How Does Mathematical Theory Explain Multiple Wave Reflections?

In summary: I should have put ##R_{23}R_{21}##.In summary, the wave moves from left to right, then reflects back twice. The first reflection is when ##Z_1## is ##R_{12}Ae^{i(\omega t - k_1x)}##. The second reflection is when the wave moves from 2 to the limit between 2 and 3, and reflects back again. The third time the wave comes to ##Z_1##, it is reflected once more. Using the geometric series, we have ##R = R_{12} + T_{12}R_{23}T_{21}e^{-i2k_2 L} + T_{12}R_{23
  • #1
Redwaves
134
7
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.
 
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  • #2
Redwaves said:
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.
 
  • #3
TSny said:
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
 
  • #4
Redwaves said:
##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.

Redwaves said:
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##]
 
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  • #5
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##
 
  • #6
Redwaves said:
##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}##?
 
  • #7
I get 1 - ##R_{12}^2##
 
  • #8
Redwaves said:
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.
 
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  • #9
Thanks! My issue was that I had put##R_{12}## on the same denominator right after the geometric series.
 

Related to How Does Mathematical Theory Explain Multiple Wave Reflections?

1. What is wave reflection?

Wave reflection is the process by which a wave bounces off of a surface and travels back in the opposite direction.

2. How does wave reflection occur?

Wave reflection occurs when a wave encounters a boundary or obstacle that causes it to change direction. This can happen with any type of wave, including light, sound, and water waves.

3. What factors affect the amount of wave reflection?

The amount of wave reflection is affected by the angle of incidence (the angle at which the wave hits the surface), the properties of the medium the wave is traveling through, and the shape and texture of the surface it encounters.

4. How is wave reflection mathematically represented?

Wave reflection can be mathematically represented using the law of reflection, which states that the angle of incidence is equal to the angle of reflection. This can be expressed using the equation θi = θr, where θi is the angle of incidence and θr is the angle of reflection.

5. What are some real-world applications of wave reflection?

Wave reflection has many practical applications, including in the fields of acoustics, optics, and seismology. It is used in technologies such as sonar, radar, and ultrasound imaging, and is also important in understanding phenomena such as earthquakes and the behavior of light.

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