Exploring the Spectral Paradox - Comments

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In summary, the article discusses the "spectral paradox" which is a dichotomy in how power spectral density is calculated between frequency and wavelength. The paper by Soffer and Lynch suggests using a logarithmic representation instead of a linear one to avoid the paradox.
  • #1
trilobite
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Exploring the Spectral Paradox
SpectralParadox.png


Continue reading the Original PF Insights Post.
 
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  • #2
Nice article! I wasn't even aware that this paradox existed!
 
  • #3
Wow! Your article was like a good invention; obvious to everyone after being explained by the inventor.

Overall, the key point to remember is that one must be careful when interpreting spectral curves, especially their peaks

Is there a general rule to follow in data analysis that would let us avoid such paradoxes?
 
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  • #4
"rubbish" was my first reaction. haha good start.
But the apparent paradox is only there because such graphs don't have a full explanation of the scale units. Δf Hz is not equivalent to a fixed step size of Δλ m so different powers will be admitted into uniform steps of frequency than in uniform steps of wavelength, as you sweep across the frequency (or wavelength) spectrum.
f=c/λ
so
df/dλ = -c/λ2
and df = -cdλ/λ
That impresses a scale factor of 1/λ across the width of the wavelength spectrum. The peaks cannot coincide. The range covers around an octave so the effect is both significant and annoying.
Thinking about it, it doesn't even surprise me that the peak of our vision is at the peak of frequency sensitivity because it's the energy of the photons the counts and that is hf. (Permission to dump on me about that last point)
Would this paradoxical behaviour show itself in any filter design that's based on wavelength (some delay line filters, perhaps)? We are so used to using frequency analysis of circuits.
 
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  • #5
anorlunda said:
Is there a general rule to follow in data analysis that would let us avoid such paradoxes?
Yes: be aware of what is being plotted against what! (Do as I say and not as I do. :wink:)
 
  • #6
This isn't a paradox; it's simply that the power-spectral-density is being calculated as per unit wavelength differential instead of as per unit frequency unit. To call it a paradox would be like saying that it's paradoxical that an apple is an apple and and orange is an orange. I want my 5 minutes back.
 
  • #7
We could avoid this dichotomy by expressing Power Spectral Density as Power Log(spectral) density, i.e., instead of

W/(m^2 * delta(nm)),

use

W/(m^2 * delta(log(nm)))

This should give the same shape curve as

W/(m^2 * delta(log(THz)))

because

log(nm) = log(c)-log(THz),

thus removing the nonlinear mapping between the wavelength and frequency based metrics.

As with the original unit, denominator is area times an interval. Here, the interval in the measurement unit would actually be the difference of the logarithms of the ends of the interval, or the log of the ratio of the ends of the interval.

We don't even have to use logarithms if we force the interval in the measurement to be constant on a log scale, i.e., a constant ratio such as a milli-octave.
 
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  • #8
NH_EE said:
We could avoid this dichotomy by expressing Power Spectral Density as Power Log(spectral) density, i.e., instead of

W/(m^2 * delta(nm)),

use

W/(m^2 * delta(log(nm)))

This should give the same shape curve as

W/(m^2 * delta(log(THz)))

because

log(nm) = log(c)-log(THz),

thus removing the nonlinear mapping between the wavelength and frequency based metrics.

As with the original unit, denominator is area times an interval. Here, the interval in the measurement unit would actually be the difference of the logarithms of the ends of the interval, or the log of the ratio of the ends of the interval.

We don't even have to use logarithms if we force the interval in the measurement to be constant on a log scale, i.e., a constant ratio such as a milli-octave.
The paper by Soffer and Lynch does make a similar point concerning the alternative of using a logarithmic representation. However, the authors state that this method has "no special physical significance" and should not be "singled out as a preferred physical ... representation" for electromagnetic spectra, even though it does cause the wavelength and frequency peaks to coincide. As a wavelength, that peak for solar radiation is approximately 720 nm, according to the paper. Note that that value is very close to Heald's median point of 710 nm, as I suspect the mathematics imply, although Heald does describe the value as "physically meaningful!"

My own opinion is that, while all of the above points are valid and interesting, they don't directly address the "paradox" at issue here. That's because this kind of paradox is primarily a psychological or pedagogical matter, not a physical or mathematical contradiction. Granted, some people may not feel the paradox, at all. But for those (like me) who do, a detailed exploration into where that sense of paradox is coming from can perhaps be helpful in dispelling it.
 
  • #9
trilobite said:
a detailed exploration into where that sense of paradox is coming from can
I would think it could be a useful exercise to find other situations in Science where a peak in a curve shifts according to the units on the x axis.
Strangely, (visible) light tends to be specified in terms of wavelength (the fault of history and the result of the prism / diffraction grating). Other members of the EM spectrum tend to be measured in terms of frequency / energy. In my opinion, because frequency is the variable that remains the same, it is a more suitable measure. Using frequency would have prevented a lot of confused questions on PF and other places about the assumed change in frequency at a boundary. (because, they say fλ=c etc.)
 
  • #10
I'm quite confused about this part:
The important point here is that you get that same value whether you measure the area under the wavelength curve (Fig. 1) or under the frequency curve (Fig. 2), even though the two curves have different shapes!

