Of Boats and Waves

Peter Rye, the creator of the hull design program Hullform, takes a look at the most testing of all sources of hull drag, and with the help of his program makes some suggestions on how it can be managed.


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When I took Hullform from its initial "amateur" format to the first professional version, one of the first items on the list of additions was a capacity to estimate the drag of a designed hull.

I was already familiar with hull drag principles, from preparing a set of lectures on the Science of Sailing, so I knew the issues which had to be addressed. The program had to be able to represent drag which originated from skin friction, from the forcing apart of the volume of water through which the hull moved (form, or profile, drag), and from the generation of waves on the water surface. Of these, the one which gave the schemes I used most problems was wave drag.

Essentials of Wave Drag

There are three main sources of the drag on a boat hull, namely skin friction (due to the roughness of the hull surface), form drag (due to the effort required to force the flow apart, as the hull moves through the water) and wave drag.

The connection between waves and drag is very different from that for other two sources. Skin friction and form drag can both be measured by the loss of energy to turbulence. However, wave drag is due energy which is radiated away, as the waves generated by the hull propagate outward.

Obviously, the lengths and speeds of waves produced by a hull are affected by its length and speed. Wave theory shows us that the length and speed of a wave are closely related; when we add this point into the drag equation, things can become quite messy!

Skipping to the later part of the mess, the basic equation of a surface wave can be written as:


speed˛ = wavelength (g / 2p)
Here, "g" is acceleration due to gravity, and "p" is the ratio of circumference to diameter of a circle (3.14159…). The exact numerical formula is not all that important for now - what is important is that the long waves move faster.

A slow moving boat will therefore sustain short waves, and as it speeds up, the waves will get longer. But there will always be a range of waves generated, with the smaller ones carrying very little energy, and the larger ones able to carry more. To keep the analysis simple, we need to focus on the longest wave produced.

To understand this long wave, we must think of what happens to the water flow around the hull. Up front, the force which the bow applies to the water, to push it out of the way, makes the water "pile up". This forms the bow wave.

At the sides, the flow around the hull accelerates, and so the level drops (due to Bernoulli's Principle - we won't linger on that here).

At the stern, the flow slows again, and the level rises again.

This rise, fall and rise again of the water is the long wave, which is very small at low speeds, but becomes dominant as speed increases.

It has a range of options - it can move forward with the hull, at the hull speed, or move out at an angle, when its speed will be a fraction of the hull speed, and its length a similar fraction of the hull length:



However, only at most one speed and length combination will obey the speed-wavelength law. At low speeds, we do find that somewhere in the range of angles from 0 to 90 degrees, there is one valid combination.

As the speed of the boat increases, this angle reduces to zero. This occurs when the speed of the boat matches the natural speed of the long wave. Beyond this point, the long wave is being dragged at a speed greater than it would naturally have. Many changes occur in the flow to handle this condition, and they all mean large increases in wave drag.

Before we finish, there is one loose end to be tied up. The short waves may not contribute much to overall hull drag, but they combine to form a readily-seen wedge shape in the hull's wave pattern. There is a lot of physics and geometry to this pattern, but it doesn't help us understand wave drag, so we can happily skip it.

Calculating the Drag Force

We can calculate roughly what the wave drag term looks like, with only a little physics and mathematics - and it doesn't take long for the message to become clear.

For simplicity, we consider only the longest wave along the hull. Its height relates to the pressure applied by the bow of the hull, to the water through which it moves. Bernoulli's Law tells us that the pressure - and so the height of the water which forms the wave - depends on the square of the hull speed. As a result, the height of the wave depends on the square of hull speed.

The period the wave takes to pass corresponds closely with the period the hull takes to pass - so varies inversely with hull speed. The speed of particles in a wave crest is determined by the ratio of its size to its period - so by the cube of hull speed. And the energy of these particles depends on the square of their speed …

So far, we've tracked down factors making the drag of the long wave depend on something like the sixth power of hull speed - and we haven't got to the end of it, by any means. Let's not go any further, but simply bear in mind that, like a sixth-power curve, wave drag rises steeply when the long-wavelength term becomes dominant.

