Friday, October 23, 2009

Signs and Canting: Cosining and Secanting

October 23, 2009. 5:00 a.m.

(this blog over written and posted over the day so the references to angle of heel differ as did our angle of heel)

Meredith is sitting at the Deep Creek Lock, northern entrance to the Dismal Swamp. She is listing 15 degrees to starboard and stuck firm in the bottom. Tide has another 2 hours to ebb.

We went to bed in 7 feet of water at low tide. At 5 a.m. we awoke to the bilge pump alarm sounding incessantly. It was hard to roust out of bed. The boat was tipped 15 degrees and I was poised to crush Connie who was sleeping on the low side. Fortunately for my diminutive wife I am pretty good at somnabulent mountain climbing.

Yesterday was a long day: Meredith left Deltaville VA at 7:00 a.m., navigated the double dogleg entrance around the various shoals and headed out into the Chesapeake southbound for Norfolk.

Per usual we were met with unexpected weather. Forecast 5 to 10 kn Deltavill winds turned out to be 20. Waves of 1 foot were in fact well disguised 4 to 6 footers on a short frequency. Wind was about 20 degrees off the nose; waves were dead on.

The 8th boat to leave the anchorage we scanned the horizon for clues as to how we should set sail. Lead boats are usually a good source of information. Not always.The first boat was a good 2 miles ahead and we could see it hobby horsing in the waves. No sails up. Those guys were taking a beating.

Everyone had followed suit. This was curious as the wind was strong and the waves had everyone on "agitate cycle".

It mystified us a bit as a bit of sail flying in these conditions is like oil on the water; it smooths the ride tremendously. We figured the wind was too close to the nose to permit the bending on of sail.

So we turned off the wind 20 degrees. Instantly the boat settled down, her clipper bow taking the choppy seas remarkably well. Of course the 20 degrees we added to the wind angle allowed us to sail not motor - well motor sail anyway.

Our boat speed shot up from 5.4 knots to 6.7. No kidding. Motoring into the waves and wind gave us 5.2 to 5.4 knots. Sailing a little bit off course added 1.3 knots to the boat speed. Of course we were no longer sailing to Norfolk but we were going fast.

How did this help us? Well, a bit of trigonometry comes into play at this point. I will forgo the lecture and just say that if you speed up enough from a course deviation you may arrive at your destination quicker than you would by "steering direct".


The Math (skip to next section if this bores you)

The formula is this: [speed while on direct course] x [secant of the angle of deviation] = the speed you must attain to meet or beat the direct course.

In our case:

speed sailing direct to course: 5.4 knots
angle of deviation: 20 degrees
secant of 20 degrees: 1.0642

existing speed x secant of course deviation = 5.4 x 1.0642 = 5.75 knots

Meredith would have to make 5.75 knots on the new course to make the same progress toward Norfolk.

We were making 6.6 so it was money in the bank.

End of Math


Not only did we make faster progress toward Norfolk but we had a smooth ride, ok, pretty smooth.

The proof was at journey's end after we changed course to correct for the deviation - we had to sail back the distance we went astray. We left Deltaville 8th in line and arrived in Norfolk 3rd. Masterful.

As sailors we are always looking for omen and this seemed a good start out of the Chesapeake.

All was not a success on the day however.

Rather than stop at Hampton Roads or Hospital Point or Tidewater Marine for an overnighter we carried on to anchor at the foot of Deep Creek Lock. It was a race as there is one bridge blocking our way and it did not open until 5:30 p.m. Sundown was 6:20. We had 2.5 nm to make after the bridge to get to the anchorage.

We made it. Barely.

Our anchor set nicely and we dined in the cockpit as low tide came and went. I remarked at low tide as to the fact that we had 7 feet at 7 o'clock.

Then came 5 a.m. and the incessant bilge alarm. Then the difficult crawl out of the berth - mountain climbing in predawn is not recommended. Meredith had a 10 degree heel.

The depth sounder, installed in the bow, read 15.4 feet and the rudder moved freely. So both ends of the boat were free of river bottom. We were hung up on our middle.

During the nocturnal high tide Meredith had drifted over and was now stuck on a bar.

We tried motoring and kedging. Nothing worked easily so we looked at the tide tables, confirmed that I had erred in not allowing for Daylight Saving Time and that the tide would rise 3 feet by high tide. We went back to bed and slept crosswise on the berth.

It is now 9:00 a.m., the heel maxeds out at 20 degrees. The tide is now rising. We anticipate getting off the mud bar presently. However we have missed the 8:30 a.m. locking and I am pissed.


Another Math Lesson

You can calculate the depth of water under our keel at low water by again calling on the principles of trigonometry, in this case using cosines.

[The depth of keel] x Cosine of [the angle of heel] = your new draft

5 feet x cosine (20 degrees) = 5 x 0.93969 = 4.7 feet.

The water was .3 of a foot or 4 inches too shallow.

This formula is very helpful in demonstrating the effectiveness or as I usually allege the uselessness of "pulling your mast over" to get off a grounding.

When a boat runs aground the first remedy offered by the anchorage water rats is to take a halyard and try to pull your boat over from the top of the mast. This, they believe will get you off the bottom.

It is of course mainly good fun for the guys in the dinghy.

Ihave never seen a boat freed by pulling on its mast. Certainly not by a dinghy.

A boat with a 5 foot draft would have to be pulled over 25 degrees to get a 6 inch "reduction in draft". Imagine a rubber dinghy pulling a 25000 lb displacement hull over 25 degrees and you begin to see the where futility sets in.

Here are some common cosines if you want to use them and the resulting draft if your boat draws 5 feet:

5 - .99619 resulting draft - 4.98 feet
10 - .98481 resulting draft - 4.92
15 - .96593 resulting draft - 4.83
20 - .93969 resulting draft - 4.70
25 - .90631 resulting draft - 4.53
30 - .86603 resulting draft - 4.33
35 - .81915 resulting draft - 4.10
40 - .76604 resulting draft - 3.38
45 - .70711 resulting draft - 3.54


We await the 11 a.m. locking

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