The Relation of the Speed of a Boat to the Water LineDate: 2/27/96 at 16:22:11 From: Anonymous Subject: math I have a math question. I understand that there is a mathematical formula or equation for calculating the speed of a boat as it relates to the water line of the boat. Can you help? Date: 3/2/96 at 14:53:31 From: Doctor Ken Subject: Re: math Hello! This is a pretty neat question. It's true that there is a relation between the speed of a boat and how low the boat is sitting in the water (relative to its zero-speed water line), but I think it would be pretty hard to calculate in general. For one thing, if you've ever been in a speedboat, you know that you can adjust the "trim" - that's the angle of the boat relative to the horizontal, which affects the water line of the boat - without affecting the speed of the boat. So we can't hope to get a direct correlation between the two. But if we simplify things greatly, we can kind of see what's going on. Most (in fact, pretty much all) boats have a curved hull, so that most of the boat's bottom is flat, and the front of the boat curves up. But if we assume that we have a boat whose bottom is perfectly flat, and slanted, like this: \ \ \ b\ \ o\ \ a\ -------------> \ t\ water coming \ \ at the boat \ \ Then we can figure out the force the water has on the boat (depending on the amount of water coming at the boat and the speed at which it comes), which will be separated into a horizontal and a vertical component (depending on the slant of the boat). The boat will rise in the water until the weight of the boat, the amount of mass displaced by an equal volume of water, and the vertical force on the boat (from the moving water) reach equilibrium. So you could kind of figure out how high this "math boat" would sit in the water. Then what you'd have to do is a sort of integration, realizing that each tiny little section of the boat's hull is actually equivalent to a flat section like the one we just worked out. Of course, I don't think you'd actually want to work this out all the way, but if you do, that's how I'd do it. -Doctor Ken, The Math Forum Date: 07/05/99 at 23:08:04 From: Colin Subject: Hull speeds of displacement hull boats This is not a question but an answer to a question. The hull speed of a displacement hull boat, (i.e. keel boat, freighter, ocean liner) is a mathematical formula that is expressed as hull speed = sqrt(length of the water line) * 1.35 Hull speed is the maximum speed that a boat with a displacement hull may be pushed through the water. Applying further force once this speed has been reached has little or no effect on the speed of the boat as the boat actually starts to sit lower in the water. It is my understanding that the end result of continually increasing the force after hull speed has been reached will result in the sinking of the boat. This does not apply to a speedboat or any boat that gets up on top of the water and planes and therefore is referred to as a planing hull. This can also be seen at the URL below. http://members.iquest.net/~jhubbard/sanjuan/archive/1174.htm I hope that you will post the math part of this answer with the question that is already on your site. Colin Date: 07/07/99 at 12:09:48 From: Doctor Rick Subject: Re: hull speeds of displacement hull boats Hi, Colin, thanks for writing! The question and answer you refer to has been there a while and I guess the original writer must not have written back saying that this was not what he meant. I have heard of the relation you state, between length of the waterline and "maximum" speed of the boat, and I'm pretty sure this is what the writer meant, not a relation between speed and _height_ of the waterline. It is my recollection that this hull-speed limitation is determined by the wavelength of the bow wave - that the distance between crests of the wave increases with boat speed until there is a crest at the bow and another at the stern, and the center of the boat is essentially unsupported. It would thus sink lower in the water relative to the wave height at bow and stern. I hunted around the Web a little and I found this note, which gives details that agree with my recollection. http://www.laser.org/archives/newlaser/laser/0289.html I'll quote it: "To the best of my knowledge there is a formula for estimating hull speed although I don't have it handy. My understanding is that hull speed, assuming monohulls, is a factor of the relative situation of bow and stern waves along the hull length at various boat speeds. As the hull moves forward both a bow and stern wave is created. As the hull moves progressively faster the stern wave can be seen cresting farther and farther aft -- all the while the bow wave remains where you would expect it, at the bow/stem. At hull speed, the aft wave is cresting at or near the furthest aft inwater section of the hull, typically near the transom. "The kicker here is that at hull speed the bow and stern wave create something like a trough, along the length of the hull, moving along with the hull in which the hull is, in effect, trapped. The amount of power, or thrust as you put it required to blow the hull out of the envelope is enormous relative to the power required to get the boat to hull speed (assuming a non-planing hull) so that locked-in condition is known as hull speed. The longer the hull's waterline, the higher the theoretical hull speed - this is why you often see strange looking hull extensions on IOR type boats - typically transom extensions at the water line." The formula you gave, stating that hull speed is proportional to the square root of the waterline length, will follow from the criterion above if the wavelength is proportional to the square of the boat speed. Since the bow wave moves along with the bow of the boat, the wave speed is the same as the boat speed. Is the wavelength of a water wave proportional to the square of the wave's velocity? I'm not an expert in water waves, but I located a Web page that says, "The speed of a deep water wave is dependent on the wavelength and/or period. C = gT/2pi or C^2 = gL/2pi. The greater the period or wavelength, the faster the wave speed." Thus the wavelength L is proportional to the square of the speed. The speed at which the wavelength matches the waterline length LWL is proportional to the square root of the waterline length. In fact, using the acceleration of gravity g = 32 feet/sec^2, I can work out the constant: 1 naut. mile 3600 sec C = sqrt(32/6.28 ft/sec^2) * sqrt(L ft) * ------------ * -------- 6076 feet 1 hour = 1.337 * sqrt(L ft) in knots (nautical miles/hour). How about that, it worked! Now all I have to understand is why the period of a wave is proportional to its velocity. (Since distance = speed * time, we have L = CT, so the formula for wavelength follows.) - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 08/09/2010 at 03:08:21 From: Mark Subject: Re: hull speeds of displacement hull boats I don't have a question; rather, I'm trying to offer additional facts that will help you answer questions like this by making them more "real world." Maximum theoretical speed formula is: LWL = Total Length of the Waterline in Feet Max Knots = SQRT(LWL) * 1.34 Max MPH (statute miles, not nautical miles) = SQRT(LWL) * 1.542 This formula is based upon the length of the wake that is generated by typical boats moving at various speeds. Any boat that is able to plane above the water can exceed these speeds in knots by a factor of 1.5 to 2 (rather than 1.34) -- or more if the boat is designed to be an ultimate racer. For typical monohull sailboats, which depend upon the wind, the average speed is typically two-thirds of the max speed. Multiplying the max speed by 0.667 gives the average speed over an extended period of time, and then multiplying again by 24 will provide the average speed in miles (statute or nautical) per day, or the approximate distance run in a day. Sailboats with crews devoted toward maximizing their speed can sometimes average up to 70% to 75% of the max speed; and racing teams sometimes arrive at 80% max speed. The averages that I'm refering to here are for long voyages (10,000+ miles), where "average" carries more meaning. Over a few lucky days, anyone can catch enough great wind to allow them to exceed their max speed. But day in and day out, 67% of max is what most crusing sailors are going to get. Date: 08/09/2010 at 21:08:34 From: Doctor Rick Subject: Re: hull speeds of displacement hull boats Hi, Mark. Thanks for the confirmation of the formula and its meaning, and for the additional information about its practical significance. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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