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The past century has brought a quiet revolution in the field of naval architecture. Today, largely through the use of the towing tank, where models are tested under controlled circumstances, we have access to an extensive knowledge about the motion of ships and the factors affecting that motion. Military and large scale commercial interests prompted most of this research although, in recent years, the same methods have been used to design the expensive water toys for the obscenely rich. Eventually what is learned in less savory pursuits filters down for more prosaic use.

Table of Contents

Frictional Resistance Effect of Length Effects of Deadwood Effect of Shape Residual Resistance Longitudinal Coefficient Displacement/Length Ratio Mid-ship Section Coefficient Bow and Stern Details

Yaw Racing Canoes Apply the Theory Design Criteria for "New Canoe" Cruising Speed Section Shape Profile and Function Fine Tuning

The surprising thing (or perhaps, not so surprising) is that the application of this knowledge to canoes has been largely superficial despite great strides in the materials used. Certainly, to the layman, the connection may not be apparent between large naval ships, racing sailboats and the canoe and it could be that the canoe industry has just been slow to recognize the connection as well.

To nature, of course, they are all just moving floating objects to be treated equally and consistently if not simply. If we know the principles that apply to one, we can, with some modification, apply them to the other. In so doing, a number of unfamiliar terms and symbols will be used. Naval architecture, like other sciences, has developed a language uniquely its own for precision and convenience. Don't be concerned. The terms quickly become familiar, and if you get stumped, the Glossary can be used for quick reference. So, if you'll be patient with a bit of hydrodynamic arcanum (and recognize that, even though the basic principles are unchallenged, there will always be debate over the details), we will explore the theory behind the motion of canoes and how dimensions and shapes affect that motion.

Modern naval architecture began some 100 years ago when the physicist William Froude proposed the elegantly simple proposition that the resistance of a floating body in motion was the sum of two parts - frictional resistance (Rf) and residual resistance (Rr), and that the two could be analyzed separately. We know now that this is not 100% true and that variations in speed cause incalculable changes in wetted surface and turbulence. Fortunately, these errors are small and from a practical standpoint can be ignored. Frictional resistance is that which occurs between the hull and the water. Residual resistance is the sum of all other resistance of which wave-making, form, and yaw are the most important. Other forces come into play under special circumstances (air resistance due to head winds and energy losses due to pitching are the most obvious) but, as they can be avoided by the skilled or prudent canoeist, they are rarely serious design considerations. We will deal first with friction.

The combined effects of wetted surface, surface condition, surface length and speed comprise the resistance due to friction and can be calculated with the formula:

Rf = 0.97 x CF x Sw x V²
where:

Rf = Resistance in pounds
Cf = Coefficient of friction
Sw = Wetted surface
V = Velocity in ft/sec
0.97 = Constant for fresh water

As the water passes, friction slows the water molecules next to the hull creating a layer of water that is carried along with the hull. This layer, called the boundary layer, is initially quite thin and the flow is laminar. As it progresses along the surface variable pressures cause turbulence. The layer gradually increases in thickness and near the stern, it breaks away into eddies. It is within this layer that friction is generated between the water molecules and not, as might be supposed, between the water and the surface. Where the flow is laminar, the coefficient of friction is quite small. It is theoretically but not practically possible for canoes to maintain laminar flow for the full length of the hull. Turbulent flow begins in near the bow with an attendant increase in friction.

The major factors affecting the frictional coefficient are the smoothness of the hull, velocity and length of the hull. Since the builder and the paddler are responsible for the velocity and surface condition the designer's influence is restricted to surface area and length. U.S. Navy studies have shown that, for conventional shapes (i.e. those that are not extreme in dimensions or configuration), wetted surface varies with length, amount of deadwood, Beam/Draft ratio and hull shape, in that order of importance.

Effect of Length

Surface area varies roughly by the square root of length and the first power of beam and the merits of increasing either must be balanced against the deleterious effects of additional area. Figure 1 shows a typical Curve of Resistance for both friction and wavemaking. It can be seen that wavemaking resistance does not become important until the Speed/Length ratio begins to exceed 0.7. Nevertheless, the greatest emphasis in canoe design has centered upon wavemaking. Some years back, a famous paddler entered a very long canoe in a marathon race. He was the odds-on favourite but, despite a superhuman effort, he lost. Since the winning speed was well below the onset of wavemaking resistance for his canoe it is a safe guess that the excessive wetted surface caused by length did him in. This balance between speed and length is a critical part of design that we will explore in more detail later. For the time being we can say that added length is not an unqualified blessing.

