Subject: Law of Superposition: if displacements add , what about the energy?

My colleagues and I are unable to find a satisfactory answer to the following 
apparent dlimma regarding the law of superposuition of waves: If the 
displacements add, and wave energy is proportional to ampltude squared, the 
energies do NOT add. Example: two sinusoidal waves of amplitude A and energy E 
interfere constructivley to add to a new wave of amplitude 2A. However, this 
should produce a wave of 4 times the original energy of one wave while the sum 
of the energies of the two waves should add to 2E, not 4E. I cannot find a 
single text that addresses this point clearly.

Re: Law of Superposition: if displacements add , what about the energy?