Reference https://www.physicsforums.com/insights/exploring-spectral-paradox/
Check out the formulae in Hyperphysics and WikipediaWe see that:

##B_{\lambda}=\frac{du}{d\lambda}\cdot g \quad ,\quad B_{\nu}=\frac{du}{d\nu}\cdot g##

where: ##\quad g=\frac{hc}{\lambda}\cdot \frac{a}{e^{\frac{h\nu}{kT}}-1}##

a
is some coefficient.How do we show that the two integrals are equivalent?

##\int \frac{du}{d\lambda}\cdot g \quad d\lambda \quad, \quad \int \frac{du}{d\nu}\cdot g \quad d\nu##
 
  • #11
It's the Definite Integral that counts. The Integral between the limits of λ will be equal to the integral between equivalent limits of ν. You have to do the job completely.
There's more to the Definite Integral than just the inverse of Differentiation.
 
  • #12
sophiecentaur said:
The Integral between the limits of λ will be equal to the integral between equivalent limits of ν.
hmm, do you have a link to the mathematical proof?
 
  • #13
tade said:
hmm, do you have a link to the mathematical proof?
Is a proof needed that the energy between the limits of frequency and the corresponding limits of wavelength is the same? It's the same energy and the same conceptual filter letting it through. I know that Maths is usually required here but this constraint comes before the Maths - not after a proof (I would have thought). :smile:
 
  • #14
sophiecentaur said:
Is a proof needed that the energy between the limits of frequency and the corresponding limits of wavelength is the same? It's the same energy and the same conceptual filter letting it through. I know that Maths is usually required here but this constraint comes before the Maths - not after a proof (I would have thought). :smile:
I managed to figure out how it works :smile:
 
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  • #15
tade said:
How do we show that the two integrals are equivalent?

##\int \frac{du}{d\lambda}\cdot g \quad d\lambda \quad, \quad \int \frac{du}{d\nu}\cdot g \quad d\nu##

The proof was basically staring at me right in the face. :DD
 
  • #16
I would like to understand better this paradox. I tried to have a mathematical approach, if I take the distribution function ##f(\lambda)## for ##\lambda## and ##g(\nu)## for distribution function of ##\nu## : then I can apply the transfert theorem :

$$f(\lambda)\text{d}\lambda = g(\nu)\text{d}\nu$$

So, I get : $$g(\nu)=|\dfrac{\text{d}\lambda}{\text{d}\nu}| f(\lambda)=\dfrac{c}{\nu^2}f(\lambda)$$

We can notice that : ##g_{max}=\text{max}(\dfrac{c}{\nu^2}f(\lambda))\neq f_{max}##

I tried to find the relation between ##\lambda_{max}## with ##f_{max}=f(\lambda=\lambda_{max})## and ##g_{max}=g(\nu=\nu_{max})##, more precisely
the relation between ##\lambda_{max}## and ##\nu_{max}## (which is not simply ##\lambda=\dfrac{c}{\nu}##).

For this, starting from ##g(\nu_{max}) = g_{max}##, I took : $$\dfrac{\text{d}g}{\text{d}\nu} = 0$$

Giving also :

$$-2\dfrac{c}{\nu^{3}}f(\dfrac{c}{\nu})-\dfrac{c}{\nu^{2}}\dfrac{\text{d}f(\dfrac{c}{\nu})}{\text{d}\nu}=0$$

Finally, I get, that seems to be wrong, the equation : $$\dfrac{\text{d}f}{\text{d}\nu}=-\dfrac{2}{\nu}f$$

So, have I got to conclude by writing :

$$f(\nu)= \dfrac{\text{A}}{\nu^{2}}$$ with ##A## a constant to determine ?

I don't know how to conclude, the last relation seems to have no sense, doesn't it ?

If I am on the wrong track, please let me know. Any help is welcome

Regards
 
  • #17
Firstly there is no paradox. ;)
The apparent problem relates to the Definite Integral and the different limits when you choose to integrate wrt f or wavelength. This has already been dealt with. I can't think how to resolve it without introducing integration. But why avoid that when it's the whole basis of Energy Density.
 
  • #18

1. What is the spectral paradox?

The spectral paradox is a concept in physics that refers to the discrepancy between the observed spectral lines of stars and the predicted spectral lines based on their temperatures. This paradox has puzzled scientists for centuries and has led to various theories and experiments to try and explain it.

2. How is the spectral paradox explored?

The spectral paradox is explored through various methods, including spectroscopy, which involves analyzing the light emitted by stars to determine their chemical composition and temperature. Scientists also use computer simulations and mathematical models to explore different theories and explanations for the paradox.

3. What are some possible explanations for the spectral paradox?

There are several proposed explanations for the spectral paradox, including changes in the fundamental constants of nature, the presence of unknown elements in stars, and errors in our understanding of atomic physics. Another theory suggests that the paradox may be an illusion caused by our limited understanding of how light behaves in extreme conditions.

4. Why is the spectral paradox important?

The spectral paradox is important because it challenges our current understanding of the universe and has the potential to lead to groundbreaking discoveries. By exploring this paradox, scientists may uncover new knowledge about the fundamental laws of physics and the nature of the universe.

5. What are the potential implications of solving the spectral paradox?

If the spectral paradox is solved, it could have significant implications for our understanding of the universe and could lead to new technologies and advancements in science. It could also pave the way for future research and experiments that could further our understanding of the laws of physics and the origins of the universe.

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