Why is wave intensity defined as displacement^2 rather than just its magnitude?

[jeebs]

pretty straightforward question. for a long time I've been blindly calculating that intensity is (displacement from zero)^2, but never questioned why.

So why is this? I get that by squaring you get a positive value, which helps in, say, quantum probability calculations, but what's wrong with just taking a magnitude of a wave at a given point and calling that the intensity?
is it just one of our arbitrary conventions in physics or is there some real reason?

 

[Fightfish]

Intensity is a classical concept, so let's leave the quantum realm alone for now.
To answer your question, we have to look back at what intensity is defined as: crudely, it is power delivered per unit area (albeit perpendicular to the propagation), which works out to energy per unit time per unit area.

Now, the energy carried by a wave is proportional to the square of its amplitude. This can best be illustrated with mechanical waves, such as a wave on a rope, in which every point is essentially in SHM. Consequently, intensity of a wave is also proportional to (amplitude^2) of the wave.