Happy Easter, everyone!
This Sunday Function is going to be short and sweet, since unfortunately I’ve got a lot to do before Monday. Let’s get down to business!
Here’s a function I’ve made up out of thin air. It’s pretty arbitrary – in fact, it’s discontinuous at x = -1/2 and not differentiable at x = 0:
Here is a huge pile of sine and cosine waves, with one bonus constant line which, if you’d like, you can think of as a cosine wave with infinite wavelength:
And here is what you get if you add all of those waves together and graph the result:
It looks quite a bit like the random crazy function I posted first, right? Sure the resemblance is a little hazy, but it is there. What I’ve done is to pick the relative sizes of those waves very carefully to reproduce my original function as closely as possible. As it is, this is pretty decent for only including a total of 9 waves. Could I make it more accurate by adding more waves of even smaller wavelength? You bet. I won’t plot the waves separately since the graph would be an illegible mess, but here’s a plot of a total of 61 waves:
Remarkable, right? By adding more and more waves I could get better and better, and it’s mathematically provable that I can make the error smaller than any arbitrarily tiny amount by simply adding enough waves.
The key thing to notice is that the original graph didn’t have anything to do with waves. But through knowledge of the mathematics of waves – of Fourier series, formally – I can find a relationship between that wacky arbitrary function and the well-known seemingly mundane trig functions. It gives insight into a relationship that otherwise would have been hidden.
I bring this up because of a feisty discussion that erupted a couple items down in the “What is light?” post. A commenter of unorthodox views is determined to believe that there must be a physical medium for light to “wave”, despite the null result of Michelson-Morley and others and the astonishing success of both the classical covariant formulation of Maxwell’s wave mechanics and the modern quantum electrodynamic formulation. This doesn’t convince him because all other waves in his experience actually displace something in position – water, sound, etc.
And of course this is true, if you define a wave as “objects undergoing positional displacement in a certain pattern”. But that’s an impoverished and stagnating view. It’s akin to insisting that Bill Gates is poor because he doesn’t physically carry much cash. The whole point in physics is to recognize and mathematically express connections that reveal different and diverse phenomena to be aspects of the same underlying principle. Sure, positional displacement is one kind of wave. It’s not the only kind. You can have waves of heat, pressure, gravitational field strength, you name it. Some of these can be thought of as mediated by a physical medium being displaced in position, and others can’t. The underlying mathematics is the same, and we now have the power to explain very different problems with simple and unified principles.
Perhaps this reads more like Sunday Sermon! Well, it is Easter, so I hope that such a thing is forgivable.