In a series of seminars, OCKAM Instruments emphasized how the polar diagrams for a yacht could be used in conjunction with the standard boat electronics to make real-time, on-the-water deciosons about how to steer, how to use windshifts, etc. The infamous "Wally" technique from the 1987 America's Cup races aboard "Stars and Stripes" is only one of the products of the OCKAM efforts. Most tactical rules OCKAM rationalized were "unconventional" to say the least, including their polar-based guidelines about how the helmsperson should respond whenever the boat's apparent wind changes. In this article, I shall concentrate on apparent wind changes due to a change in the wind speed - without a change in the wind direction. I am concentrating on wind speed changes because this is what we regularly face where we race, on the Delaware River off of Philadelphia.
I know how the apparent wind changes by heart: the boat's apparent wind (AW) is the sum of the Sailing True Wind (STW) in which the boat is sailing and the wind generated by the boat's own forward movement, by its own boatspeed (BS). Let's call this wind "Boat Wind" (BW). BW is in the opposite direction of the boat's heading, and of equal speed to BS. So, you add the STW "vector" to the BW "vector" to get the AW "vector". Remember how you add vectors?? If not, consult Fair Upwind Legs Under Current in the Instruction Hotline. It may also be clear from the discussion below.
With a sudden change in the wind velocity, the boat does not change its speed as suddenly because of its mass and inertia. Obviously, a light-weight Laser will respond to the change in the wind speed much faster than an America's Cup boat. But at least for a while, the STW vector will be of a different length than before while the BW vector is the same or almost the same, thus causing the change in both apparent wind strength and apparent wind direction.
To understand exactly how the direction of the apparent wind changes in a lull or a gust, let's do the vector addition: visualize a boat going upwind, say on starboard tack. The STW vector is at some 40-45 degree angle off the starboard bow, while the BW vector is along the centerline of the boat, pointing aft. To add the two vectors, position the BW arrow so that it starts right at the tip of the STW arrow. Then, join the beginning of the STW arrow to the tip of the SW arrowhead with a straight line. You just added two vectors! The line you found is the AW arrow, with the arrow head pointing downwind. (See the diagram to left.)
Now, when you reduce the size of the STW arrow, the sum vector (i.e., the AW arrow) will pivot to the left. That is, the apparant wind will "back". The boat on starboard tack will experience this as a header. This is what people call a "velocity header", a header not due to a change in the wind direction but due to a cange in STW speed. (So, its actually terrible terminology since "volocity" implies bot speed and direction.)
Similarly, you can see that if STW increased in strength, the AW vector would pivot to the right, and this would be a "velocity lift" for the starboard tacker. Visualizing a port tack boat results in the same observations: a decrease in wind speed yields a "velocity header"; an increase in wind speed yields a "velocity lift". (See the diagram below)
OCKAM U suggest that (as we hear in dinghy circles) when a gust hits upwind, you just "EASE, HIKE, AND TRIM!" That is, suppose you are going upwind, and there is a sudden increase in wind strength, a velocity lift. You see the lift on your telltales or steering tufts on the sail: the leeward telltales start stalling all of a sudden. OCKAM U says: Going upwind, do NOT head up in a gust because your sail is stalled! When the gust hits and you start heeling, do NOT feather, do NOT pinch! Instead, fall of a few degrees (???), ease out the sheets a bit (???) to stop the stall and to attach the airflow to the leeward side of the sail again, and after the boat accelerates - and only after the boat accelerates - head back to the optimal (???) heading for the new wind condition!
Now, there are a lot of question marks above because of my imprecise wording. But forget about the precision for now. Consider the general directive. Don't you lose to windward when you bear off? Isn't the sailor who is feathering instead gaining to windward? How do we know that the increase in speed due to bearing off more than compensates for the loss to windward? These are the tough questions, and I will try to answer them. (But my answer may change after I find my "OCKAM U." notes. So you are hereby warned! Treat the next section with a grain of salt! It's just me, the mathematician, speaking...)
Target Boatspeeds & Target Angles
For instance, for a J-24 (looking at the "polar diagrams" for the J-24), the upwind target boatspeed at 10 knots STW is about 5.3 knots, while the target in 15 knots STW is 5.6 knots. You achieve your target for 10 knots STW by sailing at 46 degrees to STW, and your target for 15 knots STW by sailing 44 degrees off of STW. (At least, theoretically...) Similarly, the J-24 downwind target boatspeeds are around 5.1 knots (at 161 degrees) for 10 knots STW, and 6.1 knots (at 172 degrees) for 15 knots STW.
