Last weekend I gathered a bunch of friends from various walks of my life to inaugurate the fifth decade for me.
Monday, May 3, 2010
Into the new decade
Monday, April 19, 2010
Empirical Feel for the Paddle Length
My first paddle was made by Canon. It had a fiberglass shaft and nylon blades. It's 230cm long. I got it to propel myself in a barge of a kayak: the folding Folbot Greenland II double. That's a beast with 34" of beam and 17" cockpit height. Folbot recommended 260cm for that boat but 230cm worked for me as long as I but a few inches of padding on my seat and was able to clear the coaming.
Since then I slimmed down. First came a single Folbot Cooper with a 24" beam. After that an old Valley Nordkapp HS—my slimmest boat at 20.5". With slimming boats and increasing appetite and skills, my paddles shrunk too. First I got an all-fiberglass 215cm Lendal, then another Lendal shaft to go with the fiberglass blades—a carbon bent-shaft 210cm. For the past 6 months I've been paddling a loaner 210cm all-carbon Werner Ikelos.
All along the way, "shorter is better and more efficient" has been the word.
This past weekend, I went for a paddle on a shallow rocky West Branch of the DuPage river. For the first time in years I've brought out my 230cm fiberglass-plastic Canon weapon. Just wanted to see how it feels and what difference 20cm in length really makes. Did not really want to bang up the primary guns against the bottom either.
First impression—the paddle felt very comfortable and efficient for casual stroking. The weight was not noticeably different from my fiberglass Lendal and the paddling was not any harder due to extra length. Very quickly I noted that the shaft was not as stiff as Lendal or Werner. It felt more like a Greenland stick as it obviously flexed with every stroke. I kinda liked that… I also liked the longer reach at the beginning of the stroke. It seemed like I could put more into each stroke and apply more power if I wanted to. So I decided to want to…
That's where things started to break apart. First, it was hard to maintain a high angle stroke as the blade went too deep under water and was harder to take out of the water. There was also no way I could generate high-frequency cadence with the longer shaft. Even though the blades on this Canon Heritage paddle had much lower surface area than my Lendal Kinetic or Werner Ikelos, the longer shaft increased the arm of the lever to the point where even these smaller blades quickly exceeded my available power. So while a fresh 4.5-knot touring pace with a stroke somewhere between a high and a low-angle felt very relaxed comfortable and efficient, sprinting with this paddle was awkward. Shorter paddle definitely has an advantage here regardless of the blade size.
Then I tried maneuvering strokes. I loved the extra extension on the stern ruddering strokes. Put the paddle parallel to the boat and work the throttle to see-saw from a pry to a draw. The boat responded wonderfully. Longer blade gives greater leverage and since the loads during ruddering are relatively low on intensity, extra length is an advantage. Something makes me think that the speed and power involved in surfing would most likely overwhelm the hands wielding a longer weapon.
Extra length does not work so well on bow rudders either. I immediately got lost with upper hand somewhere in the sky, lower blade deep under water and the amount of strain on the body noticeably greater than with a shorter paddle. Positioning the blade was also more lethargic, fine control more elusive. This could be, in part, due to the fact that I have not used this paddle for a long time but the difference was so stark that I tend to dismiss this argument of disuse.
Sweep strokes with a longer paddle—you guessed it—are more efficient. Here the more power and leverage you have the better. Since the blade is farther away from the boat, both the turning and the supporting momentums benefit.
In summary, longer paddles have come in disrepute lately. After a few hours of paddling a low-tech cheap long paddle, I could see no prohibitive disadvantages to using it for casual paddling. Some maneuvering strokes and bracing may be a bit slower with the longer paddle; however, this lag is more than compensated by the additional leverage that a blade gains when it is used farther away from the hull. As long as the blade is not too big, a longer paddle seems perfectly appropriate for a non-technical paddler. Lower paddling angle is also known as less demanding on the upper body strength and seems more appropriate for people who are less physically fit.