Figure 1

Effects of Deadwood

Figure 2

At the bow and stern there may be vertical sections of hull that, because they lie below the waterline, do not affect residual resistance. A convenient, although not completely accurate, term for these areas is "deadwood". There has been a modern trend towards straighter keels and more deadwood to provide directional stability but the price paid is in additional wetted surface. The rather minor change in profile A to B in Figure 2 results in a 1.5% reduction in the surface area of a typical 16 foot canoe without affecting wavemaking resistance.

It is worth noting that cutting away the bow may improve both the steering and directional stability . Canoes turn with the stern describing a greater arc than the bow and once the turn begins pressure build on the outside of the bow effectively locking the bow in place and accelerating the turn. The unpleasant aspect of this phenomenon is particularly noticeable in broaching conditions when the amount of hull buried in a wave trough becomes critical. Removing forward deadwood minimizes the hull's influence and increases the effect of forward control strokes. Deadwood at the stern, however, can be desirable. It acts like a skeg and resists the lateral movement of the stern. An analogy can be drawn between the canoe and an arrow: the feathers of the arrow are on the back of the shaft to provide stability; not on the front, where they would cause erratic flight.

Effect of Shape

Ask almost anyone what hull section provides the least wetted surface and they will answer "Round". While this is true, its importance has been greatly exaggerated. In fact, for normal shapes the Beam /Draft Ratio has the greatest impact and the waterline should be as narrow as possible within the confines of maintaining acceptable stability. Another factor affecting girth, and consequently wetted surface, is the fullness of the maximum section (area; not shape). This can be represented by dividing the section area by the area of a rectangular section having the same beam and draft to give the Section Coefficient (Cx). The best Cx lies between 0.94 for fine-ended hulls and 0.88 for full-ended hull. The difference between ideal and typical values is about 4 percent. Aesthetic and handling considerations generally prevent designers from ever achieving the ideal.

The great disappointment for the designer is that, after reducing friction to a minimum, the paddler is unlikely to notice the effect. A five percent decrease in wetted surface is worth bragging about, but a single year's scratching and banging can easily double CF from 0.004 on a new fiberglass canoe to 0.008. This more than offsets the designer's efforts. The cavalier attitude of most canoeists towards their boats is evidence that a 50 percent resistance increase is not often noticed if only because the onset of its effect is so gradual.

Residual Resistance

As the hull plows a furrow through the water, two wave patterns are formed. The first, the divergent waves, fan out from the bow and stern and their significance is minor. The second, the transverse waves, also form at the bow and stern but their crests lie at right angles to the direction of travel. These waves are the visible evidence of energy lost pushing water out of the way at the bow and suction at the stern pulling it back to its original level. The length of these waves (crest to crest) is equal to the natural length of a wave traveling at the same speed as the hull. About 100 years ago, William Froude determined that the speed of waves in knots was equal to 1.34 x L1/2 in feet. At low speeds there will be a large number of waves along the hull but as speed increases the number of waves decreases until the hull lies cradled between wave crests at the bow and stern. At this point, the so called "hull" speed has been reached. For heavy displacement craft, this marks the maximum practical speed attainable and higher speed is possible only with extraordinary power increases. Figure 1 shows the relationship between speed and wavemaking and, since two similarly shaped hulls of differing length will create the same wave profile and have the same resistance per pound of displacement it is possible to predict the resistance of any size hull from such a graph. This discovery of Froude's revolutionized naval architecture turning an art into a science.

It would seem from this that, for increased speed, we need only make the hull longer. This is far from the case. There are other considerations which, in their order of importance are:

Length
Longitudinal Co-Efficient
Beam
Midships Section Co-Efficient
Shape details at the ends

Longitudinal Coefficient

The Longitudinal Coefficient is a convenient number for expressing the distribution of volume along the hull. It is determined by dividing the immersed volume in cubic feet by the volume of a parallel-sided solid having the same maximum section area and length as the hull. The result usually lies between 0.48 for fine ended and 0.63 for full ended hulls. Figure 3 shows the ideal Cp for a given speed/length ratio. The important point is that fuller ended hulls have less resistance at speeds above S/L 1.2 (here Froude number is used. The Froude number can be converted to S/L ratio by dividing it by .298) due to their ability to create waves with crests that are father apart, and so, the water "sees" a longer hull. The price for improved performance at high speed is increased resistance at low speeds and the selection of a suitable Cp must be matched to the expected speed and power.

