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he Cosine Wherry

One man's approach to boat design.
Design and commentary by John A. Hartsock.

John's description of the Cosine Wherry first appeared in Wooden Boat Magazine, May/June, 1991, pp. 100-103.  He is an engineering consultant specializing in wall systems.  John has raced sailboats on San Francisco Bay, Long Island Sound, and Galveston Bay.  In 1976 he and his family moved to Edmonds, Washington, and his interest turned to open-water rowing.  He can be reached at 619 Sater Lane, Edmonds, WA 98020. 

It was at a dinner in Vancouver, British Columbia, one evening in 1976.  One of the guests, an Englishman, was a hydrodynamicist.  I asked him why attack submarines had bulbous bows.  His answer - I don't recall just how it applied to the question - was, "Of course, displacement should follow a cosine curve.."  This was the key to something for which I had been searching.

I learned to row with the Sea Scouts while growing up in Palo Alto, California, studied chemical engineering at Berkeley, and learned elements of programming a computer in the 1960s.  By the early 1970s I was convinced it should be possible to loft a superior rowing boat entirely from a set of equations.  But I had made a mistake early on by working from water-lines, much as a draftsman draws lines on paper.  That night in 1976, I realized I should have been working from displacement.

My reasoning went something like this: Work is force through distance, and force is mass times acceleration.  Acceleration is both linear and angular.  A moving hull forces water to accelerate - both linearly and angularly.  The elements of water will take the direction of least resistance.  They will follow, a three-dimensional curving path relative to the hull before returning to rest.  Displacement, unlike waterlines, is three-dimensional ....

The term "cosine" comes from trigonometry.  As shown in Figure 1, the cosine of an angle is the ratio of the adjacent side of a right triangle to the hypotenuse.  Transitions are very gradual when cosine is plotted against angle.  This is my reason for using a cosine curve for displacement when applying trigonometry to boat design.  As shown in Figure 2, waterline length is the horizontal axis and the area of sections the vertical axis.  Displacement is represented by the area under the curve.

I thought about the path that elements of water follow as a boat moves, but soon I realized this was too complex a subject on which to spend much time.  Similarly, I did not give a great deal of attention to frictional drag and turbulence.  Rather, I gambled on making flow as smooth as possible by controlling the displacement curve and, where convenient, reducing surface area.  Later the gamble would pay off, and the Wherry would prove to leave little wake.

By 1977 I was working in Seattle, Washington, and had developed a hull-design program for a Texas Instruments programmable hand calculator.  Then I met Tom Whitaker, who was helping build a Perry-designed sailboat.  Tom had learned cold molding in New Zealand after teaching in Thailand for a number of years.  He became interested in my project, and Cosine I - a 16' x 30' cold-molded, single recreational shell - resulted.  The following year I sold her and designed Cosine II - an 18' x 36' shell - which Tom also cold molded.  I rowed that boat around virtually all of the islands in Puget Sound and raced her in the open-water rowing events that were being held at that time.

In 1982 Bob Pickett of Flounder Bay Boat Lumber in Anacortes, Washington, asked me to design a 14' strip-planked rowboat.  Bob was selling cedar strips as a result of mention in David Hazen's book, "The Stripper's Guide to Canoe Building."  Bob would eventually supply materials for a prototype that I would build and keep.  The result of this arrangement was the Cosine Wherry.

Through the years I have seldom passed up an opportunity to row any boat and had often rented Whitebar skiffs and other traditional boats from Dick Wagner's Old Boat House on Lake Union.  I soon decided the Wherry should have a traditional hull shape.

Final dimensions came from my own experience and careful study of lines collected by John Gardner and Willits Ansel.  Gardner's work had been published in National Fisherman and later appeared in his book, "Building Classic Small Craft (International Marine Publishing Company, 1977).  Ansel's, "The Whaleboat," was published by Mystic Seaport Museum in 1978.

Pickett had specified the boat's length.  Beam was decided mainly on the basis of spread between oarlocks, particularly at the forward and after rowing stations.  Depths came largely from dimensions shown by Gardner and were greater than I otherwise would have used.  The shape of the "midship" section - with appreciable deadrise and rounded bilges - was influenced by Ansel's descriptions of whaleboats.