I'll call the angle at which the target boatspeed is achieved the "target angle". As the J-24 examples show, for most boats your target speeds increase with the wind speed, your upwind target angles get smaller, and your downwind target angles get wider (except for some exceptions).
How do you find the target boatspeeds and the target angles? We need to talk a bit about "polar diagrams" and "VMG maximization" to figure this out.
The boatspeed is plotted along an axis emanating to the left from the center of the vertical axis in the direction of the boats' heading. (Three example boat-speed/heading axes at 45°, 60°, and 90° are shown in the diagram on the left in black.) That is, for a starboard tacker at 90° degrees to the wind, you have a boat-speed-axis at 90° to the wind (i.e., you have the usual horizontal axis, going from right to left). But for a boat heading at 60° to the wind, you visualize a speed axis to the left that makes a 60° angle with the vertical axis. Note that the vertical axis is merely the two speed-axes for headings 0° and 180° off the wind...
You plot the boatspeeds achieved on various headings measuring along these angled axes. So, all points at any angle on a given circle centered on the middle of the vertical axis correspond to the same boat speed. Such circles for 2, 4, 6, and 8 knots of boatspeed are shown in the diagram in red. (Note that it is as if you are looking down from above the North Pole to the various parallels of latitude. All points on the same latitude circle are equally distant from the Pole... and hence the term "polar coordinates".)
Once you determine the boat speed on all different headings in, say, 15 knots of STW, connecting the speed points you plotted yields the "polar curve" for 15 knots. Three such speed points are plotted in red in the diagram above. The polar curve for 15 knots STW is shown in bold black. Typically, such curves look like a the letter "C" for boats on starboad tack (and like a horizontally "flipped" C for boats on port tack).
Now, the wind is coming from the top of the graph page. If you want to go upwind, then you would get upwind fastest by chosing the sailing angle that maximizes your progress towards the top of the page. The tradeoffs on the water are the standard ones: you can foot off too low, but at a high speed; or you can pinch too high, but at a low speed. And somewhere in the middle is some sailing angle that is just right, that gets you upwind the fastest.
How do you determine from the polar curve what this correct angle is? You look at the "C"-shaped curve, and at the very top of it, it first goes up towards the top of the page and then it curves down. The ideal sailing angle is where the "C" reaches its highest point, the point closest to the wind, or the point closest to the top of the page. (Similarly, the angle that gets you downwind the fastest is the one where the "C" reaches its lowest point.) So you take a ruler, and draw a "tangent" to (a horizontal line that barely touches) the highest point on the "C". When you connect that point on the "C" with the center of the vertical axis, this line gives you the heading the boat has to be on to get to upwind the fastest. This is the "target angle" with the STW direction. (In the diagram above, the blue horizontal lines are the tangents. The corrresponding upwind and downwind "target angles" are also shown with blue lines emanating from the center.)
If you measure the boatspeed on the speed axis right along a target angle, this gives you the corresponding "target boatspeed". This is the boat speed achieved by the yacht sailing at the VMG-maximizing angle. This is the VMG-maximizing boat speed. For instance, in the polar diagram above, the upwind target boatspeed is 4.9 knots, and the downwind target boatspeed is 6.5 knots for 15 knots STW (for the fictitious polar curve shown) .
So, what did we learn? That the "target boatspeed" is by definition the speed achieved whan you sail at the VMG maximizing angle (the "target angle") to the wind. And we know how to determine it. Furthermore, looking at the diagram above, we see that if you are sailing upwind faster than your target, you must have been footing off. If you are sailing at a lower speed, you must have been sailing at a tighter angle. Similarly for the downwind case: if you are sailing faster, you must be sailing tighter than your target angle; and vice versa. These are the "on-the-water tradeoffs" I referred to above being reflected in the polar diagram.
Finally, consider the "C" curve for another wind speed. For instance, consider the curve that results when the wind speed increases from 15 knots - as when a gust arrives. We know from experience that as the wind speed increases, the boat goes faster (almost at all angles to the wind). So, the "C" curve for, say, 20 knots STW will be outside the "C" for 15 knots. The "C" for 10 knots will be inside the "C" for 15 knots (like the polar curve in green in the diagram above), and so on. Furthermore, if you compare the highest and lowest points of the black and green polar curves in the polar diagram above, you will note that the relative positioning of these "C" curves implies that:
Why fall off in a velocity lift upwind?
