What the Wild Goose Knows
Sophisticated puzzle submitted by 
Alan Niquette in 1999
 


Copyright 1999 by Paul Niquette.  All rights reserved..

 
Why is one arm of the 'V' formation always longer than the other?

In a simple search of the Internet (geese AND migration or "V formation"), you will find an explanation for why, with each dawn of spring and dusk of autumn, we are privileged to observe those honking echelons in the sky.

    More than one explanation, in fact. As many as a hundred explanations. All exactly alike. An exclamation point might seem appropriate for the previous sentence. Let's save it for later.
According to Museum of the Canada Goose, Angeles Arrien gave a speech entitled "Lessons from Geese" at the 1991 Organizational Development Network based on the work of Milton Olson. Soon after, it was transcribed and began circulating among Outward Bound staff throughout the United States.
    The piece has been enthusiastically scarfed up in paste buffers and republished (usually verbatim) on websites all over the Internet, either attributed to "author unknown" or given a self-asserted by-line.
With minor variations, "Lessons from Geese" has been replicated throughout the world and will inspire collaborative behavior in perpetuity.

All renderings of "Lessons from Geese" include the same technical explanation for the V formation: "As each bird flaps its wings, it creates an 'up lift' for the bird following."

    At first glance, one might suppose, that such an explanation would have seemed ironic to at least one of the website owners, inasmuch as most people probably presume that flapping wings create a downward flow of air, claiming the benefit of whatever "up lift" there may be for the bird doing the flapping -- decidedly not "for the bird following." Indeed, the aerodynamic term is 'downwash,' a phenomenon readily observed in wind-tunnel photographs showing smokey streaks that droop from the trailing edge of experimental wing foils.
Sophisticated puzzle solvers have at least an elementary understanding of how things fly. Many have even studied Bernoulli's Principle and know that the air flowing over the top surface of a wing must necessarily be at a lower pressure than the air flowing under the bottom. That difference in pressure produces 'lift.' At the extreme ends of each wing, whether bird or plane, the high pressure air underneath pushing upward tends to leak around the wingtip onto the top surface where there exists a relative vacuum. The resulting air flow actually spoils part of the wing's lift. It does something else, too.
    While the air is flowing around the tip of the wing, the tip of the wing is moving forward, leaving behind a circulating pattern of air called a 'wingtip vortex.' It continues to spin and expand, forming a cone behind each wingtip, clockwise on the left, counter-clockwise on the right. In aviation, the phenomenon, wrongly described as 'wake turbulence,' resembles a pair of horizonal tornados.
Invisibly swirling off the wingtips of huge aluminum birds those vortices create dreadful hazards in the sky for following aircraft (see Sky Below on this website). In feathered flight, it is the wingtip vortex that provides the "up lift" exploited by migrating birds (see illustration below).
 

The ubiquitous essay "Lessons from Geese" is not nearly so remarkable for what it teaches us about geese but for what it teaches us about us. Consider the passage that reads as follows:

By flying in a V formation, the whole flock adds
71% greater flying range than if each bird flew alone. 

Critical thinking will surely be stimulated by the precision of that numerical assertion -- not "approximately 70%," not "more than 2/3rds," but "71 % greater flying range." You might try turning your favorite search engine loose on that phrase: "71 % greater flying range" and see what you find.   Here's a selection of -- well, 71 websites that Google found for me in 1999...

