by Haris

Confused yet? I was! So I decided to work it out for myself once and for all. I think that, in the process, I found a way to inject some method into this madness. I have not seen this method anywhere in the literature before so I hope this will help clear things up for folks.

If you know the definitions of the Rules, you can readily skip that section and find the disambiguation in the**Q&A**part. The

**Math**section of this presentation will go into the mathematical details of how I figured this for myself. If you are a math type, you may find several hours of amusement therein. If you're not, please don't read it—it may confuse you right back to where we started.

## DEFINITIONS

Here's a typical example of two rules mixed up into one: "The Rule of Thirds states that relative to the total maximum current speed, the current jumps 50% the first hour, 90% the second hour, 100% the third hour. The current then decelerates to slack in the same order." You can find this on page 90 of *Sea Kayaking: Rough Waters* by Alex Matthews.

Let's ask Gordon Brown—a recent kayak reference volume from a highly reputed BCU coach. On page 168 if his Sea Kayak (2006, Pesda) book he gives this for the **Rule of Thirds**: "… over the period of the first hour the current will flow at one third of its maximum rate …, for the second hour it will flow at two-thirds and the third hour at … three-thirds." Hence the ubiquitous 1:2:3:3:2:1 abbreviation of the rule which stands for 1/3 in the first hour, 2/3 in the second hour, 3/3 during hours three and four, and then down to 2/3 in the fifth hour and 1/3 during the last hour of the cycle.

Now **50/90 Rule** gives you "… the speed of the current at the end of each hour." Counting from slack, the current will flow at 50% of its maximum speed at the end of the first hour, 90% at the end of the second hour and full 100% or maximum speed at the end of the third hour and then back to zero with the same steps. The full rule should be stated as 0/50/90/100/90/50/0.

I could not find a reference to the **Rule of Twelfths** in Brown's volume so let's go to page 178 of David Burch's Fundamentals of Kayak Navigation (1999, Globe Pequot): "The **rule of twelfths** [is a] method of determining the tide height between high and low waters." In the six-hour period that separates the low and the high waters, the overall level will rise/fall 1/12^{th} in the first hour, 2/12^{ths} in the second hour, 3/12^{ths} in the third hour. We know from the 50/90 Rule that after three hours the current reaches is maximum speed and starts to slow down. For the rest of the flood/ebb cycle the rise/fall of the water will be 3/12^{ths} in the fourth hour, 2/12^{ths} in the fifth and 1/12^{th} in the last sixth hour. Here's a jewel or you, though, the Rule of Twelfths is also frequently presented as 1:2:3:3:2:1—same as Rule of Thirds! Fonfusing? Well, yes!!!

Burch also has definition of **50/90 Rule**** **on page 226 as a "rule to estimate the effect of changing tidal current on net progress." Upon a quick scan, he does not seem to have the Rule of Thirds in his authoritative reference guide but there is a table that gives "the constant current speed that is equivalent to the changing current of the cycle."

## Q&A

**Q**: Are rules of Twelfths, Thirds, and 50/90 all dealing with the same thing? And, if so, can they be used interchangeably?

A: No! and No!

**Q**: I've heard that the Rule of Twelfths is different from the Rules of Thirds and 50/90?

**A**: Yes! The Rule of Twelfths is about the rise and fall of the water levels at various stages of the tide while the latter two deal with current. Think vertical versus horizontal dimension.

**Q**: That helps. So the Rules of Thirds and 50/90 are basically interchangeable, right?

**A**: No! Although I've seen write-ups using the two without distinction.

**Q**: But you said they were both about the current. What is the difference between them, then?

**A**: Think of it this way: Rule of 50/90 is used to estimate current speed at the end of each hour of the six-hour tidal period, while Rule of Thirds is used to calculate distances that the current travels in one-hour increments or drift. The first one describes the speed of the current at a single point in time, while the latter helps estimate what happens over an entire hour.

**Q**: Wait, I don't get it. Is there a graph for this or something?

**A**: Great idea! Let's try a graph.

- Rule of 50/90:
instantaneous
speed of the current observed at each of the six hours in the tidal period ON THE HOUR.
- Rule of Thirds:
cumulative
distance the current has travels DURING THE ENTIRE LENGTH OF EACH OF THE HOURS of the six-hour tidal period.
- Rule of Twelfths: cumulative change in the height of the water DURING THE ENTIRE LENGTH OF EACH OF THE HOURS of the tidal period.

**Q**: I think I get it but can you give an example?

**A**: But of course. We need to look up some input information before the illustration can work. Let's say that High Water (HW) is at noon and Low Water (LW) is at 6pm. Reference materials specify that between noon and 6pm the water level will drop from 12' to 0' and the maximum ebb current speed will be 6 knots.

