REFLECTIONS ON THE SPEED OF A 10 FEET CRUISER IN BIG OCEAN WAVES. SEE PREVIOUS POST.
The following discourse might for some be a bit abstract. It has so to be because circumnavigating the world nonstop in a ten feet boat is not as yet an every day occurrence. Such an activity has previously only been done in big boats. To change size is like stepping into another dimension. One has to forget what one has learned and start all over, going back to the fundamental principals.
Having 50 years experience with small boats helps me, but even so, this new project makes me a child in a new world.
Froude numbers are used to determine the resistance and speed of a partially submerged object moving through water. They permit the comparison of objects of different sizes. They are calculated by dividing the speed by the square root of the product of gravity and the objects waterline length.
Froude numbers works well for towing tanks were there are no waves and at sea when the object is a ship; big enough not to be influenced by waves. It is regrettable that they are not applicable for accelerated systems like a ten-foot cruiser in big ocean waves. The real heavy weather situation is to complex for Froude numbers. They do not give a true picture comparing a small boats resistance and speed with other objects.
To surmount that shortcoming I have created a wave theory of my own. My trick was to use acceleration instead of gravity in the above formula. That makes my new theory universally applicable.
Unfortunately because the acceleration of a small boat at sea changes constantly by its speed and the wave amplitude, a function instead of the gravity constant, must be used to describe how resistance and speed changes over time.
Regrettably I do not have neither the knowledge nor the resources to give a continues description of what is happening.
Still the two extreme values are easily calculated. The result is very interesting and all the other values fall between them. Much is therefore illuminated.
It is a well-known fact that particles in the surface water describe a vertical, circular movement. From this follows that objects floating in this environment are subject to a centrifugal force – modified of course by the objects foreword movement.
The gravity and the centrifugal accelerations combine to create a new force that is always at right angle to the waters surface.
Thus unlike a skier, a slow ten feet cruiser cannot slide down a wave surface, also the same boat does not need energy to climb the water surface from the trough to the crest. This is of course a paradox. It seems that potential energy is created out of thin air. It is not so. The lifting force comes from the centrifugal force. Heavy logs floats with ease up against gravity from the trough to the crest of a wave.
At the top of the wave the centrifugal acceleration and gravity works against each other. At the bottom of the wave they combine.
Accelerations with values of 4,9 m/ sec squared at the top of the wave and 14,7 m/sec squared at its bottom are not extreme. The corresponding Froude numbers for a boat with 9 feet waterline length are 0,27 at the crest and 0,16 at trough. Thus, in the trough, in a following wind, there is, luckily, a striking speed increase through the water, just where it is most needed.
Another important consequence of being small in big waves is that a ten feet cruiser moving slowly through the surface water will at the crest become very light due to the centrifugal acceleration. In the above case its weight at the trough is three times as heavy as at the crest. Consequently its stability is also three times as big in the trough. And this case is not extreme as can easily be calculated.
The boat being so light at the crest may then get enough power from its sail to be lifted it out of the Archimedes cavity it is trapped in. Of course inertia still works, as it always does, even in gravity free space.
At the crest the surface water is mowing in the direction of the wind at a speed of several knots.
(Because the boat is so light at the crest, it there loses much of its stability. The consequently sudden heeling lead the ignorant person to believe that it is more wind at the top of the wave. The right answer is. The boat heels over because the acceleration is less at the crest than at the trough. But the boat still floats at the same waterline because the weight of the water at the crest is also less then in the trough.)
At the trough the boat meets surface water that is mowing against the wind at the speed of several knots. The boats speed through the water therefore increases even though its speed over the ground is the same. As we have seen, luckily the boat is now moving at very low “Froude numbers” this decreases its resistance for a given speed.
Think about it in this way, sailing at the crest is a bit like sailing on the moon. There gravity is low. Everything happens slowly. Consequently sailing on the moon will be slower than on earth.
On the other hand, sailing in the trough is like sailing on Jupiter. There gravity is strong. Everything happens fast. Consequently sailing on Jupiter will be faster than on earth.
During my planned circumnavigation I calculate with an average speed of 2 knots. This will give me a very low Froude number. Thus about 75% of my boats resistance will come from friction. Wave resistance is of lesser importance. Therefore the bow can be made very blunt. This gives my boat more displacement and enables me to have a very high prismatic coefficient .65 to .7. A side effect of this is that it gives my small cruiser very good stability for a given beam.
Clearly it is paradoxical that a 10 feet boat can circumnavigate on a small Froude number, but the hull speed of a boat is in proportion to the square root of its waterline. Now there is something funny about the graph of that function. It is of course the mirror image of the square graph, the common parabola as reflected in the bisector of the x-y axes. Now the parabola when its x-values are small keeps very close to x-axis and then as it gains momentum quickly rises to the sky and becomes nearly vertical. The square root graph does the opposite, at small values for x (in this example a 10 feet cruising boat with a short waterline) it follows the y-axis rising quickly towards the sky, but as x gets bigger, as for big boats, it flattens out and becomes almost horizontal. Consequently a few more feet waterline on big boats does not increase their speed significantly. This mathematical relationship favours the small boat and explains why I can circumnavigate on small Froude number.
These are just some reflections on one aspect of small boat sailing to show that the behavior of a small boat is not evident. That landlubbers and ignorant big boat sailors better not condemn my endeavor before they have educated themselves.
Regards Yrvind