What's a wave?

So, what's a wave? In his deservedly ubiquitous undergrad electrodynamics textbook, David Griffiths emphasizes the fact that the whole idea is pretty nebulous. Any rigid definition is likely to exclude things that are usually thought of as waves or to include things that aren't. He suggests that one possible vague definition is "a disturbance of a continuous medium that propagates with a fixed shape at a constant velocity". He goes on:

Immediate I must add qualifiers: In the presence of absorption, the wave will diminish in size as it moves; if the medium is dispersive different frequencies travel at different speeds; in two or three dimensions, as the wave spreads out its amplitude will decrease; and of course standing waves don't propagate at all.

I might add that the medium of propagation doesn't need to be continuous either - phonons are probably the most obvious example. But in any event, say you have a wave, whatever it happens to be. Take a snapshot of it with your camera. Assuming the wave exists in one dimension, its shape is given by some function f(x). In two dimensions, frozen as a snapshot in time, maybe it's a function f(x,y) which is more complicated but conceptually identical:

Alternately, you might put a cork in the water at a fixed position and watch it bob up and down. In some sense this is a snapshot in space, with time flowing by in the normal way. In this case, your wave is given by some function g(t), which describes the up-and-down motion at a fixed point in space:

(Both images from Wikipedia) Now, as Griffiths and I have been at pains to point out, the essence of a wave is the connection between these two descriptions. We've talked about f(x) and g(t), but since a wave holds its shape as it propagates, it must be true that for a wave moving at a speed v where vt = x, the function describing the wave has to stay identically the same if x is displaced by vt. In other words, the function has to be of the form f(x - vt).

But this description is usually really inconvenient for calculations. Physicists usually prefer to use one of the one-variable descriptions where the function is just written as f(t) or f(x), with the other variable held fixed. This doesn't matter so much here, but if the wave is dispersive - i.e., if v is a function of wavelength the two descriptions take on some important subtitles.

On Sunday we'll take a look at the mathematics of this, and the following week I'm going to spend some time going into a little detail about wave propagation in a dispersive medium. I have a paper coming out I'd like to blog, and it needs a bit of ground work to reduce the difficulty of nontechnical explanation. But it will be my first first-author paper, and at risk of immodesty, I can't resist explaining it at length on the internet. ;)