Except for the stern rudder and sweep strokes, a longer paddle will most likely interfere with technical paddling strokes and intermediate-to-advanced maneuvers. Acceleration is sluggish, bow rudder is awkward, braces are slower, draw strokes seemed less efficient due to sinking blade. It also seems like a longer paddle would be more of a hindrance than help when rolling in rough water. Extra leverage will be nullified by the difficultly of maneuvering the blades into proper position.
At the end of the day, I expect that longer paddles will be back in vogue in the next decade or so.
Saturday, April 3, 2010
Standing Waves on the Mighty West Branch DuPage River
With the water about a foot above average, there are some surfable standing waves on the normally placid West Branch of DuPage river just north of Butterfield Rd. in Warenville.
PS Here's a picture of the spot from the other side at an average flow.
When it is that low, I can paddle through the constriction. GPS speed with average water levels was just a hair over 3 knots.
Tuesday, January 26, 2010
THE TALE OF TWO BOW RUDDERS?
I am in love with the bow rudder but I don't know her name! There, I said it. To me it's the sexiest most efficient and effortless move in a kayak. More than that, the way the boat spins under you when motions align just right feels almost magical. The turning momentum seems to perpetuate itself. It's like the boat starts to give back what put into it.
For a decent free skill description and illustrations of the stroke with moving images see this Atlantic Kayak Tours web page. Search also for "Bow Rudder" among these videos by Doug Cooper if you'd like to see a short movie of the skill in action. If you don't have them already Doug Cooper and Gordon Brown's books are great resources for updating your kayaking skills. They both also contain good consistent descriptions of the Bow Rudder. Gordon has a DVD companion to the book out as well.
Gordon Demonstrates cross-bow rudder
Then, there's another detailed and well-illustrated description of the skill by the same name by Derrick Mayoleth. You may not even notice much difference between the two versions but, in my opinion, what separates them is so critical that two different names should be used to label them. Derrick's is the way the bow rudder looked when I met her but she is definitely not the one I fell in love with. Yet, it is the skill as illustrated by Derrick that seems to fit the name. That stroke actually attempts to pull the bow in the direction of the blade. Applying the same name to the turning maneuver presented by Atlantic Tours, Brown and Cooper (ABC) seems to me like a long stretch. More than surface semantics, both of the words in the name obscure the little ABC gem from the sights or well-meaning students of the sport. What's your name, darling?
Derrik demonstrating lean-forward bow rudder
ABC version of the 'so-called' bow rudder is neither done at the bow nor is it really a ruddering stroke. ABC way, the active blade is planted perpendicular to the surface of the water at the paddler's knees, next to the gunwale. The placement is much closer to the kayak's longitudinal center of gravity than its bow. Furthermore, there is really no reason to move it in that direction in principle. If anything, I am curious it if wouldn't work even better applied a foot aft. The live blade placement is distinctly different in the forward-reaching skill as illustrated in Derrick's post. Here the paddler needs to lean distinctly forward and advance the rudder toward the bow. The body rotation in the two versions is in the opposite directions. For ABC you "face the work" on inside of the turn with maximum torso rotation. In the alternative, you are rotated with the opposite shoulder facing the bow as the outboard hand extends as far forward as possible. The term 'rudder' in the name implies that the paddler should be trying to concentrate the action on the end of the boat. After all, 'rudder' is a steering contraption always found at the (rear) end of the boat. Substituting 'bow' for 'stern' does nothing to the implicit suggestion that the stroke should be performed as far away from the center of the boat as possible. The first term in the name—'bow'—does the same thing: it tells you to reach for the bow with the paddle.
Atlantic Kayak Tours version of bow rudder
The role of the active blade in the ABC incarnation is not intended to move or anchor the bow. Here the paddle in the water serves the role of a pivot point around which the kayak swings–bow moving in one direction stern in the opposite. In that sense, although it is clearly meant for turning the boat, the move feels more like a draw-on-the-move or a side-slip than a rudder. Cooper gives an apt analogy when he writes that bow rudder should feel like a runner grabbing on to a stationary post with one hand and spinning around. The forward-reaching version of the bow rudder emphasizes anchoring the bow, releasing the stern so that it could slide around. Leaning forward, weighing the bow, taking the weight from the stern, and sticking the blade near the bow to anchor it are mobilized to that end. The ABC version treats bow and stern on equal terms. Instead, it capitalizes on finding the most efficient pivot point on which to spin the kayak around its longitudinal center axis not unlike a table-top. The paddler remains fully upright throughout the turn. Finally, the lower elbow is fully extended in the former and tucked into the pelvis in the latter--seems a bit safer to me.
