Is there something as instanteneous intensity of a wave?

[Coffee_]

At the moment I'm revising some interference and diffraction basics, and there is something that bothers me slightly and I can't quite figure it out.

The intensity of a wave over some area $dA$ is in general is $I=\frac{1}{dA} \frac{dE}{dt}$. Clearly for an electromagnetic wave falling on a surface, the part $\frac{dE}{dt}$ is not constant and depends on time. So intensity should be a function of time.

In every text I encounter they seem to DEFINE the intensity as being the average over 1 period of $c|A(t)|^{2}$ where $A$ is the deviation of the wave at the area of interest. Are they simply using more practical definitions, and technically I'm correct above in a general sense OR am I missing something?

 

[andrewkirk]

The instantaneous rate of energy transport is going to be a function of the instantaneous electric and magnetic fields.

Here's a link with more:
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/emwv.html

It gives the energy transport vector as

\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B} where \vec{E},\ \vec{B} are the electric and magnetic field vectors. Note that the direction of energy transport, being a cross product, is perpendicular to both, as one would expect.

By the way, that's the formula at a point. To get the energy transfer rate over an area, you'd need to integrate that formula over the area.