Naval architects use schemes based on the formal equivalents of the above analysis, to estimate the drag of their designs. Below is one example, for a heavy-displacement, 10 metre long hull, showing the relationship between drag force and hull speed. (This was calculated using Hullform's "Gerritsma" drag scheme)



The vertical shaded line is drawn at


speed˛ = hull length (g / 2p)
and illustrates that the steep rise of drag force occurs when hull length and wave length match.

You may also see small oscillations in the curvature of the graph, due to the changing effects of shorter waves as hull speed changes.

Controlling Wave Drag

Before you worry about the wave-induced drag of your design, you should firstly decide whether the hull is operating in the regime where wave drag becomes important. Smaller vessels often are, but large ships never are.

Having decided that wave drag will be an issue, you must minimise the effect of the hull-length wave. This means you must minimise its size. However, your options here may be limited.

The wave is basically a result of the water pushed aside as the hull moves through. For a given hull displacement, there is not much we can do about the amount of water pushed aside. We can, however, have some effect on the amount of the flow distortion which goes into the long wave, and how much goes into shorter waves.

This point is best introduced by considering the waves generated by a barge. When it moves, a large bow wave and stern wave are generated, but little occurs in between. Obviously, a barge is not a low-drag hull form, but it does what we want - if, perhaps, in the wrong way. Short waves are generated bow and stern, but with straight sides along most of the hull, there is little to generate the long wave. In short, for high-speed operations, we need a "full hull".

We express the "full hull" property by the prismatic coefficient, which is the ratio of volume displaced to the product of waterline length and maximum cross-sectional area. A craft like a barge has a prismatic coefficient close to 1.0, while one with very slender ends can have a value of about 0.5 or less.

The two simple hull-like forms below show what prismatic coefficient means in terms of hull shape. The top one has a coefficient of 0.67 (unusually large), produced by maintaining the draft of the hull from stem to stern, with very rounded plan form. The lower form has a more common value of 0.55, and shows more tapered stem and stern, with greatest draft amidships.



By way of illustration, the design used for the previous drag curve had a fairly large prismatic coefficient, of 0.62. A modification with thinner ends giving a prismatic coefficient of 0.53, showed lower drag below the limiting hull speed, but larger drag above:



The benefit of the Gerritsma drag calculation scheme is that it estimates the best prismatic coefficient and the best centre of mass position, for any operating speed. Two crucial terms needed to get the best performance out of any displacement hull become available "up front".

Putting It All to Work

For a 10-metre hull, the Gerritsma scheme tells us that the best prismatic coefficient and centre of mass positions vary according to:

Speed 5 kt 6 kt 7 kt 8 kt
Prismatic coefficient 0.43 0.51 0.57 0.61
Centre of mass
(% back) 53.3 52.9 53.5 53.5

Because of the relation between wave length and wave speed, lengths scale with the square of the speed - for example, you can double all speeds for a 40 metre hull, halve them for a 2.5 metre hull. There's little in the centre of mass variation, but from 5 to 8 knots we go from a very skinny-ended hull, to a pretty fat one. And as you can see by the graph above, there are real drag and speed benefits from getting the prismatic coefficient right, when the hull is operating just below its wave-limiting speed.

Shallow Water Effects

A point of particular interest in yacht racing is that the speed of a wave decreases in shallow water. This means that the limiting hull speed is less.

There are two effects. The obvious one is that there is more drag at speeds just below the normal limiting hull speed. Less obviously, light displacement yachts can reach planing conditions at a lower speed. This means that the slower increase of drag with speed in the planing mode starts earlier.

The condition where this occurs corresponds to water depth about one sixth of hull length - for example, one metre for a large, six-metre dinghy. While this is hardly likely to be a depth of concern for offshore sailors, those using inland waters will very often encounter such depths.

If You Want to go Further

Any interested readers should note that drag calculations are exclusive to the professional versions of Hullform. The "student editions" (MS-DOS and Windows) can perform the important design work (including estimates of prismatic coefficient, wetted surface and other factors affecting hull drag), at a fraction of the price. Contact Blue Peter Marine Systems, 92 Dyson St, Kensington, W.A. (09 474 1288, phone and fax), or e-mail at bluep@iinet.net.au.

Is the short answer the length of the dragonboat over 6, or is it the length of the dragonboat in the water over 6?????????

Somewhere between 6 and 8 feet?

Does that mean a course 8 feet deep is just as quick as a course that is 100 feet deep????