I am yet to confirm this in the field, but I would guess that the ABC way would not work quite as well as the lean-forward version for turning into the wind. In practice, I only know that I can easily turn 180° or more on calm days but was barely able to do 90° with strong beam wind using ABC. The latter is much more efficient at producing quick radical changes of direction in tight quarters will less body contortion … not to mention it looks much more elegant and also makes your legs shake with excitement J
So what do you think—are these two different versions of the same skill or two different skills? Are we doing those who are learning the skills any favors by misdirecting their attention from the knee toward the bow and toward ruddering instead of pivoting and spinning? How about something like "gunwale swing" or "beam spin" instead? The way the kayak dances alongside the planted blade could almost pass as a dos-á-dos dance move. Whatever you call it, give them both a try and see if it changes your relationship as much as it did for me. Whatever you decide, I'll always go to my spinning version to get a smile on my face and a tingle in the belly.
While we're at it, how about a cross-bow bow rudder that is initiated by bringing your paddle across the deck rather than the bow, has the blade planted at the cockpit rather than the bow, and uses the paddle as a pivot point rather than a rudder? Any takers?
Tuesday, December 29, 2009
Is Visibility Statute or Nautical? Decide for Yourself
In a kayak, it's often good to know the distance to some destination or object. The human visual system is pretty bad at this task. I can easily recall many instances of paddling for hours without any apparent gain on large distant targets. It always feels that progress should be faster…
Kayak navigation books such as Burch or Ferrero provide some help. They give a simple formula for calculating distances at which objects with given elevations above the sea level will be seen. The range of visibility in miles is equal to the square root of elevation as measured in feet. For example, a light house 100 feet above the sea level will be visible from 10 miles away. The formula is for an observer at the sea level.
In a kayak, you are not at sea level. For an observer in a kayak cockpit, you can add 1.5 miles to the visibility range of an object as derived from the formula. One-and-a-half miles is the square root of the eye-level elevation above the surface of the water. On average, paddler's eyes are just over two feet above the surface. Here are some immediate applications—if you can see beach goers feet as they walk close to the water, you're roughly within 2 miles off-shore. That's 1.5 plus a little bit added for the elevation of the beach. Two kayaks will lose sight of each other when they are separated by three miles—one-and-a-half miles of visibility from each side.
This is where I got tripped up. The formula for the visibility range is supposed to give you statute, not nautical miles. Burch, then, suggests that the visibility range as determined this way is underestimated by about 15%. What this means is that 10-mile estimate for an object with 100' of elevation should actually be 11.5 miles instead. Funny, that just happens to be the distance in nautical miles. So what is going on here? Exactly how much off is the square root approximation?
Time to dust off high school trigonometry books. First, let's track down the mathematical solution to the problem. Then, we will have grounds to decide if the limits of visibility are statute or nautical and how precise they are.
Generalized shape of the Earth is basically a sphere. For our purposes, the sphere can be further reduced to a circle. An average radius from the center to the surface is approximately 3,440 nautical or 3,959 statute miles.
Any line that just touches the outside of the circle can be used to mark the range of visibility—objects whose height is above the line will be visible, the ones below will be obscured by the horizon. Another line can be drawn from the point where the visibility line touches the outside of the circle to the center of the Earth. This line will have two known properties: (1) its length will be Earth's radius and (2) its angle to the visibility line will be 90 degrees. The Earth's radius and the visibility line can be viewed as two sides of a right triangle. The hypotenuse of this triangle is equal to the Earth's radius plus the elevation of the object above the sea level. What we have here is a simple case of Pythagorean Theorem with two known sides of the right triangle. We can solve for the unknown third side. In our context, a2 = c2 - b2
can be written as Visibility2 = (Radius + Elevation)2 – Radius2.
