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  Course to steer at a given speed.
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Course to steer at a given speed.
A while back, I was reading the chapter in Bowditch dealing with dead reckoning. I was especially interested in the section dealing with course and speed made good. Well, last night on my lunch break (I work nights), I decided to solve a quick problem I made up:
"If I were on a ship which was 10nm due East of port, with a set of 360° and drift of 3 kts., and I needed to arrive in port in exactly one hour, what course and speed should I use?" I figured the answer to be 253° at 10.5 kts. But then I started to wonder: That's all well and good for a vessel under engine power, but what about a sailboat? I know the way to figure course to steer at a given speed, but wouldn't turning a sailboat into the current (as in my example) actually slow the speed through the water enough to change the necessary course to steer? Or is this error so small as to be negligible? Or is there some other way to factor that in? Regards, Sean 
Re: Course to steer at a given speed.
TVMD_C tall virigins make dull companions true course, variation, magnetic, deviation, compass.......... then you configure your speed....... CaptG

Re: Course to steer at a given speed.
Magnetic observations made by explorers in subsequent decades showed however that these suggestions were true. But it took until the early nineteenth century, to pinpoint the magnetic north pole somewhere in Arctic Canada (78° N , 104° W). From then on the angle between the true North and the Magnetic North could be precisely corrected for. This correction angle is called magnetic variation or declination.
It is believed that the Earth's magnetic field is produced by electrical currents that originate in the hot, liquid, outer core of the rotating Earth. The flow of electric currents in this core is continually changing, so the magnetic field produced by those currents also changes. This means that at the surface of the Earth, both the strength and direction of the magnetic field will vary over the years. This gradual change is called the secular variation of the magnetic field. Therefore, variation changes not only with the location of a vessel on the earth but also varies in time. The correction for magnetic variation for your location is shown on the nearest! nautical chart's compass rose. In this example we find a variation of 4° 15' W in 2009, with an indicated annual correction of 0° 08' E. Hence, in 2011 this variation is estimated to be 3° 59', almost 4° West. This means that if we sail 90° on the chart (the true course), the compass would read 94°. Another example: let's say the compass rose gives a variation of 2° 50' E in 2007, with a correction of 0° 04' E per year. In 2009 this variation is estimated to be 2° 58', almost 3° East. Now, if we sail 90° on the chart, the compass would read 87°. Correcting for variation These overlayed compass roses show the difference between true north and magnetic north when the magnetic variation is 10° West. From the image we find: tc = cc + var in which “cc” and “tc” stand for “compass course” and “true course”, respectively. To convert a true course into a compass course we need first assign a “” to a Western and a “+” to a Eastern variation. Note that this makes sense! because of the clockwise direction of the compass rose. Here, the inner circle is turned 10° anticlockwise, hence 10°. Now, use the same but rewritten equation: cc = tc  var 235° = 225°  (10°) So, to sail a true course of 225°, the helmsman has to steer a compass course of 235°. To convert a compass course into a true course we can use the original equation. If we have steered a compass course of 200°, we have to plot a true course of 203° in the chart if the variation is 3° East or a true course of 190° if the variation is 10° West. Magnetic deviation Magnetic deviation is the second correctable error. The deviation error is caused by magnetic forces within your particular boat. Pieces of metal, such as an engine or an anchor, can cause magnetic forces. And also stereo and other electric equipment or wiring, if too close to the compass, introduce errors in compass heading. Furthermore, the deviation changes with the ship's heading, resulting in a deviation table as shown below. The vertical axis states the correction in degrees West or East, where East is again positive. The horizontal axis states the ship's heading in degrees divided by ten. Thus, when you sail a compass course of 220°, the deviation is 4° W. (Note, that on most modern sailing yachts the deviation is usually not larger than 3°). When a compass is newly installed it often shows larger deviations than this and needs compensation by carefully placing small magnets around the compass. It is the remaining error that is shown in your deviation table. You can check your table every now and then by placing your boat in the line of a pair of leading lights and turning her 360 degrees. Correcting for both deviation and variation Converting a compass course into a true course, we can still use our equation but we need to add the correction for deviation: cc + var + dev = tc Example 1: The compass course is 330°, the deviation is +3° (table) and the variation is +3° (chart); 330° cc + 3° var + 3° dev = ?° tc giving a true course of 336° which we can plot in our chart Example 2: The compass course is 220°, the deviation is 4° (table) and the variation is still +3° (chart). 220° cc + 3° var + 4° dev = ?° tc giving a true course of 219°. Example 3: The compass course is still 220°, therefore the deviation is still 4° (table) but let's use a variation of 10° this time. 220° cc + 10° var + 4° dev = ?° tc giving a true course of 206°. Converting a true course into a compass course is a little less straight forward, but it is still done with the same equation. Example 4: The true course from the chart is 305° and the variation is +3° (chart), yet we don't know the deviation; ?