Trying to understand a derivation in a paper

In summary, the conversation is about a clarification question for a project regarding the derivation of the effective permeability of two stacked media. The conversation involves discussing the notation and solution of equation (9) in the paper "Electromagnetic Properties of a Finely Stratified Medium" by S.M. Rytov in 1956. The conversation includes suggestions on how to solve the quadratic equation and clarifications on the symbols used in the equation.
  • #1
Hello PF,
first of all I don't know where to put this post as it's not exactly a homework question but a clarification question for a project.
I'm going through the derivation of the effective permeability of two stacked medias, given the polarization of an incoming EM wave but I'm stuck at the point shown in the picture.
The title of the paper is "Electromagnetic Properties of a finely Stratified Medium" by S.M. Rytov in 1956 (I can share it if it's needed).

Homework Statement


HnjA9DP.png

Homework Equations


$$\frac{sin(x)}{cos(x)} = tg(x)$$

The Attempt at a Solution


I don't understand the notation they use and how they solve equation (9).
Does it mean $$\frac{tg\frac{b\alpha_2}{2}}{tg\frac{a\alpha_1}{2}}$$ is equal to [itex] -\chi [/itex] and [itex]-1/\chi[/itex]? Or how is it supposed to be understood? :oldsmile:
Regarding how to solve the equation the closest I get is this: $$\frac{1+\chi^2}{2\chi}\ \ tg(a\alpha_1)\ \ tg(b\alpha_2)+(cos(a\alpha_1)\ \ cos(b\alpha_2))^{-1} = 1 $$
 

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  • #2
AwesomeTrains said:
the closest I get is this
(Is that how you solve a quadratic equation ?
(9) is of the form ##a\kappa^2 + b\kappa + c = 0 ## and you are supposed to be able to solve for ##\kappa## :rolleyes:
 
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  • #3
I think I got it to work.
Suggestion: Let ##a \alpha_1=\theta ## and ## b \alpha_2=\phi ##. You then get:
## x=\frac{-2(1-cos(\theta)cos(\phi)) \pm 2\sqrt{(cos(\theta)-cos(\phi))^2}}{2 sin(\theta)sin(\phi)} ##.
(This was obtained with the quadratic formula, (## x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} ##, when starting with ## ax^2+bx+c=0 ##), and a little algebra/trig to simplify the terms inside the square root).
Next step is to let ## \theta=2 u ## and ## \phi=2v ##, and expand the ## cos(2u) ## and ## cos(2v) ## and ## sin(2u) ## and ## sin(2v) ## etc.
(Remove the square root sign of course and work with the two separate solutions).
For the "+" sign, I believe you get ## x=-\frac{tan(u)}{tan(v) } ##, and for the "-" sign, ## x=-\frac{tan(v)}{tan(u)} ##.
 
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  • #4
BvU said:
(Is that how you solve a quadratic equation ?
(9) is of hte form ##a\kappa^2 + b\kappa + c = 0 ## and you are supposed to be able to solve for ##\kappa## :rolleyes:
Thanks for the response. Sorry for the bad quality of the picture, the font is not so clear. It's a chi and not a k, k is in the alphas.
I should have written all the definitions in the first post to clarify, my bad :oldeyes:
[itex]\alpha_1 = k \sqrt{n_1^2-n^2}[/itex], [itex]\alpha_2 = k \sqrt{n_2^2-n^2}[/itex] and [itex]\chi=\frac{\mu_2 \alpha_1}{\mu_1 \alpha_2}[/itex] [itex]\\ (n_1^2 = \epsilon_1 \mu_1, n_2^2 = \epsilon_2 \mu_2)[/itex]
 
  • #5
##\kappa## or ##\chi##, it's still a quadratic equation :rolleyes:, only this time in ##\chi##, so you solve for ##\chi## :wink: .
 
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  • #6
Much simpler just to call it ## x ##. Keeping the symbols as short as possible makes it much easier to type up the solution.
 
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  • #7
Thanks for the help! I should be able to get it now, I was trying hard to solve for the k in the alphas :oldeyes:
 
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1. What is a derivation in a scientific paper?

A derivation in a scientific paper is a logical and step-by-step explanation of how a result or conclusion was reached. It is often used to support a theory or hypothesis by providing evidence or mathematical calculations.

2. Why is it important to understand a derivation in a paper?

Understanding a derivation in a paper is crucial because it allows readers to evaluate the validity and reliability of the results presented. It also helps to replicate the study or apply the findings in other research.

3. How can I understand a derivation in a paper if I am not familiar with the equations or methods used?

If you are not familiar with the equations or methods used in a derivation, it is best to start by looking up the definitions and explanations of those terms. You can also seek assistance from colleagues or consult a textbook or online resources for further clarification.

4. What should I do if I still do not understand a derivation in a paper?

If you are still having trouble understanding a derivation, it is recommended to reach out to the authors of the paper for clarification. They may be able to provide additional explanations or resources to help you better understand the derivation.

5. Are there any common mistakes or pitfalls to avoid when trying to understand a derivation in a paper?

Some common mistakes or pitfalls when trying to understand a derivation in a paper include not fully understanding the assumptions or limitations of the equations or methods used, skipping steps in the derivation, and not seeking help or clarification when needed. It is important to take the time to carefully review and understand each step in the derivation to avoid these pitfalls.

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