Plugging in the numbers and plotting the results of this equation we get the following picture. If the result of the square root solution is read in nautical miles, then it underestimates the actual distance by 6%. In statute miles, the distance is underestimated by just over 18%.
There it is clear as day: mathematically, the square root of elevation (measured in feet) is much closer to geographical visibility expressed in nautical rather than statute miles.
So why, then, is the formula is presented as offering the solution in statute miles? I don't know the answer to this one…
Here are some thoughts. Mathematical solution gives the distance at which the top of the object will reach the line of visibility. So the question is if you be able to see the top as soon as it reaches the line? To me, it depends. If the object is a mountaintop, the answer is most likely "No!" The tip of the mountain will have to get above the line at least somewhat to be seen. A mountain that is 400 feet above the sea level will reach the line of visibility when the distance is 21.3 nautical miles or 24.5 statute miles. Square root formula tells you that, when you first see the tip of the mountain, you are 20 miles away. By the numbers, when you are actually 20 nautical miles away, the mountain is already 47 feet above the line of visibility. When you are 20 statute miles away, the mountain will extend by 133'. When will you actually see the mountaintop? I don't know but, personally, I would rather be closer than I think I am. To me, nautical mile interpretation is a more conservative one and seems to be more appropriate.
If you are paddling at night and it is a light that you're looking for, you will see the glow well before it reaches the theoretical line. The moment the actual light emerges from beneath the horizon will be determined more precisely than in the previous scenario. Again, if the height of the light is known, it's square root is more appropriately interpreted as the distance in nautical, not statute, miles.
What do you think?
Kayak navigation books such as Burch or Ferrero provide some help. They give a simple formula for calculating distances at which objects with given elevations above the sea level will be seen. The range of visibility in miles is equal to the square root of elevation as measured in feet. For example, a light house 100 feet above the sea level will be visible from 10 miles away. The formula is for an observer at the sea level.
In a kayak, you are not at sea level. For an observer in a kayak cockpit, you can add 1.5 miles to the visibility range of an object as derived from the formula. One-and-a-half miles is the square root of the eye-level elevation above the surface of the water. On average, paddler's eyes are just over two feet above the surface. Here are some immediate applications—if you can see beach goers feet as they walk close to the water, you're roughly within 2 miles off-shore. That's 1.5 plus a little bit added for the elevation of the beach. Two kayaks will lose sight of each other when they are separated by three miles—one-and-a-half miles of visibility from each side.
This is where I got tripped up. The formula for the visibility range is supposed to give you statute, not nautical miles. Burch, then, suggests that the visibility range as determined this way is underestimated by about 15%. What this means is that 10-mile estimate for an object with 100' of elevation should actually be 11.5 miles instead. Funny, that just happens to be the distance in nautical miles. So what is going on here? Exactly how much off is the square root approximation?
Time to dust off high school trigonometry books. First, let's track down the mathematical solution to the problem. Then, we will have grounds to decide if the limits of visibility are statute or nautical and how precise they are.
Generalized shape of the Earth is basically a sphere. For our purposes, the sphere can be further reduced to a circle. An average radius from the center to the surface is approximately 3,440 nautical or 3,959 statute miles.
Any line that just touches the outside of the circle can be used to mark the range of visibility—objects whose height is above the line will be visible, the ones below will be obscured by the horizon. Another line can be drawn from the point where the visibility line touches the outside of the circle to the center of the Earth. This line will have two known properties: (1) its length will be Earth's radius and (2) its angle to the visibility line will be 90 degrees. The Earth's radius and the visibility line can be viewed as two sides of a right triangle. The hypotenuse of this triangle is equal to the Earth's radius plus the elevation of the object above the sea level. What we have here is a simple case of Pythagorean Theorem with two known sides of the right triangle. We can solve for the unknown third side. In our context, a2 = c2 - b2
can be written as Visibility2 = (Radius + Elevation)2 – Radius2.