Figure 3

Displacement/Length Ratio

There are a number of methods for expressing the fineness of the hull. The Displacement/Length ratio is but one of them and produces a nice round number that designers seem to prefer. Typical values are: 25 to 30 for marathon racers, 40 to 50 for recreational canoes, and 50 to 60 for long distance tripping canoes. Below S/L 0.7 there is little effect but above S/L 1.0 the lower D/L ratio is vastly superior. In fact, because of their light weight and length, marathon racers can easily exceed their "hull" speeds while still in a displacement mode. Some confuse this with planing buy it isn't. True planing is only achieved when the hull is supported by dynamic loading. Canoe hulls are neither shaped properly for this, nor do humans possess the required horsepower. High speeds for canoes are only made possible through their having excellent Displacement/Length ratios and narrow beams. The two combine to produce very small waves which are the major resistance at speeds above S/L 1.34.

Figure 4

Since the displaced water volume equals the weight of the boat (Archimedes discovered this tidbit) any increase in displacement means more water must be pushed out of the way. An analogy with the wedge is appropriate; the more gradual the displacement of water, the less power one needs to do the job. An important point to recognize is that the waterline on its own is not a good indicator of water displacement and a curve of areas provides a better graphic representation of how the water is being moved. Figure 4 shows the curve of areas for asymmetrical and symmetrical hulls. The more gradual displacement of water is easily seen between the two. Both D/L and hull shape vary with loading, and one cannot expect a canoe to perform properly when over- or under-loaded. The "capacity" quoted by most manufacturers is meaningless. A far better figure is "Designed Displacement" which is the displacement intended for best performance given the canoe's purpose. Beam Theoretically the effect of beam on wavemaking varies as the square of the beam and the first power of length (Rr = B2L). This is not ironclad, but closely approximates experimental results. Given the narrow range of canoe dimensions the effect is minor but it reinforces the benefits derived from narrowing the beam to reduce wetted surface. In general the only thing good we can say about increased beam is that it increases stability and capacity.

Mid-Ship Section Coefficient

As with beam, the best practice for reducing residual resistance is consistent with that for reducing frictional resistance and the best Cx lies between 0.80 and 0.95. For canoes, the ideal is rarely achieved since seaworthiness and aesthetics dictate a finer section. Typical values fall between 0.70 and 0.80. The fact that the variations in resistance are small for changes in Cx does not deter builders from waxing poetic about the virtues of their shapes and so long as the buyer takes it all with a grain of sand no harm is done.

Bow and Stern Details

Nowhere has the boat builder's imagination shown more freedom than in the ends of the hull. Almost every conceivable shape has been tried at one time or another. For speeds below S/L 1.1, slightly hollow forward waterlines appear best but the amount of concavity does not seem adaptable to rule. For those who like rules, 0.15 x the square root of the span is a reasonable limit. Aft, the lines can be somewhat fuller and this is often the case for asymmetrical hulls. The common wisdom for this is that the fuller lines reduce squatting with each stroke. Another reason, and possibly a better one, is that asymmetrical hulls pitch less in waves. Guaranteed to provoke an argument is the subject of the angle of entry. For some reason, the half angle is the one most frequently given, and rarely are they much smaller than 7 degrees or larger than 25 degrees - which leaves a lot of latitude for artistic expression. (The angle is determined by a line intersecting the centerline at the bow and tangent to the waterline.) Test data and logic supports the use of increasingly smaller angles as speed increases but how small is too small? High speed Navy ships hover around 7 degrees, and we could hardly go wrong following their lead.

Yaw

With each stroke, the canoe is propelled forward, but, because power is applied off-center and at an angle to the centerline, the canoe does not track perfectly straight. This deviation from the straight and narrow is called "yaw" and is most evident in the meandering of beginning paddlers. But even the experts have the problem and the energy lost as the canoe angles its way forward can be substantial. In the past, the "fix" was an external keel, but more recently we have gravitated toward straight keel lines. Both increase wetted surface. By observing the canoe as it turns we can see that the bow describes a smaller arc than the stern from which we can deduce two things: 1.) That increased lateral plane aft would be advantageous in preventing the stern from swinging, and 2.) That reduced lateral plane forward would allow the bow to describe a larger arc with less amplification of the turn. The appropriate analogy is with the arrow, which has feathers on the back to stabilize its travel. Were it to have feathers on the front, the slightest variation in breeze would send it careening off in a new direction. Under broaching conditions, the additional lateral area forward can be genuinely catastrophic. Anyone who has attempted to steer a canoe with bow-down trim has had first hand, albeit exaggerated, experience with the phenomenon. Cutting away the forward profile below the waterline has no effect on wavemaking and, in fact, there are some types of cutaway bow that reduce resistance. This is worthwhile avenue to explore in light of current trends.

