Why are the calculated energies for two oscillating waves not equal?

[thuanthuan]

Hi everyone,
I have a question about waves.

Suppose there are two waves that make one water atom vibrate.
The first equation of oscillation is: x1= A1sin(wt+p1), energy propagated to the atom is (1/2)m.(w^2)(A1^2)
and the other: x2=A2sin(wt+p2), E2 = (1/2)m.(w^2)(A2^2)
so the total energy (mechanical) of the atom is (1/2)m.(w^2)(A1^2+A2^2)

On the other hand, the total displacement is x=x1+x2=A(sinwt+p), where A=A1^2+A2^2-2A1A2cos(p1-p2)
so the energy = (1/2)m.(w^2)(A^2)

but A^2 doesn't equal to A1^2+A2^2

so the two energy calculated are not equal

what's wrong with my calculation ?

Thank you.

 

[thuanthuan]

A similar question, Suppose we have a thin film of glass with an index of diffraction is n
The upper and lower surfaces of the flim is parellel. Light is incident to the flim.

So a portion of light is reflected at the upper surface, the rest is refracted into glass. Light continues to the lower surface, then one portion of it is refracted back into air, one portion is reflected.

Someone say that to minimize the light loss due to reflection (that is, the final light traveling back to the air is maximized), then two reflected line must destructively interfere with each other. Is that true ? Can you explain why ?