Plugging in the numbers and plotting the results of this equation we get the following picture. If the result of the square root solution is read in nautical miles, then it underestimates the actual distance by 6%. In statute miles, the distance is underestimated by just over 18%.
There it is clear as day: mathematically, the square root of elevation (measured in feet) is much closer to geographical visibility expressed in nautical rather than statute miles.
So why, then, is the formula is presented as offering the solution in statute miles? I don't know the answer to this one…
Here are some thoughts. Mathematical solution gives the distance at which the top of the object will reach the line of visibility. So the question is if you be able to see the top as soon as it reaches the line? To me, it depends. If the object is a mountaintop, the answer is most likely "No!" The tip of the mountain will have to get above the line at least somewhat to be seen. A mountain that is 400 feet above the sea level will reach the line of visibility when the distance is 21.3 nautical miles or 24.5 statute miles. Square root formula tells you that, when you first see the tip of the mountain, you are 20 miles away. By the numbers, when you are actually 20 nautical miles away, the mountain is already 47 feet above the line of visibility. When you are 20 statute miles away, the mountain will extend by 133'. When will you actually see the mountaintop? I don't know but, personally, I would rather be closer than I think I am. To me, nautical mile interpretation is a more conservative one and seems to be more appropriate.
If you are paddling at night and it is a light that you're looking for, you will see the glow well before it reaches the theoretical line. The moment the actual light emerges from beneath the horizon will be determined more precisely than in the previous scenario. Again, if the height of the light is known, it's square root is more appropriately interpreted as the distance in nautical, not statute, miles.
What do you think?
Tuesday, November 24, 2009
New Blog
Saturday, October 24, 2009
Rules for Tides: Thirds, 50/90, Twelfths
Confused yet? I was! So I decided to work it out for myself once and for all. In the process, I found a way to inject some method into this madness. I have not seen the rules differentiated this way anywhere in the literature so I hope this will help clear things up for folks.
If you know the definitions for these rules, you can readily skip the next section and find the disambiguation in the Q and A part. The Math section 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, it is likely to confuse you right back to where you started or worse.
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 of the max current rate 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.
The 50/90 Rule gives you "… the speed of the current at the end of each hour." Starting 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. It will then slow down in the same steps: 90% at the end of the fourth hour, 50% fifth and back to slack at the end of the 6-hour period. The full Rule of 50/90 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 1/12th of the full tidal range in the first hour, 2/12ths in the second hour, 3/12ths 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 3/12ths in the fourth hour, 2/12ths in the fifth and 1/12th in the last sixth hour. Here's a crown jewel or you: the Rule of Twelfths is also frequently presented as 1:2:3:3:2:1—same as Rule of Thirds! Fonfusing? Well, yes, you should be!!!
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 in his text which gives "the constant current speed that is equivalent to the changing current of the cycle."
All of these definitions help me separate the Rule of Twelfths from the other two. Rule of Twelfths is about the height of the water while 50/90 and Thirds are both about the current. Horizontal and vertical dimensions. Burch helps some by introducing the term 'net progress' but I think he has it associated with the wrong rule!
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 then, right?
A: No they are very different, although I've frequently 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. The Rule of Thirds, on the other hand, is used to calculate distances that the current travels in full 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 to an object affected by the current over a period of one hour.
Q: Wait, wait ... I don't get it. Is there a graph for this or something?
A: Great idea! Let's try a graph.
Q: I think I get it but can you give an example?
A: We need to have 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' above the chart datum 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 1/12th of the total change in water levels between HW and LW. Since 1/12th 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 2/12ths—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 3/12ths and will stand at 6'. Then another 3/12ths by 4pm = 3'. Another 2/12ths by 5pm = 1'. The final 1/12th 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 flow or 3 knots (6*50%=3). At 2pm the current will flow at 90% of its maximum level or 5.4knots (6*90%=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 1/3rd of the maximum current speed or 2 miles (nautical). During the second hour of the ebb—between 1pm and 2pm—the current will travel 2/3rds or 4 miles. Between 2pm and 3pm the current will drift the most or 3/3rds = 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: That's great but my current speeds are in kilometers per hour. Can I still use the Rule of Thirds or 50/90?
