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

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thuanthuan
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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.
 

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