    Anti-Smoking Site, Archives Page, Australia's Fair Dinkum, Aviary Website, Berkeley Resort: Deep Thoughts, Best of Excel News, Bethel Church Website, Brotherly Love, Relief and Truth, Business Quality Network, Candlelist Digest, Christian Harmony- Illustrations, Church of God in Christ Website, CLub Loopy, Collection of AA Writings, Community Healing Project, Discussion Group, Mumbai Central, Elementary School, E-zine and Magazine, Fifth Realm Thoughts, FlyBusiness Women, Geese - on the Radio, German Poems, Happy News, High School Alumni Email Forum, High School Journalism Site, Human Resources, Human-Relations, Independence, Inspirational Page, Inspirational Page at a Multi-Level Marketing Site, Inspirational Site, International Prayer Fellowship, Islamic Network Discussion Archives, Islamic Voice, Jehovah's Witnesses Official Website, Junior High School Teacher's Website, Lessons From Geese, Lessons From Geese, Lessons From Geese, Lessons from the Geese, Lessons From the Geese, Lessons from the Geese, Make a Joyful Noise, Martial Arts Website, Mom's Cocoon, New Life Churches of Australia, New Zealand Site, Newsletter for Hewlett-Packard Computing Professionals, Newsletter-Clean and Green, Newspaper Editorial, Nursing College, Personal Home Page, Personal Website, Philosophy, Plum Island Surfcasters, Poultry Newsletter, Precision Aire, Right to Life, S.A.F.E./Literature, Scoutmaster's Minute, Sermon for Ordinary, Several Newspapers, Slide Show for a Faculty Meeting, Some Schools, Stories Sites Galore, Strategic Studies, Superintendent s Message, Teams Make Performance, The Goose Story, The Motley Fool, Things Are Look'n Up, Website for Stories, Weekly Bulletin from a Catholic Church, and World Association for Persons with Disabilities.
...all containing the exact same text:
     
    By flying in a V formation, the whole flock adds
    71% greater flying range than if each bird flew alone. 
So eager are people to believe this numerical assertion, multitudes have apparently suspended all analytic judgment. That indeed may be a sentence at the end of which an exclamation point belongs.

erodynamics won't generally be regarded as inspirational. That's because aerodynamics is mostly about drag. Drag?  What the sophisticated solver knows is that four forces act on a body in flight: weight and lift, thrust and drag.

  • Weight acts upon all objects on earth. In flight, weight is overcome by lift. Lift is provided by wings on aircraft and on birds. Lift overcomes weight. In steady flight, lift equals weight.
  • Thrust is provided by engines in aircraft and from the flapping of wings in birds. Drag acts upon all objects that move through the air. Drag tends to retard motion and must be overcome by thrust. In steady flight, drag equals thrust.
The ratio of lift to drag (L/D pronounced &LOV&RDE) is a key aerodynamic parameter. In steady flight, L/D is equal to W/T, the ratio of weight to thrust. For a goose flying at migration speed, a generous estimate of L/D = 20, which means W/T = 20. The sophisticated solver will regard that as a kind of 'mechanical advantage' (like a 20-to-1 lever) and might be heard to exclaim "Hoo-hah, the goose requires thrust amounting to only 5% of its weight to stay aloft."
    The wing of a bird provides lift whether it's flapping or not. Flapping produces thrust. The shape of the wing changes during the downstroke to force the air underneath to flow toward the rear. To summarize: Thrust overcomes drag, which gets multiplied by L/D to produce lift, which overcomes weight.  Huh?
If each goose flying in the V formation is to add "71% greater flying range," as alleged, 71% less drag must be experienced "than if each bird flew alone."  We begin our analysis with the lead goose, the only bird in the formation not receiving any benefit of "up lift" from the wingtip vortex of a preceding bird. The lead goose necessarily faces the full force of drag as if flying alone. The sophisticated solver knows what the wild goose knows -- that there are two different kinds of drag.
  • 'Parasite' drag is common to all objects that move through the air; it is the 'price' paid for motion. Parasite drag increases with speed.
  • 'Induced' drag is common to all objects that fly through the air; it is the 'price' paid for lift. Induced drag decreases with speed.  Counter-intuitive, that.
In order to achieve the greatest flying range, the lead goose is well advised to maximize L/D, which can be accomplished only by minimizing drag. There necessarily exists what is called 'minimum drag speed' -- flying faster requires more thrust to overcome parasite drag, and flying slower requires more thrust to overcome induced drag. At the minimum drag speed, thrust is least, range is most, and parasite drag just equals induced drag.
  • Parasite drag has a common name: 'wind resistance.' It is easy to see that 'streamlining' reduces wind resistance. The wild goose is well streamlined.
  • Induced drag is determined by the contours and dimensions of the wing, which shapes the air flow to provide lift.  You need a wind-tunnel and smoke trails to see that.
Nota bene, the V formation does nothing to reduce parasite drag.  To maintain the V formation, of course, all the birds must fly at the same speed and experience the same wind resistance as if flying alone.  Not so with induced drag.  Except for the lead goose, every bird in the formation reaps the benefit of "up lift" from the wingtip vortex of the bird ahead. That "up lift" translates into a reduction only in induced drag experienced by the bird that receives it. The question is, how much?

