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Redwaves

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- 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.

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.

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