So far, we have only discussed recreational canoes that have moderate to low high Displacement/Length ratios. Sprint racing canoes are another tale. Typical D/L ratios are:

C-1 - 19
C-2 - 25
C-4 - 24

The extremely low displacement and long length mean that these canoes generate very small wave systems. At S/L 1.0 the trim begins to change as the canoe sinks into the trough of the bow and stern waves it creates. At S/L 1.7, the canoe still displaces water equal to its own weight but the stern wave crest is now well aft of the hull. Above S/L 1.7 the hull is in a state of semi-planing and is supported by a combination of static and dynamic pressures. The point at which planing actually takes place depends upon hull shape with wider hulls with flat sterns planing sooner than narrow hulls with round or V'd sterns, the lower range being around S/L 2.0 and the upper range as high as S/L 3.0. It is extremely doubtful if canoes ever plane under human power. The hull shapes are just not suitable. Canoes, however, are capable of very high semi-planing speeds of above S/L 2.0. We have examined the fundamentals of how water and hull shape interact. Next, we will follow the design process as each factor is applied to the creation of a new hull shape.

Applying the Theory

For centuries, boats and ships were built by eye, the product of accumulated skills and knowledge handed down to and built upon by succeeding generations. Improvements came slowly. Only within the past 100 years has science played a major role in boat design. Even today however canoes are rarely "designed". They are more often adaptations or modifications of earlier shapes. Given the apparent success of this method, many would question the need for a more scientific approach. The value, of course, lies in the plodding nature of trial and error and the preponderance of failure over success. The designer, by applying hydrodynamic principles developed through experimentation (and, of course, trial and error), is able to improve the breed more rapidly while minimizing mistakes and risks. The process used is rarely inspirational (advertising hype not withstanding), and begins with a set of parameters for the proposed new canoe that might look something like this.

Design Criteria for "New Canoe"

1. Primary purpose is tandem wilderness tripping of up to two weeks in duration, but most extensive use on weekends.
2. Some whitewater capability, but not a priority.
3. Intended for canoeists of intermediate to advanced capabilities.

These rather simple criteria are remarkably enlightening. First, we can ascertain displacement by adding the expected weights of the paddlers, the desired weight of the canoe, the expected gear weight and 1/2 the food weight for the longest trip. (Half the food weight is to compensate for the diminishing of supplies as the trip progresses and for lightly loaded weekend trips.) For a design example, let us say that these are 320 lbs., 60 lbs., 58 lbs. and 28 lbs. respectively, totaling 466 lbs. The criteria also suggests some things about the hull shape. Since the target market is skilled extreme beam for stability is not needed nor must there be a straight keel with the attendant increase in wetted surface for directional stability. (The occasional whitewater use mandates some rocker.) We also know that length must be moderate to fit a variety of uses such as bushwacking and puddlehopping as well as charging across vast areas of flatwater.

Cruising Speed

Before the first line is put on paper, the arbitrary decision concerning the anticipated cruising speed is made which can make or break the design. There is an ideal range of lengths and shapes for every speed and to vary widely from this range will result in substandard performance. For this case we will set the cruising speed at 4 mph from which we can determine the length. Figure 5 shows a plot of frictional, residual and total resistance for a canoe tested by the author and extrapolated for waterline lengths from 13' to 19' at 4 mph. Two things are immediately apparent. As the length increases there is a gradual increase in frictional resistance and a more rapid decrease in residual resistance.

Figure 5

When the two are combined for total resistance, we discover that length is not an unmitigated blessing and the ideal length at 4 mph is 15.5'. This is not to say that we should use 15.5'. There are times when we might overload the canoe or wish to paddle much faster, and so, we should choose 16.5' for its increased capacity, potential for higher speeds and negligible increase in frictional resistance. Given length and speed, the Speed/Length ratio is determined and from that and the ideal Longitudinal Coefficient. In this case the S/L is 0.85 and the "best" Cp is 0.51. From this we calculate the area of the largest section by the formula;

Cp = Disp (cu. ft.)/vol. of the prism = 7.48/x = 0.51,

therefore x = 14.67
and the area of the largest section = 12.47/16.5 = 0.89 sq. ft.