et us make an initial -- extreme -- supposition that each bird gets 100% of its lift from the wingtip vortex of the bird ahead. In other words, that "up lift" equals 100% of the bird's weight. The following bird would not have to produce lift for itself and therefore would not have to overcome induced drag. It still has to flap its wings, though. Every bird must provide sufficient thrust to overcome parasite drag. With an induced drag of zero, the following bird would have to overcome only 50% of the drag faced by the lead goose.

Our extreme supposition was generous to the max and probably unrealistic. Nevertheless the widely asserted value of 71% greater flying range has already been demolished, for we must conclude that a bird in formation will have no more than 50% greater flying range than if it flew alone.
A more realistic supposition is that while flying in the V formation each bird gets 50% of its lift from the wingtip vortex produced by the bird ahead and the remaining 50% of its lift provided by its own wings, which reduces its induced drag to 50% of that faced by the lead goose -- that plus the parasite drag that every bird must overcome.
Our more realistic supposition reduces the estimated available "up lift" such that a bird in formation will have no more than 25% greater flying range than if it flew alone.
Each bird has two wings (there's a First Principle for you) with which to provide sufficient lift to overcome what we have postulated to be half its own weight and, through flapping, to overcome half the parasite drag. The following bird can reap the benefit of only one wingtip vortex generated by the bird ahead, which cuts the available "up lift" in half.
Taking into account the fact that only one wingtip vortex provides "up lift" from the bird ahead, a bird in formation will probably have no more than 12.5% greater flying range than if it flew alone.
The entire motion of the wingtip vortex must also be taken into consideration. Since the circulating air from the 'horizontal tornado' is moving upward during only about half of its circulation, the available uplift is diminished accordingly.
Taking into account all these effects, we might reasonably conclude that a bird in formation will have no more than 6.25% greater flying range than if it flew alone. But there are more effects to evaluate.
The structure of the wingtip vortex is conical. This is a consequence of the resistance to its inertial motion by the surrounding air, which slows the circulation speed with increased distance from the bird ahead, reducing its momentum and expanding its radius. The "up lift" will be adversely impacted by an unknown factor, but probably more than 50%.
Inasmuch as the wingtip vortex weakens behind the bird ahead, we must adjust our estimate such that a bird in formation will have no more than 3.125% greater flying range than if it flew alone.
The ability for the following bird to capture the "up lift" for its own benefit depends on a number of complex variables, including interaction if the wingtip vortex from the preceding bird with the body of the following bird, relative flapping phases, and various body dimensions. These factors -- and the need to allow for the birds to take turns serving as lead goose -- will further reduce the benefit of the V formation to each bird.
Setting aside the specious precision in our own numbers (four significant digits? why not five?), we might reasonably conclude that a bird in formation will have no more than 1.5625% greater flying range than if it flew alone (not "71%").
To call attention to that conclusion, an exclamation point appears at the end of this sentence!

f the benefit of the V formation is really only a 1-2% greater flying range, one might wonder why that behavior would have ever evolved.  Richard Dawkins in Climbing Mount Improbable (W. W. Norton 1996) provides abundant evidence that even the smallest advantages of a trait will assure its perpetuation through natural selection.

Ah, but there are other advantages to flying in the V-formation, as described in Wikipedia. That's what the wild goose knows.

    The wild goose knows something else, too: not to fly directly behind the bird ahead. There's a double-dose of 'down-draft' lurking back there (see diagram above) -- a real drag for the goose and possibly a 2% penalty in flying range.
Groaner Alert: What we see in the V formation is each individual bird avoiding a wild goose chase.
Almost forgot...



Home Page
Puzzle Page
Math and Models
The Puzzle as a Literary Genre