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The Physics Of Sailing

Boats Before Airplanes

It may not come as any great surprise that humans have been getting around on the water using boats for thousands of years. Airplanes are a relatively new innovation that was inspired by watching birds cover great distances in very little time. Thanks to Archimedes principle (circa 246 BC), boats displace a volume of water equal in weight to the weight of the boat. This allows watercraft constructed of various materials (including wood, aluminum, steel, cement, and lead) to float and maintain their stability.

Many wonder how airplanes stay aloft just as birds soar above a ridge full of updrafts. Airplanes are designed with a fuselage that carries passengers (or cargo), wings that generate lift, and appendages that provide control. The lift force must be at least as large as the weight force in order to maintain or increase the altitude. This is possible because of a principle called the Kutta condition. The formal definition is that the velocities leaving the trailing edges of the appendages (wings, tail, and rudder) must be finite. The practical implication is that the wind must leave these "foils" in a smooth pattern - without making any sudden turns. This results in the wind flowing around the upper side of the wing travelling a longer distance than the wind flowing over the lower surface. Again, we look (this time only as far back as 1752) to a brilliant mathematician named Daniel Bernoulli. He theorized that as a fluid moves faster over a surface, the pressure must decrease in order to satisfy the Kutta condition.

The above equation describes his theory in mathematical terms where rho is the fluid density, v is the fluid velocity, g is the gravitational constant, z is the distance below the "surface" of the fluid, and p is the fluid pressure. In this relation, the faster the fluid at any one point, the lower the pressure. On the flip side (literally), the slower the fluid at any one point, the  higher the pressure. This (finally) completes our quick discussion on what keeps planes from falling out of the sky.

Source: https://studiousguy.com/bernoullis-principle-examples/

Wind Versus Water

Sailboats "fly" in two distinct medium - wind and water. They float above the water, but use this Kutta condition to slice through the air and through the water. The layperson may think that wind and water are completely different worlds. But, to the aspiring mathematician they are are governed by the same set of general equations. The only real difference between the two is the density of the fluid and the compressibility of the medium. The density is simply the mass per unit volume. We all know that air can be compressed to a higher pressure inside of a combustion engine or around the nose of a supersonic aircraft. But, water is also "compressible" at very high velocities. Our discussion here will only focus on the low speed dynamics of sailing.

The Squeeze Play

We already talked about the pressures and forces that allow an airplane to stay aloft because of their wings. The same principles work in the world of sailing. The following diagram shows an overhead view of the forces working against each other to push a sailboat through the water. There is a force vector on the sails that is mostly sideways, but that has a (sometimes) slight direction pushing forwards. There is also a force vector working underwater that is due to the lift generated by an underwater wing called a keel (or centerboard, on smaller boats). The underwater vector is pointing almost exactly sideways to the centerline of the sailboat. The sum of these two vectors creates a "squeeze play" force propelling the sailboat forwards. This is similar to the idea of squeezing a slimy watermelon seed between our fingers and watching it shoot across the picnic table at our unsuspecting sibling.

The Balancing Act

Just as the sails and the keel work together to push the sailboat forward, they also offset each other to keep the sailboat from tipping over into a capsize. Increases in the wind velocity provide not just a driving force forward, but also a sideways force that tries to tip - or heel - the sailboat. These forces increase as the square of the velocity (thank you again, Daniel Bernoulli), but with a slight "snag" to the equation. As the sailboat tips, some of the wind starts spilling off the top of the sails and the heeling vector becomes less of a direct lever trying to capsize the boat in the water. At the same time, the heavy keel begins to exert its own influence by creating a "righting moment" trying to keep the sailboat from completing the capsize. The diagram below shows the balancing act by these two opposing forces.

The Total Drag

Sailing is all about harnessing the forces of nature to get to a desired  destination. Weight is a downward force that is offset (we hope) against an upward buoyant force. The driving lift from the sails is used to push against the sideways force from the keel. The righting moment from the keel helps to keep the sailboat upright and the crew from acting on their mutinous thoughts. The last thing we need to cover is the drag force that keeps the sailboat from reaching escape velocity (sorry to burst  your bubble all you fans of the children's book "The Wreck Of The Zephyr"). The drag force from the wind flowing over the sails and past the hull is important - but minimal - in relation to the drag forces created on the water side of the total picture. As with any discussion of fluid flowing past streamlined objects, there is an "easy" portion of the drag to estimate - namely frictional drag. Then there is everything else that gives mathematicians and physicists headaches. These "residual" drag forces are generated from turbulent swirls in the wake and the wave patterns displayed as the sailboat slices a path through the water. The total drag force is mostly friction at lower speeds and mostly waves and turbulence as we achieve higher velocities.

Stocks Versus Bonds

This discussion would not be complete without bringing us back to the reality of the investing world. Just as the sails on a boat  provide the driving force (or the excitement) to the experience, so too does the stock portion of an investment portfolio. The keel on a sailboat is similarly akin to a bond allocation helping to right the portfolio during stormy markets and smooth out our ride. The table below shows how the stock index funds tend to have what is called a negative correlation to bond index funds. This means that the two types of investment assets typically move in slightly different directions over time. The green coloring to the cells shows this negative correlation with the strongest "righting moment" currently being provided by the combination of iShares Core S&P Small-Cap ETF (IJR) and iShares 1-3 Year Treasury Bond ETF (SHY). 

The benefit to investors is that combining both stocks and bonds into a portfolio is exactly like pairing the sail and keel on a boat. The end result is that we can get to our destination without our crew jumping ship - just when we need them the most.  The chart shown below highlights this benefit in the form of a smoother ride over the past several years by combining an equal portion of stocks and bonds. We can adjust the portfolio allocations between stocks and bonds to suit investor risk preferences or current market conditions. Similarly, we can "shorten" sail or ask our crew to "hike" out on the weather rail when the wind velocity increases or we hear complaints about upset  stomachs.

See you out on the high seas!

Resources

https://en.wikipedia.org/wiki/Archimedes%27_principle

https://en.wikipedia.org/wiki/Bernoulli%27s_principle

https://en.wikipedia.org/wiki/The_Wreck_of_the_Zephyr

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