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    A rate of change rule
    From: Frank Reed
    Date: 2016 Oct 30, 19:15 -0700

    We have many phenomena in navigation that vary more or less as a "sine curve" over some period of time. A quantity reaches a peak, descends slowly at first, accelerates to some maximum rate in the middle of the period, and then settles in gradually to some minimum level. Examples:

    • The Sun's declination: runs from 23.5° on June 21 down to -23.5° on December 21. Nearly a sine curve on the run from June to Dec.
    • The equation of time: the Sun flips from being early by nearly 15 minutes to late by 15 minutes in the run from the beginning of November to early February. Graph it out, and that stretch resembles a sine curve.
    • Tides at a port with ordinary semi-diurnal (twice daily) tides. water level goes from 4 feet above MLLW (e.g.) at 3:00pm to a foot below MLLW at 3:30am. The level in between resembles a sine curve when plotted, so long as there aren't significant "over-tide" components, as are common in estuaries.

    In any of these cases, it's nice to be able to work out the maximum rate of change from top to bottom, fast to slow, high to low. The average rate is easy. You take the total change and divide by the time. So in the tidal case above, the water level changes by 5 feet in 12.5 hours, giving a mean rate of 0.4 feet per hour. Or in the declination case, the mean rate is 0.26° or 15' per day. These mean rates are useful, but often we want to know the maximum rate. How fast is the water level actually falling at full ebb, halfway between high and low tide? How much does the Sun's declination change at the maximum rate around the equinoxes? There's a simple mathematical rule for this, not hard to prove: take the mean rate and multiply by pi/2 which is 1.57 nearly enough. In the tidal example, the maximum rate of change is thus 0.63 feet per hour. For the declination example, the maximum rate of change of the Sun's declination is 24' of arc per day --equivalent to a north/south speed for the sub-solar point of very nearly 1.0 knots. I'll leave the equation of time case as "an exercise for the reader": what is the average rate of change of the equation of time from early November to early February? And what is the maximum rate of change, using this simple estimation method?

    It's just a trick for estimating things in your head without looking up the details. Any time you have some quantity that varies over time in an approxmately sinusoidal fashion, take the average rate of change from max to min and then multiply by pi/2 or 1.57 to get a very good estimate of the maximum rate of change over that period.

    Frank Reed

       
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