° cc + 3° var + ?° dev = 305° tc Luckily, we can rewrite this so this reads: cc + dev = 305° tc  + 3° var = 302° In plain English: the difference between the true course and the variation (305  + 3) = 302 should also be the summation of the compass course and the deviation. So, we can tell our helms person to steer 300°, since with a cc of 300° we have a deviation of +2° (As can be deduced from the deviation table above). Example 5: The true course from the chart is 150° and we have a Western variation of 7 degrees (7°). We will use the rewritten equation to get: 150° tc   7° var = cc + dev = 157° From the deviation table we find a compass course of 160° with a deviation of 3°. Voilà! Magnetic course © sailingissues.com The magnetic course (mc) is the heading after magnetic variation has been considered, but without compensation for magnetic deviation. This means that we are dealing with the rewritten equation from above: tc  var = cc + dev = mc. Magnetic courses are used for three reasons: To convert a true course into a compass course like we saw in the last paragraph. On vessels with more than one steering compass, also more deviation tables are in use; hence only a magnetic or true course is plotted in the chart. Bearings taken with a handheld compass often don't require a correction for deviation, and are therefore useful to plot in the chart as magnetic courses. Note, that the actual course lines the navigator draws in the chart are always true courses! These can subsequently be labeled with the true course or the corresponding magnetic or compass course if appropriate. In the next chapter we will be plotting courses in the chart. To summarise, we have three types of “north” (true, magnetic and compass north) like we have three types of courses: tc, mc and cc. All these are related by deviation and variation. Glossary Maps with isogonic lines: World  overview 2000 World  detailed 2000 World  detailed 2005 World  animated in timeVariation: The angle between the magnetic north pole and the geographic north pole. Also called the magnetic declination. Secular variation: The change of magnetic declination in time with respect to both strength and direction of its magnetic field. West () , East (+): Western variations or deviations are designated with a negative sign by convention due to the compass card's clockwise direction. Deviation: The error in compass heading caused by electric magnetic currents and or metal objects. Deviation table: A table containing deviations in degrees versus the ship's heading (compass course) in degrees. Usually plotted in a graph. True course: Course plotted in the chart i.e. course over the ground or “course made good”. The course corrected for compass errors. Compass course: The course (ship's heading) without the correction for compass errors. cc + var + dev = tc: This equation shows the connection between the compass course, its errors and the true course. It can also be read as: tc  var = cc + dev. Use the logo to navigate through this course, ...or go to the next chapter, or download the complete navigation course as PDF  including practice materials, exercises and answers. Register Email: Tell a friend Email: Search Books·chartsSitemapAZContactRSSCharterguidePDFs Top Charter guide Yacht charters Greece Turkey Dolphins Chapters: 3 2 1 6 5 4 9 8 7 10 5 May 2012 © 2000 – 2012 Diederik Willemsen  Site map  AZ index  
Re: Course to steer at a given speed.
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Determining the course to steer when underway in a sailing yacht is an iterative process. It can be done "on the water" based upon ones actual speeds achieved orwith somewhat lesser accuracywith one's Polars, although Polar plots don't account for variations in sea state which can effect ones speeds regardless of what the Polars indicate (hence they are actually relative speed diagrams rather than actual). The solution can be done mathematically although a discourse on the trig would go over a lot of heads. A graphical method, tho' is pretty easy. One begins with ones point of departure, the rhumb line to one's destination, one's assumed speed, and the average set and drift of the currents across the rhumb line. One first sets ones dividers to the distance of one's assumed speed along the rhumb line. (For example, if the scale of the chart is such that an inch = 2 miles, and ones assumed speed is 6 knots, one would set the points of one's divider 3" apart). One then strikes a line through the point of departure at the angle of the average "set" of the current across the rhumb line and marks a point that is the distance along the set line equal to the "drift". (For example, the average "set" might be 45º, with an average "drift" of 2.5 knots, hence one would place a mark along the set line that is 11/4 inch from the point of departure.) One then swings the divider in an arc (a "speed arc") from the "set point", to the rhumb line, marking the point of intersection. A line from the "set point" to the intersection of the speed arc with the rhumb line is the course to steer. (This is where Polars come in handy.) If the course to steer is closer to the wind, one might find that ones speed is somewhat less than the speed one assumed in making one's "speed arc", hence the radius of the speed arc needs be reduced. Likewise, if the course to steer is further off the wind, such that one's speed is greater than the speed assumed in making the speed arc, one would increase the radius of the speed arc. In either case, one then marks the intersection of the "adjusted" speed arcaround the set pointwith the rhumb line and defines an adjusted course to steer. How many iterations of the foregoing are necessary depends upon the shape of the Polars of one's own yacht (in the sense of how much one's yacht's speed with vary relative to changes in the apparent wind). The foregoing process can be done without Polars, simply by observing ones actual speed through the water, which is the reference