The shape of the midship section is important, not because of its effect on resistance, but because it influences the shape of all other sections (i.e. A full midships section generally results in full sections towards the ends). The two characteristics that also effect performance are waterline beam and, to a lesser degree, the midships coefficient (Cx). Studies show that the least wetted surface for hulls having Cp's below 0.56 is obtained when Cx is 0.94. Most canoes fall somewhat below 0,94 which represents a rather full section. The trade-off is in seaworthiness as finer sections have a more forgiving motion in waves. A large influence on resistance is the Beam/Draft ratio and increased beam should never be more than necessary for stability. How much stability is "necessary"? Only the paddler knows and the designer can only hope that his guess is right. The ultimate stability of a canoe, unlike other types of boats, lies with the passenger and a successful design will take this into account.

On this hypothetical canoe, an elliptical section of 32" waterline and 5.1" draft fits our parameters. (This means, of course, a 32" waterline at a draft of 5.1 inches.) The Cx of 0.79 is not too far out of line and the beam is sufficiently narrow for good performance. Do not confuse "beam" as used here with the "beam" seen most in canoe literature that is taken at the 3" or 4" waterline or at the widest point of the hull! While this number may give vague indications of hull shape if you know where it's taken it has no significance unless it happens to be the waterline beam.

Section Shape

The shape of the maximum section has a profound effect on stability. A wide, flat section produces greater initial stability and a quicker, more pronounced motion in waves. A rounder section has less initial stability but a more predictable motion. However, since the entire waterplane contributes to stability it erroneous to consider the midships section in isolation from the remainder of the hull. The trend toward longer and finer ends in modern canoes carries with it a loss in stability which is not always warranted in recreational canoes. By the same token, the very full ends used to increase stability and please the mass market are an equally great sin. The designer, unless pressured by some special consideration, will compromise. Slightly concave waterlines forward and a long entry are known to reduce resistance and improve the canoe's action in a seaway while fuller lines aft are acceptable. Should the stern be filled out too much directional stability will suffer and, if it is too fine, there will be a loss of control in following waves. There is a subtle balance here and few firm rules due to the complicated nature of turbulent flow near the stern and compromises made for maneuverability and seaworthiness.

Profile and Function

The profile is determined next. In this case, rocker is incorporated to improve maneuverability (remember the whitewater). A fringe benefit is that rocker reduces the hull's tendency to "hog" and so, is of structural benefit. The degree of rocker is usually arbitrarily set based on past experience. Unfortunately, too much for one may not be enough for another and the debate will enliven campfires for years to come without resolution. The designer makes his choice and hopes for the best while proclaiming to all who will listen that he alone is following the path of true enlightenment.

Where rocker ends and deadwood begins is arguable and the term "deadwood" is used here more as a convenience than in a technical sense. For our purposes, a workable definition is: "that portion of the profile lying below a fair curve drawn from the waterline to a point 2' from the bow or stern". At the bow the deadwood can be cut away severely for reasons mentioned earlier. The stern, however, utilizes lateral resistance to resist swinging and, because of the turbulent flow the added area is less detrimental. Once again we have the subtle balancing act of pros and cons for which there is no perfect answer. For whitewater, both ends should be cut away severely since the bow is not always the bow nor the stern always the stern relative to the water flow.

Fine Tuning

Now a process of adjusting the shape to provide the proper displacement and form begins. Some designers will draw a curve of areas (similar to those in Figure 4) and then draw each section to fit the curve. More often a few sections with the desired shape are drawn and the lines faired to suit. (All this is very fast using a computer with the proper software). A few iterations may be needed to achieve the desired displacement and form. Much is made of sectional shapes in advertising that attributes or implies some mystical importance to a particular shape or combination of shapes. In fact, subjective evaluations of these features are all we have, and their reliability is highly suspect. Indeed, to determine the best sections would involve testing an infinite variety of shapes, which is simply not possible. In a way, this is a blessing, as the designer can be as arty as he pleases without doing much damage.

The topsides are generally drawn at the same time as the underbody and offer the same freedom of expression. The benefits of one configuration over another are specific, and are exercises in compromise and occasional gratuitous variations. High ends and freeboard will keep out waves but increase wind resistance. Tumblehome can make paddling easier in the area where beam is reduced but allows more slop to come in and reduces ultimate stability. Flared ends will turn waves away but might make paddling more difficult. As yet, no universally perfect shape has evolved although there are "good" shapes for specific purposes. This may be as far as most designers go since calculating the center of buoyancy and stability can be tedious without a design program. All that remains then is to build and evaluate the prototype but that is another topic.

For the time being, let's leave our designer under the delusion that he has made a breakthrough in canoe design. Being more pragmatic and less emotionally involved, we know the truth: breakthroughs are few and far between.