The sails are set to the apparent wind, which is always closer than the true wind. The apparent wind moves further forward the faster the yacht sails. The formulae for calculating the apparent wind involve the boat speed (BS), the true wind speed (TWS), and the true wind angle (TWA):

Apparent wind angle is given by ArcTan((SIN(TWA)*TWS)/(BS+COS(TWA)*TWS));
Apparent wind speed is given by SQRT((SIN(TWA)*TWS)^2+(BS+COS(TWA)*TWS)^2)

The spreadsheet model (about 24kb) calculates the apparent wind given these parameters. It is interesting to see how much the apparent wind angle increasingly differs from the true wind as the yacht moves from a beat onto a reach, and how the apparent wind speed dies as the yacht moves onto a run. The graph shows that, closehauled, at the sort of speed that an IOM makes in the middle of its No.1 rig, the apparent wind is about 30 degrees off the bows when the true wind is about 45 degrees off.

A very interesting aspect of the apparent wind and how it changes due to wind gradient for different points of sailing is shown in the page on "Twist due to wind gradient".

But what is an appropriate value for the boat speed? Under all but the most extreme conditions, the IOM sails in "displacement mode". Frank Bethwaite (High Performance Sailing, Waterline Books, 1996) has a particularly useful explanation of the limits on yacht speed, and it boils down to the wave generated by the hull as it passes through the water.

As the yacht speeds up, the wave crests become more spaced out. The wave is created in the first place by the hull moving through the water, and it is one of the curious facts of nature that the crest-to-crest wavelength of a uniform wave train depends upon the speed at which the waves travel -- longer waves travel faster. As the yacht speeds up, the first crest builds underneath the bows, but the second, aft crest moves away from the stern, ceasing to support the hull. The yacht then has to sail "uphill", and usually cannot, so is stuck at a speed which corresponds to a wavelength exactly the length of the hull. The formula for the speed (in knots) of a wave, (ie the limiting "hull speed" of the yacht) is 1.32 times the square root of the wave length (ie the hull length) in feet. For the IOM, this equates to a top speed of about 4 ft/sec.

As an aside, Ross Garrett (The Symmetry of Sailing, Adlard Coles, 1987) tells us that the speed (in ft/sec) of a wave in shallow water is different, and is equal to 5.66 times the square root of the water depth (in ft). I thought this exciting, imagining that my boat could sail faster in the shallows, but Marc Rabaud has kindly pointed out that, if I used the full formula on page 247, I would see that the hull speed always decreases with decreasing depth... Ah well.

2006-07-10

©2024 Lester Gilbert