The answers to these questions are now more obvious:
When the wind increases, your target boatspeed increases and your target angle gets narrower. You want to get on both your new target angle and your target boatspeed as soon as possible. But if you first head up to your new target angle and then try to get up to speed, it takes you a long time to increase your speed to the target boatspeed. However, if you first achive the new target boatspeed fast by bearing off, it takes you little time to head up to your new target angle. So:
So, in the end, when the gust hits, everything boils down to how much you are bearing off and whether your boatspeed starts exceeding the "target boatspeed". But nevertheless, it is clear that you have to bear off!
So, this is the "ease, hike & trim" rule of dinghy sailing... On a Laser without any electronic true wind angle or boatspeed instruments, you can't check whether you are above or below your targets. But assuming you were steering optimally before the gust hit, it is clear that you have to ease the sheet and bear away a bit, and then start heading up after you approach your (currently unknown to the Laser sailor) target speed for the new increased wind.
Incidentally, why don't you continue beating at the wider angle? Why are you supposed to head back up? Is it just because of the fact that your upwind target angle is narrower for the gust. No! It is also because as you quickly accelerate, the BS increases, the BW vector gets longer, and the gust that gave you the lift in the apparent wind is now turning into a header in the apparent wind, the header now being induced by the increase in your boatspeed... (See the second diagram, and consider the case where STW stays the same but BW increases.)
Head up in a velocity header upwind?
I look at everybody else in our fleet, and all of them ease the sheets and fall off when the wind dies going upwind. When the lull arrives, you feel a header, a velocity header. So, this is the conventional wisdom for both headers and light air. Your sails have to be looser for a light wind. And you can't point as high in light air, so you need to reach a bit rather than sailing close hauled. But I don't follow this convention. When a lull arrives, I head up! Those of you who call me "king of light air", harken! This is also what OCKAM U suggets for a lull going upwind...
Why should you head up? The wind is down, but the boat has inertia! Your target speed is lower, but your actual speed is too high compared to the target boatspeed for the lull. I think OCKAM U called this "extra energy to burn" or "speed to burn", or something like that. How do you burn it? You sheet in and head UP to gain to windward. As you slow down to the new target speed for the new wind because you are pinching, you then start falling off and trim the sheets to the new wind.
Again, why do you start falling off? Because of two reasons: one, the upwind target angle for the lower wind is wider, and two, the lull that gave you the instant header in the apparent wind is now turning into a lift in the apparent wind, the lift now being induced by the decrease in your boatspeed...
Similarly, a gust downwind gives you an instant velocity lift. In a lift, normally you would bear off towards the leeward mark, but in a velocity lift downwind, you should head up before falling off! The gust increases your target speed and widens your downwind target angle, but to get at both targets in the shortest possible time, you need to head up first to accelerate fast.
1) Regarding the OCKAM "head down in the lull downwind" rule, note that the rule actually is "head to the mark in the lull". The Laser sails efficiently by the lee. So, if you were by the lee when the lull arrived, the rule actually implies "head up to the mark" initally if you are sailing by the lee. After burning the extra speed, what do you do? You should gybe! 2) I find myself practicing the "head up in the gust" rule unintentionally on the Laser (the gust hits and the boat rounds up by itself if you can't hold her down), but I seem to lose rather than gain to leeward. So, I have to think about whether if you have a keel under you or not makes a difference in the optimality of the OCKAM's "up in gusts downwind" rule. Does the increased heel when you head up hurt more than your gain in instant boatspeed?
Furthermore, when you get the velocity lift with the gust, if you bear off to start sailing by the lee instead (which is easy on the Laser with the free-standing rig), you also gain instant speed (and more boat control since as you heel the Laser to windward, the center of effort on the sail and center of lateral resistance on the daggerboard get aligned). Obviously, this is true only if you were sailing dead downwind or close to it when the gust hit. The point is, the OCKAM "up in gusts downwind" rule is based on polars for boats that can not sail by the lee! The "C" curves on all polar diagrams end right at 180° off the wind (see polar diagram above). However, the Laser sails fast at, say, 220° off the wind!
In retrospect, what we need is only the polar diagrams for the Laser. The analysis procedures outlined in this article are relevant for the Laser and all other dinghies, too. But the downwind conclusions such as "head up" or "head down" are not necessarily correct. If you put a little "Speedmate" and "Skymate" on your Laser, you could plot a half-reliable polar curve for each approximate wind speed, and figure everything out for yourself. So, what are you waiting for?