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The Shape of the Canoe
Part 2: Residual Resistance

by John Winters

As the hull ploughs 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 sterm 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 travelling 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 L^1/2 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 displacement craft, this marks the maximum practical speed attainable and higher speed is possible only with extraordinary power increases. (NOTE: Canoes with very low Displacement/Length ratios are a special case and will be discussed separately.).

If you still need help with the terms, see the Glossary for quick reference.

Figure 1 shows the relationship between speed and wavemaking for a canoe of 16 foot waterline. The resistance in pounds is shown along the left and the ratio of speed to length along the bottom. 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:

          1. Length
          2. Longitudinal Co-Efficient
          3. Beam
          4. Midships Section Co-Efficient
          5. Details of shape towards the ends

Longitudinal Co-Efficient

The Longitudinal Co-Efficient 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 2 shows resistance curves for hulls of the same generic shape but having differing Cl's. The important point is that fuller ended hulls have less resistance at speeds above S/L 1.2 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 Cl must be matched to the expected speed of the hull.

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. The significance of these numbers can be seen in Figure 3.

Below S/L 0.7 there is little effect but above S/L 1.0 the lower D/L ratio is clearly 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; it isn't. True planing is only achieved when the hull is supported by dynamic loading. Canoe hulls are neither shaped properly for this to occur, 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.Since the displaced volume of water equals the weight of the boat (Archimedes discovered this, not Froude), any increase in displacement means more water is being 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.

Since the displaced volume of water equals the weight of the boat (Archimedes discovered this, not Froude), any increase in displacement means more water is being 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 three hulls with the area of each section represented as a percentage of the maximum section. "A" is a longitudinally symetrical hull and "B" and "C" are assymetrical with "C" having its greatest area forward of midships. It is sometimes claimed that a hull similar to "C" "replaces" water better but test data has not yet been found to support this at any but the lowest speeds. In fact, for speeds above 1.34 x L^1/2 there is a good argument for a transom stern, even for canoes.

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.


Theoretically the effect of beam on wavemaking varies as the square of the beam and the first power of length (Rr = B^2L). This is not ironclad, but closey 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 Co-Efficient

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 seaworthyness 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 boatbuilder'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 assymetrical hulls. The common wisdom for this is that the fuller lines reduce squating with each stroke. Another reason, and possibly a better one, is that assymetrical 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 deg. or larger than 25 deg. - 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 deg., and we could hardly go wrong following their lead.


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 meanderings 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 the cures increase wetted surface and may have a similar drawback. 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 apropriate 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 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 relatively high Displacement/Length ratios. Marathon racing canoes are another tale. Typical D/L ratios are:

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

The 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 as the water cannot fill in behind as rapidly as it is pushed away at the bow. 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, 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, but they are capable of very high semi-planing speeds of above S/L 2.0.

In this and the previous article, we have examined the fundamentals of how water and hull shape interact. In the next, we will follow the design process as each factor is applied to the creation of a new hull shape.

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