Incoherent radiation---Cross term

[davon806]

Homework Statement

Please see the attached,this was extracted from my notes.(Left side is the 1st page).I was trying to prove the sentence underlined (on the right page). Given ε(r,t) , how can I prove that the cross-terms average to 0? Actually I don't really understand why there are cross-terms. Would be much appreciated if someone can elaborate the paragraph on right page in detail !

Homework Equations

The Attempt at a Solution

To calculate the intensity and energy density, using the formula on left page, I suppose we need to find ε(r,t) ^2 by taking its dot product. However,this would give me a double integral which I don't know how to proceed...

 

[Charles Link]

If you sum up and have a bunch of different frequencies (or take an F.T. and get a sum of $E(\omega)exp^{i \omega t}$ over all frequencies), when you square the expression, the terms involving two frequencies (the cross terms) each time average to zero.