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The given state:
Let $\psi$ be a wave that passes from medium $a$ to medium $b$.
Let $A$ be the amplitude of $\psi$.
Let $R$ be the amplitude ratio of the reflected wave $\psi_r$ and the original one, and $T$ the amplitude ratio of the transmitted wave $\psi_t$ and the original one.

The question: What is the intensity of the transmitted wave?

An attempt:
The condition at the boundary of the two media demands that $1+R=T$.
The intensity of a wave is $I_\psi=|\psi|^2$.

On one hand we have $I_t\propto T^2=1+2R+R^2$.
But on the other hand, assuming that the intensity is preserved and that intensity is additive, we also have $I_t=I_\psi - I_r \propto T^2=1-R^2$.

Assuming that $R\neq 0$ there is a contradiction between the 2 answers.
Which one is the right answer?

Thanks.

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Why did you say $1 + R = T$ in the first place? –  Muphrid Feb 4 '13 at 21:07
    
Also, it really depends on your boundary conditions...what are they? The second equation is usually true because it represents a quantity proportional to energy, or sometimes to probability. The amplitudes can not sum to 1 for example, for refraction of light for example... –  daaxix Feb 4 '13 at 22:17
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