Analysis of a 1941 Paper Titled: The Relationship of Wind Correction Angle to Drift Angle
1941 Authors: W. S. ALEXANDER AND 0. M. KLOSE, Boeing School of Aeronautics
We are in an airplane and want to fly from A to B, what heading should we fly? Assuming we know the coordinates of A and B, we can easily compute and fly the course heading needed to get from A to B. Except, if there is wind, and there is almost always wind. For any wind not 0° or 180° relative to our desired course heading, we will be pushed left or right. The angle of this push is called drift angle. So, pointing our airplane directly on the desired course heading to B will not do, we will never get to B! We need to compute a modified heading that factors in the wind if we want to get to B. This modified heading is found by computing what is called a wind correction angle and adding or subtracting the wind correction angle to or from our desired course heading. Thus,
So now a related question. Is the wind correction angle the inverse of the drift angle? For example, if we are getting pushed 8° to our right by the wind, should we fly 8° to the left of our desired course heading? The answer is, no, the wind correction angle almost never the inverse of drift angle.
With that preamble, a diversion from our hypothetical flight and on to reviewing this 1941 paper: The Relationship of Window Correction Angle to Drift Angle which is the only reference I’ve been able to find that explains the math. The fact that drift angle is almost never our desired wind correction angle should be obvious, but for me it was not. I stumbled upon it, and I had to work hard (including re-learning math I had long forgotten) to figure this out. Where would I be without the internet, generally, and especially YouTube?
For historical context, the paper was published by Boeing just before or after the United States entered World War II (Pearl Harbor was attacked Dec 7, 1941). Getting from point A to point B, bombing the heck out of point B and then safely returning to point A or diversionary point C would have been top of mind of the authors. I doubt that GPS/GNSS could have even been a dream at the time (first satellite was launched in 1957). There were radio navigation systems in the 1940s but highly unlikely that radio nav was available over enemy territory! So, pilots and navigators had to rely on dead reckoning and visual observation.
At the time, an important navigation task whilst airborne was to calculate actual wind speed and angle. They did this using the knowns of airspeed, ground speed and wind drift angle, the latter two measured optically. With this information in hand, they could compute the necessary wind correction angle to stay on the desired course (and also where their bombs were likely to hit). The handy-dandy tool used for these computations is still in use today and is called an E6-B Flight Computer.
With this context in mind, the paper states:
THE PURPOSE of this paper is to provide a definite and comprehensive answer to the long-standing question among aerial navigators concerning the exact nature of the relationship between wind correction angle and drift angle.
So, they were primarily solving a navigation problem but also likely solving a bombing accuracy problem. Perhaps the authors were building navigation instruments, including mechanical computers? Perhaps they were working on a bombsight? Or a combination of the two? Regardless of precise motivation, the paper derives a set of equations that define the relationship between an aircraft’s drift angle and wind correction angle as a function of airspeed, windspeed and the incident angle of wind relative to desired course. For reasons I have not figured out, yet, over half the paper is devoted to developing equations and graphs showing the difference (subtraction) of wind correction angle and drift angle for ratios of windspeed over airspeed between zero and 1 and incident angles of wind from 0° to 180°. Further, the paper derives differential equations used to calculate maximums and minimums of the difference.
Diagram and Definitions
Here are the definitions from the paper and a cut/paste of the diagram they used. They draw two wind triangles, one for correction and one for drift and they define θ such that the trig identity of sin(180°-θ)=sin(θ) may be used to simplify equations.
a = the true airspeed of the aircraft w = the speed of the wind θ = the wind angle D = the drift angle C = the wind correction angle r=w/a, ratio of the wind speed to the true airspeed of the aircraft
ALT Image Text: The diagram shows two wind triangles, one for drift and one for wind correction. Triangle OA’B’ represents the wind correction triangle and triangle OAB represents the drift triangle. Vector OA of length a is desired course. Vector OB’ is co-linear with vector OA and has a length that is ground speed after wind correction applied. Vector OA’ of length a is wind-corrected course. Vector A’B’ of length w is the wind vector forming the wind correction triangle. Vector OB has a length that is ground speed is course after drift is applied. Vector AB of length w is the wind vector forming the drift triangle.
Derive Equations for Angles C and D
The paper first develops the equations for angle C, wind correction angle, using the law of sines. Law of sines is used in most every other reference I have found. The only twist in this paper is that the authors use the sin(180°-θ)=sin(θ) trig identity to simplify.
And because w/a=r
Now, the drift angle
With law of cosines, we know that:
∴
Intermediate Result with an Example
Stepping away from the paper, we now know all we need to set our heading and compute the related values of drift angle and ground speed with correction and with drift.
We know that:
And
We have previously derived the ground speed with drift as
Similarly, the ground speed with wind correction is (interior angles of a triangle add up to 180°).
Working all this into an example. Let’s say that our hypothetical point B is at a bearing of 75° from our current position, point A. Let’s further say that our plane is flying at a = 100kts and we have a wind blowing at 30° with velocity of w = 20kts. Note these three things:
Navigation bearings are 0° north and clockwise, which is not how trig functions are defined!
θ = ∠OA – wind direction angle.
Wind direction in aviation weather reports is “from”, so the wind vector is the reciprocal.
In our example, θ = 75°-30° = 45°, ∴
Now whilst it is true that the angle formed is 8.13° and true to the diagram, this is not quite what we need for wind correction. Recall, wind is reported “from” and not “to”, so 20kts at 30° is not our wind vector. Our wind vector is the reciprocal. We can manage this fact in one of several ways, e.g. flip the sign of the result, flip the sign of the input value, or add 180° to the input value. We can use any of these methods so long as we are clear which we choose, explain that choice and how to use the result.
In our example, we want to subtract 8.13° (flip the sign) from our desired course heading, i.e. we want to fly a 67° heading to achieve our desired 75° course heading. The corresponding ground speed will be:
To summarize,
Given plane = 100kts @ 75° and wind = 20kts @ 30°
That’s all for now, I’ll update this post or create a new one with the remainder of the paper. Preview is:
It should be apparent that angles C and D will be equal only when
Or simplified (square both sizes and divide both sizes by w)
Reference Sites and Videos
All 6 Trig Functions on the Unit Circle – video
Online E6B Computer / E6B Emulator – online calculator
luizmonteiro - Time - Speed - Distance / E6B Emulator – online calculator
https://mediafiles.aero.und.edu/aero.und.edu/aviation/trainers/e6b/ - University of ND online E6B
How to solve a word problem using vectors (youtube.com) – Drift and GS using component vectors
Aviation Formulary V1.47 (edwilliams.org) – Ed William’s all things E6B calculations
E6B - Wikipedia – E6B Wiki
Mathematics of Flight Crosswinds.pdf (af.mil) – US Air Force Museum paper, math and sample problems
https://www.geogebra.org/calculator/rurdakax - GeoGebra model I built