Interesting, now what is the impact when lanes differ from 4 feet to 8 feet?????

Always another question.

No - i read that to mean at one sixth the hull length you can reach planing conditions at low speeds. I don't think you can even dream about making a DB plane unless you have a motor attached to it.

Typical depth stated is half the waterline for the usual wave displacement to work. You'd probably need a course around 6m deep before depth stops mattering to a DB. But it's only an issue if the lanes aren't equally shallow.

In marathon racing we call this " poping the boat ". In shallow ( less than 3 ft. or so )water IF you can sprint into the shallows the boat sort of surfs its own bow wave and relativly high speeds can be acheived. If you do not get the boat popped , it is very slow.Factors other than water depth and boat speed come into play i.e. crew weight ( it is easier for a light crew ), stroke rate ,trim, and very small differences in hull shape.
Water that is a bit deeper , 3ft. to 6 ft. , is just slow suck water.
I do not know if it is possible to pop a DB. Huron kayak tried it at Statford a couple of years ago.Most of the crew knew how to pop a marathon boat. We increased the stroke rate and sprinted to the sand bar shallows.It did not work. Perhaps another technique would work , or better team , but I doubt it.

A practical study was done in Britain and indeed they found that drag became insignificant only after 6m.

"In marathon racing we call this " poping the boat ""

Hee. Does that mean the boat gets a really tall hat and its own popemobile?
Paddling: the pontiff himself approves.

um, never mind.

And now in English?

In the stuff I've been reading by John Winters he writes that smaller performance craft (like marathon and sprint canoes) have fast enough hulls to overcome their wave systems and achieve some kind of semi-planing. Most remain in displacement mode, but they just do it well enough that it's possible to break through their smaller waves. However these types of craft are the exception to the rule.

My interpretation is that most paddle-powerd craft (including dragon boats) are prisoners of their own wave systems or so-called "hull speeds." Somthing the size of a dragon boat displaces too much water and creates too much friction to break free of its wave under human power. That's my understanding of it anyway (please feel free to set me straight if I've read this wrong).

I'd be curious to know the approximate hull speed of the typical dragon boat. Is it possible for crews to push dragon boats close to this ceiling?

Lets say you can typically get up over 8 knots after a start. Sometime after transition, boat speed probably drops a knot or so. We talk about being "on plane" and "gliding" at these cruising speeds.

If you assume that a dragon boat's hull speed is higher than 7 or 8 knots, I wonder what "glide" is? Is it some intermediary state between displacement and planing? Is there even a relationship between "glide" and hull speed?

Does turbulence created by the act of paddling have an effect on the wave?

Can you affect hull speed by biasing weight distribution towards the rear to get the bow "up"?

Fascinating stuf but it raises so many questions!

If you get the bow up you're paddling "uphill".
Weight goes to the front, so you can paddle "downhill", especially if you have a cox that can surf.

Thought this was covered in another thread, but the math for hull speed is not difficult,easy to look up online. It's a function of the square root of the waterline length.

HS=2.5*LWL^(1/2), HS in knots, LWL in metres (if I got the right formula)

Crunch the numbers, taking an absolute best case of 12.5 m which is the length of a regulation DB, you get about a 1m50s 500m time.

I've only ever seen a handful of teams go sub-2 on a regulation course (no, Milton doesn't count) so it looks like there's still some room for improvement!

/runs to the ERG for a couple quick pulls

Ooh - I like Q's numbers, combined with Seat#5's questions about glide.

So, on average, to exceed the hull speed, a dragon boat would have to go sub-2 minutes. Not likely. However, the speed of a dragonboat is not constant. Everytime you paddle, there's a surge, followed by slowing down during the air time (recovery). So, perhaps, when you feel "glide", the surge has been great enough to temporarily exceed the hull speed. Hmmm.... Does this suggest that long deep strokes might in fact be more efficient that short, quick ones?

take into account that the boat needs about 10-15 seconds to reach top speed from zero, and the 2 minute time becomes about the fastest, which seems to validate all the math from you smart guys.

Actually my number represent the upper limit for the hull speed. 12.5 m is the tip to tail measure of the boat I believe.

If you assume a more reasonable 11m water line the hull speed averages to something like 1m57s over 500m. This also assumes the nirvana of 6m or more of water. I think many top teams are approaching the hull speed and may be exceeding it occasionally. It's not impossible, you just get much less return for effort after that point. I think any stroke rate that maximises the energy your crew puts into the boat is good.

