I'm pretty sure I've seen the answer to this somewhere before but I can't seem to reason through it again... hopefully you don't mind helping me out.

Suppose I have two EM plane waves w/ electric field magnitudes of E1 and E2 respectively. For convenience, assume that E1=E2=E. We know that the power of each of the respective waves is proportional to abs(E)^2. Now, if we assume that the two waves are in phase and we apply superposition we should get a net power proportional to abs(2*E)^2 or 4*abs(E)^2. In other words, the superimposed wave has 4 times the power of the original wave. This is contradictory to my intuition... it seems like if I put two waves together, both with a given power, I should end up with twice the power not 4 times the power.

So, ultimately the question is, what's going on here? Am I correct, should I really be ending up with a superimposed wave that has 4 times the power as either of the original waves?

Thanks in advance for the help.