# 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. ðŸ˜‰

1. #1 joemac53
April 7, 2011

Can’t wait. You go!

2. #2 Juice
April 7, 2011

If a wave requires a medium then what is the medium for electromagnetic waves?

3. #3 complex field
April 7, 2011

Electromagnetic waves do not need a medium

4. #4 Wigie
April 7, 2011

If a wave requires a medium then what is the medium for electromagnetic waves?

Answer 1: The EM wave itself. A sinusoidally varying electric field produces a sinusoidally varying magnetic field, which produces a sinusoidally varying electric field, which…

Answer 2: In a quantum field theory picture, the very *non*-empty “vacuum”. All those virtual particle-antiparticle pairs being created and annihilated mediate the propagation of photons.

5. #5 Pseudonym
April 8, 2011

A wave is any phenomenon whose evolution or state satisfies a wave equation.

No, I don’t have a definition of a wave equation. But you’re welcome.

6. #6 lordaxil
April 8, 2011

If you think of a wave as being a fairly non-local excitation of some underlying quantum field, then this picture includes both phonons and photons (and any other particle or quasi-particle you care to consider). All of the wave propagation mediums are discontinuous at some scale.

As a postscript to Wigie’s answer 2, it is important to realise that the quantum field in which EM waves propagate transforms in a Lorenz-invariant fashion, so that it is never possible to define a state of absolute rest with respect to the “medium” in which the wave propagates.

7. #7 In Hell's Kitchen (NYC)
April 9, 2011

“Answer 2: In a quantum field theory picture, the very *non*-empty “vacuum”. All those virtual particle-antiparticle pairs being created and annihilated mediate the propagation of photons.”

So we can calculate the permittivity and permeability of vacuum (the “medium” properties) from this quantum field theory picture ? I would love to see a reference to this calculation !

8. #8 RM
April 9, 2011

Pseudonym @5 – It’s not that hard: A wave equation is simply an equation whose form describes the evolution or state of a wave.

It’s like Justice Stewart’s definition of obscenity: “I know it when I see it”. — wave is a four-letter word, after all.

9. #9 Juice
April 9, 2011

Electromagnetic waves do not need a medium

Posted by: complex field

Then David Griffith’s definition is bad.

10. #10 complex field
April 10, 2011

cf. “possible”, “vague” and “nebulous” in Matt’s first paragraph. Moreover, the Michelson-Morley experiment demonstrated that EM waves do not require a material medium through which to propagate. Which is not to say that they *can’t*; clearly they can (eg, water, glass at optical wavelengths). They just do not *require* it (eg, radio waves in outer space).