How Does Mathematical Theory Explain Multiple Wave Reflections?

[Redwaves]

Homework Statement

Impedance $Z_1, Z_3$ are separated by $Z_2$ with a thickness $L$.
$\psi_r = R\psi_i$
Show that the global reflection is $R = \frac{R_{12} + R_{23}e^{-i2\omega L/v_2} }{ 1 + R_{12}R_{23}e^{-i2\omega L/v_2}}$

Relevant Equations

$R_{12}$ means the wave is reflected at the boundary between 1 and 2, moving from 1 to 2.

I know for a wave moving from left to right, $\psi_i = Ae^{i(\omega t - k_1x)}$

The first reflection where $Z_1$ is $R_{12}Ae^{i(\omega t - k_1x)}$

The second reflection. The wave moves from 2 to the limit between 2 and 3 then reflect...
Thus, $T_{12}R_{23}T_{21} Ae^{i(\omega t - k_1 x - 2k_2 L)}$. Where $L = \frac{\lambda_2}{4}$ and $\lambda_2 = \frac{2\pi}{k_2}$ so $2k_2 L = \pi$

The third time the wave comes to $Z_1$. $T_{12}R_{23}R_{21}R_{23}T_{21}e^{i(\omega t - k_1 x -i4k_2L)}$

We have $R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} + T_{12}R_{23}R_{21}R_{23}T_{21}e^{-i4k_2 L} + ...$

$R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L}(1 + R_{23}R_{21}e^{-i2k_2 L})$

$R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} \sum_{n=0}^{\infty} (R_{23}R_{21}e^{-i2k_2 L})^n$

$R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} \sum_{n=0}^{\infty} (-R_{23}R_{12}e^{-i2k_2 L})^n$

Using the geometric series

$R = \frac{R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L}}{1 + R_{23}R_{12}e^{-i2k_2 L}}$

The dominator is right, but not the numerator. I really don't see where I made a mistake.

 

[TSny]

We have $R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} + T_{12}R_{23}R_{21}R_{23}T_{21}e^{-i4k_2 L} + ...$

$R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L}(1 + R_{23}T_{21}e^{-i2k_2 L})$

There is a mistake in going from the first line to the second line where you factored out $T_{12}R_{23}T_{21}e^{-i2k_2 L}$. Check the last term in the parentheses in the second line.

 

[Redwaves]

There is a mistake in going from the first line to the second line where you factored out $T_{12}R_{23}T_{21}e^{-i2k_2 L}$. Check the last term in the parentheses in the second line.

It's a typo, I have $R_{21}R_{23}$ on my sheet. For some reason, I don't have any more preview while I type. It's the second term that shouldn't have the T's

 

[TSny]

$R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} \sum_{n=0}^{\infty} (-R_{23}R_{12}e^{-i2k_2 L})^n$

This looks good.

Using the geometric series

$R = \frac{R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L}}{1 + R_{23}R_{12}e^{-i2k_2 L}}$

The dominator is right, but not the numerator. I really don't see where I made a mistake.

Shouldn't this be $$R = R_{12} + \frac{T_{12}R_{23}T_{21}e^{-i2k_2 L}}{1 + R_{23}R_{12}e^{-i2k_2 L}}$$
You should be able to reduce this to the result stated in the problem. You will need to know the relation between $T_{12}$ and $T_{21}$ and the relation between $T_{12}^2$ and $R_{12}^2$. [EDIT: I should have said that you need the relation between $T_{12}T_{21}$ and $R_{12}^2$]

 

[Redwaves]

I don't know if that is what you mean.
$T_{12} = 1 + R_{12}, T_{21} = 1 + R_{21} = 1 - R_{12}$
However, I have $1 - R_{12}^2$

 

[TSny]

$T_{12} = 1 + R_{12}, T_{21} = 1 + R_{21} = 1 - R_{12}$

Using these relations, what do you get for $T_{12}T_{21}$ expressed in terms of $R_{12}$?

 

[Redwaves]

I get 1 - $R_{12}^2$

 

[TSny]

I get 1 - $R_{12}^2$

OK.

Using $T_{12}T_{21} = 1-R_{12}^2$ you should be able to reduce $R = R_{12} + \large \frac{T_{12}R_{23}T_{21}e^{-i2k_2 L}}{1 + R_{23}R_{12}e^{-i2k_2 L}}$ to the desired result.

 

[Redwaves]

Thanks! My issue was that I had put$R_{12}$ on the same denominator right after the geometric series.