Is there a general rule to follow in data analysis that would let us avoid such paradoxes?Overall, the key point to remember is that one must be careful when interpreting spectral curves, especially their peaks
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!"We could avoid this dichotomy by expressing Power Spectral Density as Power Log(spectral) density, i.e., instead of
W/(m^2 * delta(nm)),
W/(m^2 * delta(log(nm)))
This should give the same shape curve as
W/(m^2 * delta(log(THz)))
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.
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.a detailed exploration into where that sense of paradox is coming from can
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!
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).hmm, do you have a link to the mathematical proof?
I managed to figure out how it worksIs 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).
One of the first (or the first?) threads in PF which discusses this paradox was started by me in 2006: "Wien's Displacement law, a paradox".