# Approximating mean daily and hourly temperature beyond Fourier series

Summary: What "well-known" and short parametrized mathematical function describes daily and hourly temperature for a given location?

If you look at the mean daily temperature graph for a given location, it looks like a sine wave with a period of one year.

Similarly, the hourly temperature for a given day for a given location also looks like a sine wave with a period of one day.

However, closer inspection (Fourier analysis) shows that they're not. There are fairly strong components of frequency 2/year and 3/year for the daily temperature, and the hourly temperature also has strong non-single-period terms.

Is there a parametrized function that reasonably describes the daily mean temperature and (a separate function) the hourly mean temperature? The parameters would be location-based.

I realize I can keep taking more Fourier terms to increase accuracy, but I was hoping for something more elegant. For example, maybe the graph is a parametrized version of sin^2(x) or some other "well-known" function.

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Interesting question, although I'm not sure whether it really falls under the scope of physics. It's basically just mathematical modeling, which physicists do a lot, but so do meteorologists and the thing you're modeling is a meteorological phenomenon. (I'm not going to close it, though, unless the community at large seems to think it's inappropriate. As I said, it's interesting, and I don't feel like the argument for closing is terribly strong.) – David Z Dec 16 '10 at 2:09
I'd be happy to move it to statistics or another appropriate site if enough people dislike it here. – barrycarter Dec 16 '10 at 2:37
These harmonics are typical for a saw-tooth signal. Try to fit your function with a sum of a sine and a saw-tooth functions, both of the same period. I would not include higher frequencies in the function since they have no physical meaning. – gigacyan Dec 16 '10 at 8:28
What physical meaning does the sawtooth have (just out of curiousity). I'll try your suggestion and report results. – barrycarter Dec 16 '10 at 15:23
@barrycarter: I think what you should really be looking for is a meteorology site. Unfortunately, there isn't a meteorology Stack Exchange yet, but one has been proposed. At this point, I'd recommend that you search the web for an existing meteorology forum and try asking your question there. (You might want to also post it as an example question on the proposal at Area 51, I think it would be recognized as a good question there.) – David Z Dec 16 '10 at 22:44