The **Rule of Twelfths** will tell you that from noon to 1pm—the first hour of ebb tide—the water level will drop 1/12^{th} of the total change in water levels between HW and LW. Since 1/12^{th} of 12' is 1 foot, the water will drop from 12' to 11' between noon and 1pm. Between 1pm and 2pm the water will drop an additional 2/12^{ths}—that's 2 feet. At 2pm the water height will be two feet less than 11' or 9'. By 3pm the water will drop another 3/12^{ths} and will stand at 6'. Then another 3/12^{ths} by 4pm = 3'. Another 2/12^{ths} by 5pm = 1'. The final 1/12^{th} will drop the water level to 0' at 6pm.

The **50/90 Rule** helps estimate current speed at the top of the hour. At noon the speed is 0% of maximum—that's slack. At 1pm the current will pick up to 50% of its maximum level or 3 knots (6*0.5=3). At 2pm the current will flow at 90% of its maximum level or 5.4knots (6*0.9=5.4). At 3pm 100% = 6 knots and then back to 5.4 knots at 4 pm, 3 knots at 5pm and another slack at 6pm.

Finally, we can use the **Rule of Thirds** to estimate how far the current will travel during each of the hours of the tidal period. Between noon and 1pm the current will travel 1/3^{rd} of the maximum current speed or 2 miles (nautical). During the second hour of the ebb—between 1pm and 2pm—the current will travel 2/3^{rds} or 4 miles. Between 2pm and 3pm the current will drift the most or 3/3^{rds} = 6 miles. Then back in descending order—by 4pm another 6 miles of drift, by 5pm another 4 miles and, finally, by 6pm the drift will add another two miles. Don't read the following mathematical sentence *in italics* if you understand the differences between the rules—it may confuse you. *For those who are not afraid of math, drift or distance traveled by the current is the same as AVERAGE current speed during the hour.*

**Q**: Wait, I just lost it, I use the Rule of Thirds to estimate what and 50/90 what?

**A**: 50/90 Rule is for estimating the current speed on top of each hour in the 6-hours tide cycle. Rule of Thirds is for estimating the distance that the current will carry an object during each of the 6 hours in the cycle. If you want to know when the current will get too fast for you paddle against, use 50/90 Rule. If you want to know how far you will drift with the current use the Rule of Thirds.

**Q**: Wow, that's great! How precise are these estimates?

**A**: Glad you asked! These are all rules of thumb. As all rules of thumb, these ones describe idealized situations. Time between HW and LW is almost never 6 hours. Semi-diurnal tides are the norm but certainly not the only kind of tide. Tidal flow is influenced greatly by shoreline dynamics and other factors on Earth as well as in the skies. Use these rules as guidelines. Importantly, slack current normally lags somewhat behind the HW/LW due to friction. Some places, it is not uncommon to find that slack is mismatched with HW by hours.

**Q**: Why doesn't Burch reference the Rule of Thirds and why does he have that complicated table to calculate drift? He is the ultimate authority in the field of kayak navigation after all. That makes me uncomfortable with the Thirds Rule.

**A**: Well, to an extent it should. Burch probably thought that the rule of thirds is too far off the actual estimates for the drift that are obtained mathematically (see the last section for details). It estimates almost 10% too much for the first hour and then compensates by subtracting between 3% and 5% during the remaining two thirds. Quite a gross approximation if you ask me but would you rather use 26/70/96 rule?

## Da MATH

For the present calculations of floods I used the following sine function to get the flow throughout the tidal cycle:

Flow = MaxFlow * sin(2*π*Period^{-1}*Time)

where Flow is the proportion of maximum current speed or actual current speed, Period is the length of the entire tidal period or 12 hours for tides and Time is the clock variable. To set the output to the actual speed of the current at any given time the MaxFlow constant can be changed from unity. When MaxFlow=1 we get proportion of maximum flow or the speed of the current when maximum flow is one knot. A constant can also be added to Time variable. This constant will allow one to manipulate the Y intercept.

Here are the results that this function yields:

Note how the numbers of 50% and 86.6% fit the 50/90 Rule pretty close.

The numbers don't work nearly as well for the Rule of Thirds. Mathematically the drift/average current speeds are 25.6%, 69.9% and 95.5% rather than 33.3%, 66.7% and 100% predicted by the rule. For 6 knot max current, the Rule of Thirds will predict 1*6/3 + 2*6/3 + 3*6/3 = 2 + 4 + 6 = 12 nautical miles of drift. The actual average current speed over the three-hour period is 0.713 so over the the entire 3-hour period the drift would come out to be 3hr*6knots*71.3% = 12.8 nautical miles. Numbers are pretty close when added up but deviate from the ideal scenario when taken hour by hour.

The average speeds for each of the one-hour intervals can be obtained by integrating the area under the harmonic curve for each hour. In this exercise they were calculated using simulated data instead. A theoretical dataset was generated using the above harmonic function with 1,000 data points for each hour. That's roughly one data point for every 3 seconds.

Let me know if this helps clear things up or if it is useless. Suggestions for improvement? E-mail karovaldas@gmail.com. I am especially curious about the graphical representation of three rules.