# Where did the false "equal transit-time" explanation of lift originate from?

It's supposedly a "widely circulated" false claim that wings generate lift because of their asymmetric shape, forcing air above to travel faster so that they meet up on the trailing edge at the same time as air below, and by Bernoulli's equation, the pressure below the wing must therefore be higher.

I've only ever encountered this "equal transit-time" theory when it's being debunked, so it seems like its more common to see it pointed out as wrong rather than alleged to be true.

Was this incorrect theory ever a common misconception? And did this misconception originate from a particular source?

• If you've only encountered it when it's being debunked, then that is because of recent improvements in education. When I was at school this explanation was virtually always used, and every single time I wondered why the top parcel of air couldn't just travel at the same speed as the bottom one, which was never explained. Aug 25, 2019 at 2:19
• During grade school, which textbook was used in physics? Do you recall? Or even earlier, the "science" books? Examine their explanations of lifting-force. (The physics class might get it right, but all other books are typically promoting this "longer path" fallacy. Sep 22, 2023 at 19:27

The popularity of the equal transit-time fallacy is a bit more complicated than a mistake spreading from a single source. It is simple, intuitively appealing (blowing over an airfoil is often invoked, along with an erroneous picture of flow lines around an asymmetric wing), and gets things done quickly. Just like the "explanation" of seasons by the Earth's changing distance to the Sun, or textbook pseudo-explanations of history, like the "crisis" over the discovery of irrationals, or Maxwell's "mathematical" reason for adding an extra term to Ampere's equation. "The truth is what works". Eastwell in Bernoulli? Perhaps, but What about Viscosity? lists the following motivations:

"• Simplified Bernoulli explanations are quick, sound logical, and make correct predictions. As Brusca (1986b) said with a sense of satisfaction, the prediction is “in complete agreement with what happens in practice” (p. 15). This would be fabulous if it wasn’t for the fact that these explanations are also wrong!
• Statements like “as the speed of a moving fluid increases, the pressure within the fluid decreases” facilitate a misunderstanding of Bernoulli’s principle and, when used in a sweeping sense and therefore out of context, are wrong.
• Viscosity, entrainment, and the Coanda effect are not to be found in lower-level literature, despite the fact that such literature deals with phenomena that rely on these concepts.
• There appears to be a desire to have a single, best explanation for an observed behaviour when in fact a combination of factors may be “at play”
"

The fallacy was likely independently reinvented by multiple textbooks authors and physics teachers. Norman Smith, who was one of the first to sound the alarm back in 1972, says the following in Bernoulli and Newton in Fluid Mechanics:

"Millions of children in science classes are being... told that Bernoulli's theorem is responsible for lift on the airplane wing and for the force that makes a spinning baseball travel in a curved path. Unfortunately, the "dynamic lift" involved in each of these items is not properly explained by Bernoulli's theorem.

[...] First, an airfoil need not have more curvature on its top than on its bottom. Airplanes can and do fly with perfectly symmetrical airfoils; that is, with airfoils that have the same curvature top and bottom. Second, even if a humped-up (cambered) shape is used, the claim that the air must traverse the curved top surface in the same time as it does the flat bottom surface (or that the molecules must meet again) is fictional. We can quote no physical law that tells us this. Third - and this is the most serious - the common textbook explanation, and the diagrams that accompany it, describe a force on the wing with no net disturbance to the airstream. This constitutes a violation of Newton's third law.

[...] The use of Bernoulli apparently began in this country some 30 years or more ago and has spread throughout school science books and popular literature to exclude virtually any mention of Newton and momentum. College-level aerodynamics textbooks generally do not use this approach, though it would not be surprising to find a few exceptions... It might be well to point out that the Bernoulli explanation is used mostly in the United States British, for example, seem to use Newton's third law in textbooks and popular literature".

Writing in 2007, Eastwell notes that things did not change much, if not gotten worse:

"Such a pressure difference is commonly justified, on the basis of Bernoulli's Principle, by statements such as “when air sweeps across a surface at high speed the pressure on that surface is lowered” (“Bernoulli Station,” 1989, p. 308) or “as the speed of a moving fluid increases, the pressure within the fluid decreases” (Mitchell, n.d., n. 1), or it is implied on this basis (e.g., Brusca, 1986b). This reasoning for the pressure difference, found not only in popular writings but also in specialist, peer-reviewed journals (e.g., see also Bauman & Schwaneberg, 1994; Holmes, 1996), is wrong."

Now that even Wikipedia has a subheading False explanation based on equal transit-time, perhaps the tide has finally turned. It is not that the use of Bernoulli's equation is entirely wrong, it is rather that it comes with a number of erroneous add-ons. As Beatty pointed out in Airfoil Lifting Force Misconception Widespread in K-6 Textbooks:

"Air-deflection and Newton's Laws explain 100% of the lifting force. Air velocity and Bernoulli's equation also explains 100% of the lift. There is no 60% of one and 40% of the other. One of them looks at pressure forces, the other looks at $$F=ma$$ accelerated mass. For the most part they're just two different ways of simplifying a single complicated subject. Much of the controversy arises because one side or the other insists that only their view is correct.

