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I am a high school student and as I've seen so far when referring to plots of two quantities A vs B, it is usually that the quantity A is on the $y$-axis and the quantity B is on the $x$-axis. Take for example, velocity vs time plots where velocity is plotted along the $y$-axis and time is plotted along the $x$-axis. However, I came across a problem today that required me to plot the wavelength of a wave against its frequency, essentially a wavelength vs frequency plot. I sketched the plot with the wavelength along the $y$-axis and the frequency along the $x$-axis but my textbook did the opposite, i.e., it has plotted frequency along the $y$-axis and wavelength along the $x$-axis. After a bit of searching around the internet I found that the independent variable is usually what sits on the $x$-axis, for example time is usually an independent variable so it will usually be plotted along the $x$-axis. But, for my problem how should I know which one is to be considered the independent variable, wavelength or frequency? What are some common quantities that would usually be considered an independent variable in this context?

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    $\begingroup$ If we are doing statistics, where the error bars on one variable is far bigger than the error bars on the other, then when we want to estimate the distribution of the one with the bigger error bars, that one has to be the y-axis, because the formulæ in statistics tend to be asymmetric between x and y. However, in physics, we generally try to keep the independent variable on the x-axis, but routinely fail to do so consistently. Both wavelength and frequency could have been engineered to be independent, so this is even worse. There can thus be no convention to help you here. $\endgroup$ Commented Oct 28 at 15:54

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When it comes to wavelength and frequency, it's hard to imagine that one is always calculated from the other. Instead they are sort of different sides of the same coin. So that's an example where it's simply not unambiguous which should be considered the independent vs the dependent variable. Most quantities have a clearer distinction which is the independent and which the dependent variable. You're right that the independent variable should be (and almost always is) plotted on the x axis.

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  • $\begingroup$ So in this case both plots are equally valid? $\endgroup$ Commented Oct 28 at 12:05
  • $\begingroup$ In general / without additional context, yes. In an application where you're asked to find one from another, the one you are given should be on the x axis and the one you find using it should be on the y axis. $\endgroup$ Commented Oct 28 at 12:06
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    $\begingroup$ What additional context exactly would indicate an independent variable of interest? $\endgroup$ Commented Oct 28 at 14:37
  • $\begingroup$ There's a large space of possible examples but here's one. Let's say you do an experiment where you measure the wavelength of a wave and know or measure its velocity. You then calculate the frequency as $f=\frac{v}{\lambda}$. In that context, the frequency would most reasonably be considered the dependent variable since it is calculated from the wavelength, not vice versa. $\endgroup$ Commented Oct 28 at 17:20
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    $\begingroup$ Of course, a different experiment could be done wherein the frequency was the directly measured quantity. So neither frequency nor wavelength is "fundamentally" the dependent variable here, unlike some other relationships where the choice of dependent is more clear. $\endgroup$ Commented Oct 28 at 17:23
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See: https://en.wikipedia.org/wiki/Abscissa_and_ordinate

The abscissa is unambiguously the independent variable, while the ordinate is the dependent variable.

Now the thing about a frequency vs wavelength plot ($c=1$):

$$ f = \frac 1 {\lambda} $$

is that it looks just like its inverse:

$$ \lambda = \frac 1 f $$

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  • $\begingroup$ I really do not like how your tone is definitive here. There are plenty of conventional physics plots where both of what you are saying is untrue. For example, you can find IV plots with axes swapped. If you meant frequency v.s. period, then the relationship is as you say, but in the case of frequency v.s. wavelength, there are dispersion relations. They are not so trivially related to each other as you portray. $\endgroup$ Commented Oct 28 at 15:50
  • $\begingroup$ well, anyone who does a dispersion relation as $\omega(\lambda)$ and not $\omega(k)$ needs their head examined anyway. $\endgroup$ Commented Oct 29 at 15:46
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    $\begingroup$ That is almost the entirety of the field of spectroscopy. Spectroscopy data predated the transition of modern physics from $\lambda\to k$, and even today the data presented in graphs are in terms of $\lambda$ instead of $k$, even if in equations they would write $\omega(k)$ $\endgroup$ Commented Oct 29 at 15:55

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