I'm sorry but I failed to mention that this book has two revisions, one of which represents a HUGE change to the book. It was entierly rewritten, every diagram redone, all information re-sourced.
The paragraph I'm speaking of does not appear in the original book. Our understanding of propegation of VLF hadn't changed much but the way it is explained, and our understanding of different extremely important phenomena, are not described in the first edition. (One of the most noticable cases of this is the only mentioning of Whistlers in the first book is in relation to Exospehric events which is definetly not the case).
I'll recomment on this when I get into work and take a photo of the specific paragraph. I'll show it to you and maybe you can better describe what it is. If you've got an email in your description I'll shoot one off when I add it.
> The variables are mnemonic (_R_eflection), vary systematically and use pictographic symbols standardized in another field, and most importantly all of this is explained.
What's the mnemonic of "X"?
> Using single-word variable names might still be a step up for readability when writing equations. But consider that when using the equation, you will end up writing those variables over and over again, with just slight modifications in parts of the formula
And? I'm fine with that. If anything, as a programmer we can write IDEs for math majors. Maybe we might actually kill of LaTeX and get something that makes some sense to fill it's place.
> But when you need the expanded form of the expression to apply some simplification, you can't really do that, short of simply not writing down the step. Mathematical notation is really better suited for this kind of frequent manipulation and rewriting.
My MathIDE can with one click transform your reduced equation into an expanded form. It would also definetly possible to turn an expanded form into a reduced form.
> Nonetheless, I think some of the lessons learned in programming concerning the use of dense formal languages might transfer to mathematics. One example is explicit declaration before use, which would probably have helped your problem with X just as well as a descriptive variable name. Another is the IDE experience, where hovering over a symbol gives you the contextual information you need to understand it.
Also sent you an email, putting my response here for posterity.
First of all I'm sorry for responding so late, HN comments are somewhat of a fire-and-forget for me.
Second, I'm not much of a plasma physicist (or any kind of physicist, really). In fact, I was mostly interested in your example because I wanted to see how well I could understand it without the relevant background. Turns out, not very well. But I think I figured out what the fs are, so maybe that will help you.
Looking at the paragraph in your photo, I noticed that it mentions three kinds of frequencies: of the whistlers, of the plasma and the electron gyrofrequency. Since f is frequently used to denote frequencies in physics, I guess that f, f_N and f_L are these three. (The units of Hertz are another hint.)
Based on the description, the whistler frequency should be the smallest one. In the example, that is f. It remains to determine f_N and f_L. Unfortunately, the names are definitely not mnemonic, so I needed some other source. Turns out Wikipedia has an article on the Appleton-Hartree equation that defines all the variables it uses. (https://en.wikipedia.org/wiki/Appleton%E2%80%93Hartree_equat...)
There, f_0 and f_H are used for plasma frequency and gyro frequency. Also not mnemonic, but at least explained. I guess the relevant equation is in the section on quasi-longitudinal propagation. I simplified it even further by using the assumption that the angle theta = 0, which means that the sine is 0 and the cosine is 1. Then the equation becomes
n^2 = 1 - X/(1 ± Y)
and since the book assumes that X and Y are much larger than unity, this can be approximated by
Further assuming that the ratio is so large that the 1 can be dropped, this gives an expression for the refractive index
n = f_0/sqrt(f*f_H)
Matching terms with the equation in the book, f_N appears to denote the plasma frequency (f_0 on Wikipedia) and f_L is the gyro frequency (f_H on Wikipedia).
Mystery solved!
Now I completely agree with you that this kind of detective work shouldn't have been necessary and the full derivation should have been in an appendix or in a cited reference.
The paragraph I'm speaking of does not appear in the original book. Our understanding of propegation of VLF hadn't changed much but the way it is explained, and our understanding of different extremely important phenomena, are not described in the first edition. (One of the most noticable cases of this is the only mentioning of Whistlers in the first book is in relation to Exospehric events which is definetly not the case).
I'll recomment on this when I get into work and take a photo of the specific paragraph. I'll show it to you and maybe you can better describe what it is. If you've got an email in your description I'll shoot one off when I add it.
> The variables are mnemonic (_R_eflection), vary systematically and use pictographic symbols standardized in another field, and most importantly all of this is explained.
What's the mnemonic of "X"?
> Using single-word variable names might still be a step up for readability when writing equations. But consider that when using the equation, you will end up writing those variables over and over again, with just slight modifications in parts of the formula
And? I'm fine with that. If anything, as a programmer we can write IDEs for math majors. Maybe we might actually kill of LaTeX and get something that makes some sense to fill it's place.
> But when you need the expanded form of the expression to apply some simplification, you can't really do that, short of simply not writing down the step. Mathematical notation is really better suited for this kind of frequent manipulation and rewriting.
My MathIDE can with one click transform your reduced equation into an expanded form. It would also definetly possible to turn an expanded form into a reduced form.
> Nonetheless, I think some of the lessons learned in programming concerning the use of dense formal languages might transfer to mathematics. One example is explicit declaration before use, which would probably have helped your problem with X just as well as a descriptive variable name. Another is the IDE experience, where hovering over a symbol gives you the contextual information you need to understand it.
Yep I totally agree.