Time, Tide and the Rule of Twelfths (or Tenths)

The Rule of Tenths is a decimal approximation of fractions from the Rule of Twelfths... the two are equivalent.
(Drawing not to scale)

Getting around with shoal draft means getting up close and personal with the bottom. Since the tide is busy coming and going, we'll want to keep our eye on it.

The Rules of Tenths and Twelfths are rule-of-thumb approximations. Of what and why are the subject of this post. For now, it's a tool that helps you know ahead of time what the tide is doing at any given moment between high and low. Don't be frightened; it's only arithmetic.
Tides run high to low in about six hours (ebb tide, or the ebb), and back again (flood tide, or the flood) in about the same span. The volume of water flowing in those six hours follows a bell shaped curve. One can divide that curve up into the six hours of the tide. In each hour, a fraction of the whole of this tide's range will come or go. Range is the difference between high and low water:

RANGE = (Height of Water at High Tide) - (Height of Water at Low Tide)

Heights of Water at High and Low Tides can be read in tide tables. They are measurements made relative to Zero Tide Datum; an arbitrary height from which all others are counted. Height of tide may be positive (above ZTD) or negative (below ZTD). Charted depths (heights of bottom), however, are shown in positive units below ZTD. This can lead to ambiguity.

For simplicity, I'm going to speak in terms of heights of water and bottom. I'll reserve depth for the distance between height of water and height of bottom at any given moment, regardless of whether or not it's immersed.



Notice that tidal range, and fractions thereof, will be expressed as some unit of height (feet, fathoms, meters). It is important to keep in mind that these figures measure the height of a change in sea level, and not the height of sea level, itself. Sea level heights are relative to zero tide datum, while range heights are not.

Our graph shows amount or volume of flow. For example the blue column in the fourth hour, for example, represents 3/12, 0.25 or 25% of the total volume of water for one tide, flowing in on the flood, out on the ebb.

At the beginning and end of the tide, no volume is flowing... water is at a stand-still known as slack tide. At high tide, the current is at high slack; at low tide, the current is at low slack.

At the middle of the tide, the greatest volume is flowing. At the middle of a flood tide (incoming), the current is at maximum flood. At the middle of an ebb tide (outgoing), the current is at maximum ebb.

The greater the tidal range, the more volume is flowing in any given hour. The lesser the tidal range, the less volume is flowing in any given hour. Tidal ranges vary according to the Moon (see Neaps Springs Eternal).

Any moment of the tide divides our curve into volume that has come in, and volume that has yet to come in.

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Okay... that's the basic picture. Now we get to the numbers. To start with, we pull a little trick to simplify things.

Since water level is rising and falling over an area's entire surface, we may ignore area (and with it volume), and concentrate on changes in height of water, flooding or ebbing, expressed in units of height.

While this trick eases our calculations, it is good to remember that, in this case, height represents volume. A big tidal range generates greater height/volume and therefore currents will be stronger than experienced during low tidal ranges.

A second point to observe is that these hourly changes in height do not represent sea level. Our graph is different than those depicting sea level during a tide cycle. While superficially similar, tidal curves climb or fall for the whole six hours, depending on whether it's flooding or ebbing.

Let's take the end of the 4th Hour of an incoming tide as an example. At that point, 75% of the tide has come in (10% + 15% +25% + 25% for the 1st through 4th hours), with 25% left to go (15% + 10% for the 5th and 6th hours). If the range is 12 feet, then 9 feet will have come in, and 3 feet will be left to go.

Whether I use percentages, tenths or twelfths, the result is the same. The tenths have the advantage over twelfths, in that most tide table heights are given in decimal units (generally feet in US waters). It makes the arithmetic one step easier if we don't have to convert.

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We typically have a good bunch of data at our disposal: our draft, heights and times for high and low water from tide tables, depth of water and time of sounding, charted depth (shown below zero tide datum).

Applying the Rules to these data, with different approaches we can answer the following, and more:

What is the height of the bottom?
What is the height of the water at a given time?
When will a certain rock show? How high will it be at a given time?

...These are useful for charting depths. Once water height is established, it's like a vast water level... everything it laps at that moment is also that height. Rock heights are often useful in navigation.

Will we ground out? If so, when?
Will we float? If so when?

...Useful for shoal drafters on a steady basis. Deep drafters benefit if, say, going on the grid. Most cruisers will be letting their tenders go dry on occasion.

How much water will come in before high tide?
How much water will go out before low tide?

...These help with anchor scope and swing calculations. Or deciding how high to drag that tender!

Think of these as puzzles, as story problems, as a challenge. Draw 'em out on paper. Have fun with it. Little more than arithmetic is involved.

With a little thought and practice, you'll be able to do it in your head.


NOTE: There are many sources that help with forming a solid picture of tidal dynamics. I've just touched on the subject here, and there are many further wrinkles to be aware of. Depending on your local, tides may behave somewhat differently than I have described. It's a fascinating subject and worth of a sailor's study! I highly recommend that you learn about educate yourself for the tides of your cruising grounds.

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Extra Credit Story Problem:

Let's say we're sneaking into a cozy, high water bight where we plan to dry out. We've arrived at the beginning of the 5th hour of the incoming tide (High was 16 feet, low was 10 feet). On arrival, we sound and find that the bottom is 4 feet below the waterline.

Q: What is the height of the bottom?

A: Let's work it out, starting with what we know...

HI = 16ft   (above ZTD)
LOW = 10ft  (above ZTD)
RANGE = HI - LOW = 16ft - 10ft = 6ft   (NOT relative to ZTD)
DEPTH = 4ft   (from sounding, NOT relative to ZTD)

By the Rule of Tenths, we add up fractions for each hour of the tide...

.10 + .15 + .25 + .25 = .75      Of the tide since Low Tide
.10 + .15 =  .25                        Of the tide till High Tide

We apply them to the range to find the change in water height since low and high, respectively...

.75 x RANGE  = 4.5ft     Height of Water since Low Tide, OR
.25 x RANGE  = 1.5ft     Height of Water till High Tide

We adjust our known heights (HI and LOW) by one result or the other, adding to low OR subtracting from high... should be the same result, either way, so you'll only actually be choosing one pair or the other for any given calculation. Let's use 'Height of Water Now' as a shorthand for 'Height of Water at Time of Sounding'...

Height of Water Now = LOW +  Height of Water since Low Tide = 10ft + 4.5ft = 14.5ft, OR
Height of Water Now = HI - Height of Water till High Tide = 16ft - 1.5ft = 14.5ft

Last, we adjust the Height of Water Now for the sounding we took earlier...

Height of Bottom = Height of Water Now minus DEPTH = 14.5ft - 4ft = 10.5ft  (above ZTD)

Voila! We note the Height of Bottom for our hidey hole as 10.5ft on our chart and have a glass of something sippy.

With practice, this will likely be done in your head in about 30 seconds. The important thing is to have a clear picture of what's going on, and go step by step. Don't hesitate to use paper, especially if tired, cold and/or hungry.