frame one is relying upon (even though the reference frame is moving itself, at the set and drift of the current) and making periodic adjustments from new points of origin as the voyage progresses. The foregoing process is somewhat more difficult to describe than to actually perform and, unfortunately, in these daze of GPS navigation, not often performed even though it can shorten the length and time of a passage (and more significantly as the distance between point of departure and destination increases). The passage between say Miami and Bimini, across the Gulf Stream, is a good example of where the process can be used to advantage. In terms of your question seeking a specific time of arrival, that can be achieved to the extent that one can slow ones progress, perhaps by "scandalizing ones rig", as one might do to arrive at a specific point at a specific time for, perhaps, a Bar passage or to save ones tide, but, of course, one cannot sail faster than ones maximum potential speed at a given apparent wind angle. In such cases ones arrival point may be unknown, but one can define a "Known Unknown" rather than allowing oneself to confront an "Unknown Unknown". For example, on passages between the Channel Islands and the Alamitos Bay Channel entrance in Long Beach, we would always steer to favor an arrival point northwest of the Channel entrance so that, if we didn't hit the Channel bang one, we knew that once the coast came into view we could merely head east along a bottom contour line to find our entrance. (I.e. we didn't know where the channel was, specifically, but we knew it was to the east of us; and, we knew if we stayed in 10 fathoms, we weren't going to bang into anything while we sailed down to ithence a known unknown.) FWIW... 
Re: Course to steer at a given speed.
This is good for a power boat and for 'testing' purposes but sailboats generally don't have the luxury of setting an exact speed, and certainly not one over hull speed.
Additionally for sailboats you have to factor in leeway, which of course depends on the boat, the wind, and which tack you are on. If the purpose of the exercise is to hit an exact spot at an exact time  again, good for tests and the USPS (US power squadron) type contests. On sailboats it just ain't practical. 
Re: Course to steer at a given speed.
Sailing isn't straight line, or constant speed. Your course will vary based on wind direction, speed through the water, etc so while your question is a good mental exercise it's not a practical application for sailing. In other words, line up the bearing line on your chartplotter with your destination and do your best to get there as soon as possible:))

Re: Course to steer at a given speed.
To answer your question if you are calculating the CTS from a known current and a known boat speed. the current triangle is a snapshot of an hour.
Plot your DR course..Or desired course to your destination. Then plot the Set and drift of the current from your starting point ( currents are always given in "True" direction) That current line is an hour long..( use miles to represent kn. ) now plot a line from the end of the set and drift line....that represents your boat speed for an hour..and swing it back to your desired track. Plot the point where it intersects and then measure the distance from the starting point along your DR. That new distance will represent your true speed or the speed you will make good..( how far you will have gone in an hour) You can then figure out how long it will take to make the entire trip, given your speed made good. Of course, this is all assumes that your boat speed and the strength and direction of the current all remain constant, which is rarely the case, especially if you are sailing. But I think this is the answer to the question you are asking? IF you're sailing with a current setting you, you may also want/need to add a few degrees of leeway into the formula depending on the strength of the wind. You can decide what you want to use as a Course to steer..( true, mag, compass etc. ) and make the appropriate calculations...just remember that currents are given in True degrees. 
Re: Course to steer at a given speed.
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Re: Course to steer at a given speed.
I'd do what my old man taught me..
"Put some 'money in the bank', head up higher than you think, it's easier go spend the money than it is to get more later on". And I'd watch the shore once I could see it... is my bow "giving up land"? then head up. "eating land"? bear off. Steady bearing? then keep on. And it won't be my headstay that's heading toward port, it'd be some point on my stbd bow that's heading towards our coursemadegood. So if the heading is 253 to make good 270, then the seabuoy will come up 17 degrees on the starboard bow. That would get me there without a lot of course change. And I could do the hypotenuse calcs roughly in my head. And by pocket GPS would give me the speed made good, so I could figure out how many minutes to get there. But I couldn't necessarily make it come out 60 minutes, if we weren't making the speed. Speed along that hypotenuse is much trickier under sail than power. We sheet 'em in, and that's our speed. So we take what we can get, we don't control speed, and luffing for 10 miles (or more, once you know what 'Csquared' is) to slow down is hard on the sails. So precise tracklinefollowing for sailboats, and precise predictedlog speed, is more seatofthepants than it is math for most of us (meaning me), this is especially true on a beat. So we'll reach port when we get there. Also sailboats sideslip more than powerboats, it's our nature. So you have to allow for a crosswind even in still current. 
Re: Course to steer at a given speed.
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BTW  am average boat speed 10.5 knots in a sailboat is really rather quick. If you really want to get it right, you also need to account for leeway. 
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