A: Absolutely! Only the units of measurement change in the Rule of Thirds. Instead of nautical miles you will get kilometers. If the speed of the current is given in MPH, then you get distances in statute rather than nautical miles.
Q: Wait, I just lost it again, 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: 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 will not be exactly 6 hours. For diurnal tides it will be around 12! Divide the time between HW and LW into six equal intervals and you can still use the rules. Tidal flow is influenced greatly by shoreline dynamics and other factors on Earth as well as in the skies. Importantly, peak current speed normally lags somewhat behind the the midways between HW and LW due to friction. It is not uncommon to find that slack is mismatched with HW by hours. Use these rules as guidelines.
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, it probably 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 is off by 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?
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 reasonably 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. A simple average of all speeds was used.
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.
If you know the definitions for these rules, you can readily skip the next section and find the disambiguation in the Q and A part. The Math section 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, it is likely to confuse you right back to where you started or worse.
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 (2007, Heliconia) 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 of the max current rate 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.
The 50/90 Rule gives you "… the speed of the current at the end of each hour." Starting 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. It will then slow down in the same steps: 90% at the end of the fourth hour, 50% fifth and back to slack at the end of the 6-hour period. The full Rule of 50/90 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 1/12th of the full tidal range in the first hour, 2/12ths in the second hour, 3/12ths 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 3/12ths in the fourth hour, 2/12ths in the fifth and 1/12th in the last sixth hour. Here's a crown jewel or you: the Rule of Twelfths is also frequently presented as 1:2:3:3:2:1—same as Rule of Thirds! Fonfusing? Well, yes, you should be!!!
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 in his text which gives "the constant current speed that is equivalent to the changing current of the cycle."
All of these definitions help me separate the Rule of Twelfths from the other two. Rule of Twelfths is about the height of the water while 50/90 and Thirds are both about the current. Horizontal and vertical dimensions. Burch helps some by introducing the term 'net progress' but I think he has it associated with the wrong rule!
Q and 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 then, right?
A: No they are very different, although I've frequently 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. The Rule of Thirds, on the other hand, is used to calculate distances that the current travels in full 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 to an object affected by the current over a period of one hour.
Q: Wait, 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 travels DURING THE ENTIRE LENGTH OF EACH OF THE HOURS in 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 in the tidal period.
Q: I think I get it but can you give an example?
A: We need to have 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' above the chart datum 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 1/12th of the total change in water levels between HW and LW. Since 1/12th 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 2/12ths—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 3/12ths and will stand at 6'. Then another 3/12ths by 4pm = 3'. Another 2/12ths by 5pm = 1'. The final 1/12th 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 flow or 3 knots (6*50%=3). At 2pm the current will flow at 90% of its maximum level or 5.4knots (6*90%=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 1/3rd of the maximum current speed or 2 miles (nautical). During the second hour of the ebb—between 1pm and 2pm—the current will travel 2/3rds or 4 miles. Between 2pm and 3pm the current will drift the most or 3/3rds = 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: That's great but my current speeds are in kilometers per hour. Can I still use the Rule of Thirds or 50/90?
A: Absolutely! Only the units of measurement change in the Rule of Thirds. Instead of nautical miles you will get kilometers. If the speed of the current is given in MPH, then you get distances in statute rather than nautical miles.
Q: Wait, I just lost it again, 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: 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 will not be exactly 6 hours. For diurnal tides it will be around 12! Divide the time between HW and LW into six equal intervals and you can still use the rules. Tidal flow is influenced greatly by shoreline dynamics and other factors on Earth as well as in the skies. Importantly, peak current speed normally lags somewhat behind the the midways between HW and LW due to friction. It is not uncommon to find that slack is mismatched with HW by hours. Use these rules as guidelines.
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, it probably 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 is off by 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. MaxFlow constant can be used to set the output to the actual speed of the current. 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 reasonably 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. A simple average of all speeds was used.
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.
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