[...] Social psychology aside, there are also several serious mistakes usually associated with the "popular" explanation described above. Those who believe the "popular" explanation are wrongly insisting that any parcels of air divided by the wing's leading edge must meet again at the trailing edge. This is incorrect. Actually it doesn't even occur: experiments easily show that the air above a wing far outraces the air below, and parcels never meet again. (In fact, if a wing is adjusted so the parcels really do merge, this is always the zero-lift configuration!) The same people also believe that wings fly only because of pressure, and that wings don't need to deflect the oncoming air downwards. Also incorrect. These and several other mistakes commonly appear in elementary science texts, as well as in popular articles about aircraft physics. These mistakes change the popular "airfoil-shape" explanation into a system of misconceptions."

• Thanks for the detailed answer, I feel like the quotes mentioned in the answer are more criticisms towards Bernoulli's principle being incorrectly taught or "over emphasised" (in the sense that Newton's 'action-reaction' law is implied as irrelevant, and some simplified diagrams of air flowing around a wing do not even depict its deflection)... Aug 18, 2019 at 12:51
• ... I'm more interested in the specific use of the argument that air parcels that start at the wing together, must join up together as they leave, and because of this, air above must travel faster (as I don't think is any more intuitive than just believing on faith that air does move faster over the top of a wing). Aug 18, 2019 at 12:51
• @eugenhu Maybe I misunderstand, but isn't the argument supposed to be simply that higher speeds mean lower pressures, according to the Bernoulli's equation, and the difference is the lift? That's how Smith recounts the textbook "explanation". Or are you interested why people thought that the split air parts must rejoin? To maintain the laminar flow, presumably, although this is usually left unsaid. Aug 19, 2019 at 0:30
• Bauman-Schwaneberg justify it as follows:"If we consider a wing passing through stationary air, the principal effect on the air is the downward thrust. To the approximation that this is the only net effect, the time required for passage over and under the wing is the same. The air above must therefore be traveling faster (relative to the wing) to keep up with the lower streamlines." Aug 19, 2019 at 0:31
• I'm not an expert (I'm just learning the topic now), but as far as I understand, it's not entirely wrong to say the faster moving air above a wing causes lower pressures (it might be more accurate to say that one does not cause the other), by Bernoulli's equation, and this analysis can be valid for simple models. I just don't understand why some explanations felt the need to justify that air moves faster above a wing because it must 'catch up' to the air below, to me, this does not seem necessary for laminar flow... Aug 19, 2019 at 1:04

Klaus Weltner and Martin Ingelman-Sundberg (from the Department of Physics at the University Frankfurt) in an ultimately unpublished paper, “Physics of Flight — reviewed”, submitted to the European Journal of Physics in 2003, argue that the original source for the misconception of “equal transit time” was, somewhat unwittingly, aerodynamics pioneer Ludwig Prandtl circa 1922.

Their article leads with a summary of “Darstellung der Aerodynamik in Schulphysikbüchern”, a book published in 1989 by Klaus Niermann:

Niermann analyzed all German language textbooks published within the last one hundred years investigating the explanations of aerodynamic lift. The main result is that explanations based on Bernoulli’s law are dominating since the 1920s. He found the same to be true for a survey of contemporary American and English textbooks.

The authors eventually go on to conclude:

Thus the origin of the path length reasoning may be found in a diagram given by Prandtl in 1922.

With this diagram Prandtl tried to show that air at the inner layers stick to the surface. In respect to one point this diagram is not correct. It indicates that the volumes remain at the same vertical position and it indicates that originally adjacent air meets again at the end of the airfoil. […]

Diagrams of this type might have misled scholars to the hypothesis that adjacent air has to meet again after passing the airfoil.

The text says 1922, which may very well refer to the original German publication, though the actual reference the authors cite is the English translation, produced by the National Advisory Committee for Aeronautics, titled “Applications of Modern Hydrodynamics to Aeronautics” and published in 1923.

The asymmetric shape (not its explanation with "equal transit time"!) comes from the first mathematical theory that explained the lifting force (Chaplygin's formula, known in the West as "Blasius theorem"). I suppose that the theory of "equal times" was developed in attempts to explain the Chaplygin-Joukowski theory to non-mathematicians:-)

Under certain circumstances this theory gives a good agreement with experiment. Since a complete mathematical theory does not exist to this day, the subject is open to all sorts of speculations on "what causes the lifting force". A sound argument against the "asymmetric theory" is that an airplane can fly upside down:-)

This explanation has probably been independently 'rediscovered' many times, and it is unlikely to be able to point to a particular origin.

Imagine that two adjacent air elements ... A and B ... are about to reach the aerofoil’s nose. Element A passes above the aerofoil whilst element B passes beneath. Eventually element A reaches the aerofoil’s tail or trailing edge and element B, according to this argument, must reach the tail at the same time so as to regain its position adjacent to element A ... this flawed explanation of flight has survived and still enjoys wide belief, probably because it has the appeal of simplicity. Its origin now seems to be lost in the mists of time although it may be that its use in aircrew training manuals gained for it a popularity which extended to school textbooks and even the occasional British A Level examination paper.