Analysis only gets us so far. We race monster trucks people. Usually in a mud pit.

So top crews probably do push boats up to hull speed? And exceeding hull speed would mean climbing up and over the bow wave, which is virtually impossible?

Does this contribute to the closeness of races at elite levels? If there's a clear boat speed ceiling, beyond which incremental increases in power are required for the slightest gains, could this in part explain why elite crews rarely break away from each other?

When you consider, for example, that in a mature sport like rowing you can still see significant margins in Olympic or World finals (yes, I know it's not the same thing) it's unusual that DB could have such parity at the top.

Keep going ahead and using your powers of observation. While a lot of us, myself included, don't know enough about fluid dynamics or marine physics or whatever, we can still observe with our senses and come to some understanding of our sport.

Okay, that's the long winded version of "see what you see".

I have often been amazed that rowing events are so often won by margins that would be the cause of absolute embarrassment in dragon boat. Different sports, I realize. 4 seconds in rowing as a margin of victory between first and second place is common, while in Dragon Boat this would be considered and absolute PASTING. This would indicate to me that the maximum hull speed of a rowing shell is much higher than most crews can achieve or maintain for extended periods of time, while it is well within the grasp of top flight dragon boat crews.

What do we learn from this "maximum hull speed" idea?

Maybe that the team that wins is the team that get the boat up to max. hull speed the fastest, and keep it there the longest. And attempts to go higher than max hull speed are essentially a waste of energy.

Tight dragon boat races may just be due to wash riding rather than everyone reaching max hull speed

When you consider that according to the True Rankings only 4 teams average faster than 2:05, it seems that few, if any, teams, elite or otherwise, can actually hit the theoretical hull speed let alone maintain it.

I'm just guessing, but it would probably take an all-star world-caliber Open crew to do it. IIRC, there was 1 Open team that went sub-2 at the CCWC on Sunday, albeit into a headwind.

Most finals at large festivals are all pretty close from say C to A with wins measured in tenths and hundredths. It seems to be a matter of boat interaction (wash, strategy or whatever) more than some speed asymptote.

The wash being ridden and the hull speed are both dependent on the boat's displacement of water. They're not unrelated.

And you can blow as much energy as you like going past hull speed as long as you don't run out before the end. Mostly I take from this "get stronger, get fitter".

If anyone has got some links to decent websites with hard math related to boats I'd be interested. Looks a lot like my old EMF courses from the few small pieces I tracked down.

I would argue that wash riding and max hull speed are unrelated, or at least very loosely related. Place a couple empty lanes between each team and you will eliminate the wash riding but you will not change max hull speed. In such a scenario I bet there would be wider margins in each race.

"It seems to be a matter of boat interaction (wash, strategy or whatever) more than some speed asymptote."

Actually I think the asymptote idea has merit. Note how it's pretty easy to get a boat up to 10-12kph, but gets progressively harder to go faster. So as you approach maximum hull speed (even if you don't reach it) you're putting disproportionate amounts of energy into small gains in speed.

It's all about diminishing returns.

Sort of agree - but I'm not sure what effect someone on your wash or shallow water has on the hull speed. It might be exactly the same but require much more energy to reach, or wave interference does something weird. Beyond my math skills or level of knowledge at this point.

Welland is an interesting example, always separates the contenders from the pretenders. But it's very wide and also much deeper than most any courses. Remember that the standard hull speed formula supposes a minimum depth.

Hull speed is simply the speed at which the length of the boat is equal to the wave length of the waves it creates.i.e. The boat is riding between crests of 2 waves.The faster waves go the farther apart they are so the longer a boat is the faster its hull speed is. Hull speed is not the maximum speed a boat can go , it is the fastest speed that can be easily achieved. To go faster the boat must push into its own bow wave. This is easier for skinny boats than fat boats .DBs are similar to racing canoes in their length to width ratio so they can exceed hull speed about as easily as some racing canoes like marathon boats , but not as easily as sprint boats. A lot of energy is required to exceed hull speed , it is a cubic function so twice as fast requires 8 times the energy. This is why a team like Missy is only about 10% faster than an good regular team , even though they are generating twice as much force.