Prismatic coefficient is defined in Figure 2.  It is a measure of the fineness of the ends of a bull.  The coefficient of Cosine I (the first shell) was only 0.500.  This meant the waterline went from the equivalent of 180 degrees to +180 degrees.  This prismatic coefficient created considerable surface area with little added displacement and reduced fore-and-aft stability.  In Cosine II, I had increased the coefficient to 0.552, and in the Cosine Wherrv, I further increased it to 0.576.

Considerable attention was devoted to thwart and oarlock locations, shown in Figure 3.  The boat needed to be rowed by one or two persons, but was too short for three oarsmen.  A single oarsman would sit on the center thwart, or two rowers would sit on the forward and after thwarts.  Thwart height was set so that, based on the center rowing station, oar grips came into the body about 12' above the thwart.  Oarlocks were set 12 abaft the after edge of the thwarts.  The boat's center of buoyancy, and luckily her center of gravity, fell at the after edge of the center thwart.  (The 100-lb Wherry can be carried by a single person facing aft with the thwart resting on his shoulders.)

Calculations used for the design of the shells had to be refined and expanded.  By now I was working on my own as a consultant and had purchased a Hewlett Packard HP-41, the- ultimate programmable hand calculator of the time (perhaps of all time).  The process, once the basic dimensions were selected, was to decide on equations for the following: Sheer, in both plan and elevation; keel and ends; mid-ship section; and transom.  It was necessary also to select factors used in the above equations.  I came to think of it as mathematical sculpting.  Because I had no CAD system, I needed to make a series of calculations on the HP-41, draw the shape on paper, and then decide if I wanted to change it.  Changes involved going back to the equations and the factors they contained.  Since the Cosine Wherry was to be asymmetrical, two sets of computations were required, one for the bow and one for the stern.  The only requirement was that the shapes had to be the same where the two halves were to meet.

A key to the design process was the selection of an exponent in the equation for the shape of the sections.  As shown in Figure 2, the displacement curve dictated the area of each section below the waterline.  Sheer and keel shapes were fixed in the early stages of design, and the curve of each section had to pass through these two points.  The fullness of the curve - and thereby the area below the waterline - thus became a function of the size of the exponent.  It was determined by a trial and error (iterative) subroutine in the program.  The area could also be varied by adjusting the keel line.  If the keel were to come up gradually, as it did toward the stern, the exponents would be larger and the bilges fuller.  Forward, the keel was deeper and exponents smaller for a sharper entry.  Full after sections made the boat more stable.  The final result was a table of x,y,z coordinates from which mold stations could be made with little or no lofting.

I built the prototype Cosine Wherry in our one-car garage and launched her in the spring of 1983.  Strip planking proved to be an excellent method of home construction of this type of boat.  It placed little constraint on the hull shape.  Few special tools or skills were required.  I also carved two pairs of 8' spoon-bladed spruce oars.  In some ways they proved to be more of a challenge than the boat.  A better length for them might have been 8' 6".

I decided to call the new boat the Cosine Wherry - not because she met any definition of a wherry, but in memory of the old Navy boat in which I had learned to row.

The Cosine Wherry proved to be better than any of us had expected.  She became the basis for "Rip, Strip, & Row" by Brown, Pickett, and Hartsock (Tamal Vista, 1985) from which several hundred Wherries have been built.  Since then I've designed boats ranging in length from 8' to 24' using similar methods.

The Cosine design method could be applied to other displacement hulls; but, for the time being at least, I prefer to stay with rowing boats.  Sailboats are complicated by rigging and angle of heel.  Powerboats require a knowledge of engines, which I do not have.  (I hate to see an outboard mounted on a boat that rows as easily as the Wherry.)  Open canoes do not perform well on the salt water near which I live.  Kayaks are excellent sea boats; however, for unexplained reasons, I do not really understand them nor am fully comfortable in them.

Creating the Cosine Wherry was a rewarding experience.  She has given me an appreciation of what must have taken place between builders and users of small boats in the 19th century.  They must have been wonderfully intelligent people with an intuitive feel for hull shape and a willingness to communicate with one another.