# Some sources say this explanation is sometimes used in flight training manuals.

The above explanation is extremely wide-spread. It can be found in many textbooks and, to my knowledge, it is also used in the RAF's instruction manuals.

Source: Holger Babinsky (2003). How do wings work?. Phys. Educ. Vol. 38. p.498.
Accessed: http://www3.eng.cam.ac.uk/outreach/Project-resources/Wind-turbine/howwingswork.pdf

Ballow speaking about his experience when learning from Air Force training manuals while a member of the Nevada Air National Guard:

The training covered the theory of flight as outlined and explained in the Air Force Training Manual [...] Lift is generated by the air passing over the wing. since the upper surface of the wing is curved and the lower surface is straight, the air passing over the upper surface of the wing has farther to travel, and so it is thinned out.

Source: Tom Ballow (2005). The New Science of Flight and Movement. p.13.

Discussion on the r/flying subreddit:

DPE used the 'equal transit time theory' of lift as the correct way to explain lift. What do?

Posts on Aviation Stack Exchange:

Why is the wrong explanation of “air travels a longer distance and creates a lift” so popular?

Principle of aerodynamic lift: are misconceptions also taught in flight schools?

# Here are some specific instances of where the "equal transit-time" argument is used.

The distance over the top of the blade with the angle of attack is greater than the distance along the bottom surface of the rotor blade. Air molecules that pass over the top must move faster than those passing under the bottom to meet at the same time along the trailing edge.

Source: Federal Aviation Administration (2012). Helicopter Instructor's Handbook, Chapter 3 Aerodynamics of Flight. p.2.
Accessed: https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/media/FAA-H-8083-4.pdf

The shape of a wing looks like an elongated water drop laying on its side ... Usually the top is curved more than the bottom making the upper surface slightly longer than the bottom. Since air passing over the top and bottom must reach the rear of the wing at the same time, the air passing over the top must not only travel faster, but also changes direction and is deflected downward.

Source: DoD 101 - An Introduction to the Military, Aircraft for Amateurs
Accessed: https://fas.org/man/dod-101/sys/ac/intro.htm

The next examples are from older journal papers.

This acceleration of the particles over the top side of "the shape" will hasten the upper air-particles toward the trailing edge, so as to make them "join up" with their immediate bottomside neighbors, from which they were separated at the leading edge. Particle pairs thus hit the leading edge together, are separated, and yet arrive at the trailing edge together—although the topside particle had to travel much faster to do so.

Source: Norwood Russell Hanson (1965). Aristotle (And Others) on Motion through Air. The Review of Metaphysics, Vol. 19, No. 1.

If we consider a wing passing through stationary air, the principal effect on the air is the downward thrust. To the approximation that this is the only net effect, the time required for passage over and under the wing is the same. The air above must therefore be traveling faster (relative to the wing) to keep up with the lower streamlines.

Source: Robert P. Bauman and Rolf Schwaneberg (1994). Interpretation of Bernoulli’s equation. The Physics Teacher Vol. 32. p.485.

# As for why "equal transit-time" ever had to be invoked.

I could not find any physical justifications for it, perhaps its surface level intuitiveness stems from being conflated with some kind of continuity equation/condition, almost as if air parcels split at the leading edge, must rejoin at the trailing edge otherwise there would be a "build up of air" on the side with the longer path (the "equal transit-time" requirement is of course not necessary to avoid this). This Quora question seems to suggest there is some confusion related to this: Why continuity equation must be followed by aerofoil?

"Hat der Körper eine gleichmäßige Geschwindigkeit in ruhender Luft, so wird auch in der den Körper umgebenden Luft eine gleichmäßige Bewegung eintreten, die im wesentlichen darin besteht, daß die Luft vor dem Körper sich auseinander thut und hinter dem Körper wieder zusammengeht."

Otto Lilienthal 1889

Which translates as

"If the body has a constant speed in still air, then there will also be a uniform movement in the air surrounding the body, which essentially consists in the air separating from the body and coming together again behind the body."

Writing about two years after you posted this question, Graham Wild, a Senior Lecturer in Aviation at UNSW Canberra, posted a preprint to the physics education section of arXiv that attempts to answer this: [2110.00690] On the Origins and Relevance of the Equal Transit Time Fallacy to Explain Lift. He traces the misunderstanding all the way back to D'Alembert in 1744, who simply assumed equal transit time (ETT). Later work, in 1752, purported to derive the ETT property from Newton's laws, applied to fluids. This incorrect derivation was subsequently confirmed by Euler in 1757, lending authority to it. An ETT result does hold for potential flows, but such flows do not produce lift. Wild's paper (which I have not fully digested) discusses whether ETT can nonetheless play a role in helping to understand the origins of lift, leaving it as an open question.