Proving that the incident intensity is not the same as the sum of others

[LCSphysicist]

Homework Statement

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Relevant Equations

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I was supposed to find the relation among the coefficients $T$ and $R$ which represent the amplitude of the reflected electric field and the transmitted electric field respectively, that is, $$E_{R} = E_{i} R, E_{T} = E_{i} T$$ as well as the coefficients $t$ and $r$, that represent the fractional part of the intensity incident, that is, $$I_{R} = I_{i} r, I_{T} = I_{i} t$$

In fact, assuming that the first medium is the air/vacuum, I was able to deduce correctly the relations $$r = R^2, t = nT^2$$ where $n$ is the index of refraction of the second medium.

After doing so, I should be able to show that the sum of the intensities of the reflected and the transmitted wave is not equal to the intensity of the incident wave. The main problem is that I used exactly this to drive my relations! That is:

$$S_i = S_t + S_r \implies \frac{B_i E_i}{\mu_1} = \frac{B_t E_t}{\mu_2} + \frac{B_r E_r}{\mu_1}$$
$$B = E/v \implies \frac{1}{\mu_1} = \frac{R^2}{\mu_1} + \frac{T^2 c}{c/n \cdot \mu_2}$$
$$ \mu_{1} \approx \mu_{2} \implies 1 = R^2 + n T^2$$

where I now call $r = R^2, t = n T^2$

So I think you can see why I am confused. How am I supposed to prove the relations between $r$, $t$, $R$, $T$; and how do I prove that the intensities are in fact not equal?

 

[haruspex]

Homework Statement:: .
Relevant Equations:: .

I was supposed to find the relation among the coefficients $T$ and $R$ which represent the amplitude of the reflected electric field and the transmitted electric field respectively, that is, $$E_{R} = E_{i} R, E_{T} = E_{i} T$$ as well as the coefficients $t$ and $r$, that represent the fractional part of the intensity incident, that is, $$I_{R} = I_{i} r, I_{T} = I_{i} t$$

In fact, assuming that the first medium is the air/vacuum, I was able to deduce correctly the relations $$r = R^2, t = nT^2$$ where $n$ is the index of refraction of the second medium.

After doing so, I should be able to show that the sum of the intensities of the reflected and the transmitted wave is not equal to the intensity of the incident wave. The main problem is that I used exactly this to drive my relations! That is:

$$S_i = S_t + S_r \implies \frac{B_i E_i}{\mu_1} = \frac{B_t E_t}{\mu_2} + \frac{B_r E_r}{\mu_1}$$
$$B = E/v \implies \frac{1}{\mu_1} = \frac{R^2}{\mu_1} + \frac{T^2 c}{c/n \cdot \mu_2}$$
$$ \mu_{1} \approx \mu_{2} \implies 1 = R^2 + n T^2$$

where I now call $r = R^2, t = n T^2$

So I think you can see why I am confused. How am I supposed to prove the relations between $r$, $t$, $R$, $T$; and how do I prove that the intensities are in fact not equal?

According to (1232) onwards at https://farside.ph.utexas.edu/teaching/em/lectures/node104.html the reflected and transmitted intensities do add up to the incident.
The notation is a little different, using R, T where you have r, t.

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