The Relation of the Speed of a Boat to the Water Line

Date: 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/