# How many waves are in the ocean? [closed]

XKCD #2741 mentions in the title text

It's cool how, when there's a number lots of people are curious about, but which isn't easy to measure, some random guess will get cited everywhere and become the universally quoted value. Unrelatedly, did you know there are 850 trillion waves in the ocean?

Let's try to prevent that random guess from spreading by coming up with a better one (or an explanation for that number). To $$1$$-$$2$$ significant figures (a Fermi estimate), how many waves are in the ocean?

Some thoughts:

• Wikipedia has a table of wavelengths for a given wind forcing. (Uncited, and with no formulas ðŸ˜ž.) If we could establish the mean wind forcing, that tells us the mean wavelength (roughly). If we can also fingure out the waves' width, then we know the waves' area, so that $$\text{# of waves}=\frac{\text{ocean surface}}{(\text{mean wind})\left(\frac{\text{wavelength}}{\text{wind}}\text{ (linearized)}\right)(\text{wave width})}$$ But I suspect wave width is strongly geometry dependent and hard to measure away from shore (if it is even well-defined outside of shoaling!). Also (if my memory of geophysics courses serves) the wavelength depends nonlinearly on the forcing, so the distribution of wind across the planet will be a nontrivial correction.
• I know geophysicists have spent a lot of time estimating how much energy the ocean waves carry in total (c.f. this paper), and the ocean's wavelength spectrum (this paper might be a good place to start; this one seems a tad old). So presumably one just needs to calculate the mean energy for that spectrum and divide, but I'm not conversant enough with the literature to do so.
• May be more suitable for earthscience stackexchange. I figured I should start here. Commented Jun 13, 2021 at 5:40
• Your question is not clear in physics terms. What do you mean by "waves" in the ocean? Just one wavelength you consider a wave in the question? Commented Jun 13, 2021 at 5:53
• Maybe a lower bound will be of limited help: there are at least as many waves in the ocean as there are ways to define what that means... Commented Jun 13, 2021 at 6:08
• in any given location there will be a number of wave trains of different period/direction going past each other. it's common to have a few sets of wind waves at 5-9 sec intervals, and then longer ground swell at 11+ seconds. Then there's also some lower cutoff you'll have to define where it becomes "ripples" or wind chop Commented Jun 13, 2021 at 6:45