The Sinusoids of Fall

This time of year, it's not unusual to get the sense that the days are becoming shorter very quickly.  

This is not just an illusion, it is actually happening.  As we make our way from long summer days to the long nights of winter, the day-to-day reduction in day length is never greater than it is near the autumnal equinox - and conversely, the day-to-day increase is never greater than the first day of spring - the vernal equinox.  

This is an excellent illustration of what scientists call a sinusoidal function; a function whose form looks like an unending series of waves (technically, 'sine waves').  

Sinusoidal functions are are all over the damn place in nature and science - from the pendulum on a grandfather clock, the vertical motion of an engine's piston as the crankshaft turns, or the up-and-down motion of a weight hanging at the end of a spring.  

This is a graph of a simple sine wave.  Notice that when the wave is at the top and bottom of its path, it is just about level.  This could represent the summer and winter solstices.  The days are at their longest, and shortest, respectively, and hardly change at all from day to day.  Then the slope gradually increases, until it is steepest when the curve is exactly at zero (equal lengths day and night).  From there, the slope gradually becomes less steep, until it reaches the other extreme, at the top or bottom, at which time, the process begins in reverse.  

We may sense the stability of the solstices; as the enjoying the long days of early summer, or the interminable wait in December and January, waiting for the longer days of spring.  In both cases, things are not changing very rapidly.  

If this hasn't bored you to tears yet, consider this further illustration.  

The animation below dynamically illustrates the relationship between day length and the rate of change in day length, using two related sine waves.  Think of the sine wave along the bottom of the graphic as day length, and the vertical wave on the left as the rate of change in day length (what calculus nerds call the first derivative).  It is when the day length is at the middle that the change is at its maximum.  

I could go on, but that would surely induce a nap, if I haven't already.