Sines of Spring

I'm a wee bit tired of consolidation, but I am happy this morning even though it's April 5th and I'm looking out the window at snow falling. (It's not sticking, yet, though there is a small patch of snow still remaining from the major snowfall of last month. Too small; our little doggie who has been looking for snow-patches to poop on, a couple of times each day, had to poop on the grass this morning.) Anyway, I am happy because I was looking at Hamilton, New York (13346) Conditions & Forecast : Weather Underground

Length of Day 12h 54m -- Tomorrow will be 2m 52s longer.

The days are getting longer, but that "2m 52s" is a few seconds less than it was at the equinox. Sure; day-length is quite close to a sine-wave, if you think of 12 hours as the "zero" point around which day-length oscillates, hitting a maximum at the summer solstice of about 15 hours in this part of the world, and a minimum of about 9 hours at the winter solstice. At the equinoctial points for spring and fall, day-length is changing as fast as it ever does: almost 3 minutes per day.

And it occurred to me, not for the first or the fortieth time, that the 3 minutes per day and the three hours offset ought to be mathematically connected. Of course.

But this time (while out with the small doggie, just before daybreak hence significantly before the sun actually gets over the eastward hill) I actually thought about it, and remembering that it's about 90 days to the solstice, so if 3 minutes per day were fixed we'd add 90*3=270 minutes, four and a half hours, to the basic 12 hours and that would be the day length. How do we get from that too-large to the actual? It's simple, divide that 270 by pi/2, i.e. multiply it by about 2/3rds, to get 180 minutes which is the three hours we actually observe. Why pi/2? Well, the three minutes per day actually shrink, being multiplied by the cosine of the number of degrees (almost equal to the number of days) we've traversed, as that cosine heads from 1.0 down to 0.0 on the 90th day[degree]. How to evaluate that?

Think about it as an ant crawling around a circle, starting at x=1,y=0 on the right and going counterclockwise to x=0,y=1 at the top. In order to crawl an upwards distance of 1.0 along this path, in which the upwards component of her motion is proportional to the cosine (and total motion-so-far to the sine), she has to crawl pi/2, right?

And for some reason I'm thinking of the rock that my eldest gave me when he was working for a landscaper; he engraved pi on one side, e on the other, and the note said "because some things really are written in stone."

And we now have an eighth of an inch of snow with more falling fast, and the doggie was happy to run in it and chase a robin -- but I called her back in before she started yapping at several deer on the golf course, just on the other side of our fence.

Of course the temperature is also a sine-wave, but it lags day-length by over a month; our average temperature is a little under 50.5F, I know because that's how it comes out of a well into the geothermal system, and so the actual temperature will average something like 50+40*cos(d-120) where d is the day-count in the year...

Or then again, maybe not.