks.

Just like cicadas, this means that every often though, all of these different things will end up piling up in the same week. Anyone want to take a stab at why I'm damn glad it's finally the weekend?