4.4.1 Vectors and the wind triangle Velocity vectors We know that an aircraft in flight is airborne, and consequently both the path it projects over the ground and its speed relative to the ground are the resultant of the aircraft velocity and the wind velocity. Those velocities are vector quantities having both magnitude (speed) and direction in azimuth, so we can add the two velocities together to produce a resultant vector representing the aircraft's groundspeed and its track over the ground in azimuth. It is common practice to do that non-mathematically by drawing scaled, arrowed lines to represent each vector quantity. The lengths of the lines represent the magnitude (speed) of each vector, and the placements indicate the application points and the directions of motion; the resultant vector represents the aircraft track over the ground and the groundspeed. For example, waypoint Beta is 150 nautical miles north-east (045° true) of waypoint Alpha and an aircraft departs overhead Alpha for Beta, maintaining a heading of 045° true while flying at the aircraft's normal cruising speed of 75 knots TAS. At the time of departure, the wind velocity at the cruise altitude is 135° true at 20 knots; i.e. the 20 kn wind is coming from the south-east. Where will the aircraft be after two hours flight? Certainly not over Beta, as it will have moved 150 nm north-east within the air mass while the air mass has moved 40 nm north-west. So we might surmise that after two hours flight its position will be about 40 nm north-west of Beta, and this is shown in Figure 1. The aircraft has drifted from its intended path or track over the ground and the 'track made good' is about 15° to the left of the 'required track'. We should note that, relative to the aircraft's course, the wind velocity normally has both a crosswind component and a headwind or tailwind component, and that headwind or tailwind component will also affect the aircraft's speed relative to the surface — the ground speed. (I have used the ISO standard symbol 'kn' for knot in the diagrams; the symbol 'kt' sometimes seen is the standard symbol for kilotonne.) Note: in the USA the term 'course' is synonymous with 'track' for air navigation, but the International Civil Aviation Organisation (ICAO) preferred usage is 'track'. The heading is the direction with which the longitudinal axis of the aircraft is aligned at any given time. This heading may be expressed as relative to true north — the true heading; or if adjusted for magnetic variation — the magnetic heading; and if adjusted for variation plus compass deviation — the compass heading. The wind triangle So, if we want to track over the direct route from Alpha to Beta we will have to ascertain both the expected wind velocity at the time of flight and a heading to fly that will provide the necessary crosswind correction angle. In the 3-vector wind triangle only the wind vector is completely known — the forecast wind velocity 135°/20 kn. We know part of the air or heading vector — the true airspeed 75 knots — but not the direction. We also know part of the ground vector — the direction (ground track 045° true) from Alpha to Beta — but not the ground speed. We can determine the two unknowns — the heading and the ground speed — by plotting scaled vectors on paper (figure 2). You will need some drawing instruments, a protractor and ruler, but a pair of dividers can be useful. • Draw a line connecting v1 and v2, marking it with one arrow to represent the heading vector and measure the line's orientation with true north with the protractor to determine the heading (060°T). Thus we have the first unknown — the direction in which to point the aircraft. Annotate the heading (060°T) and TAS (75 kn). Also note the wind correction angle [WCA] — the difference between the required track (045°T) and the heading (060°T) — is 15°, and the drift will be to the left — also known as port drift. Note: the wind correction angle is the angular difference between the required track and the heading, intended to ensure that the track made good will equate with the required track. Note that the terms 'crab angle' and 'drift angle' are very often used instead of 'wind correction angle'. But the latter term is more precise; crab angle and drift angle do have slightly different meanings or associations. Drift angle is measured in flight, and is the angle between the heading and the track made good. Crab angle is the preferred term when associated with crosswind landing. • Now measure the distance between Alpha and v2, which is the distance (72 nm) moved over the ground during one hour. This is the second unknown — the ground speed. Annotate the ground speed (72 kn) adjacent to the bearing (Figure 3). • We can now calculate the sector flight time from overhead Alpha to overhead Beta; this time is called the estimated time interval [ETI]. ETI (minutes) = Distance (nm) / ground speed (kn) × 60 = 150/72 × 60 = 125 minutes. It is interesting to note that even though the wind is a full crosswind to the track required, the ground speed is less than TAS and thus the ETI is a bit greater than you may have expected. This is because the heading of 060° would now include a small headwind component. Direct headwind/tailwind If the wind is aligned directly with the required track then of course it is not possible to construct the triangle, as there is no wind correction angle and the ground speed is the TAS ± wind speed. However, just as an illustration that the wind triangle still provides the correct answers, I have repeated the previous Alpha to Beta plot with winds that are only 10° off the required track; i.e. nearly full headwind and tailwind components. It may be thought that if an out-and-return trip is flown where the wind is directly aligned with the required track, the headwind encountered in one direction will be offset by the tailwind in the reverse direction; thus the total flight time will be equivalent to that in nil wind conditions. Not so — the greater the wind speed the greater the flight time on an out-and-return flight, no matter what the wind direction. Imagine a flight Alpha–Beta–Alpha in nil wind conditions. The ground speed on both the 150 nm outward and return legs would equal the TAS (75 kn) and each leg would take 120 minutes for a total flight time of 240 minutes. Now let's factor in a 25-knot north-east wind. The ground speed on the outward leg would be 50 kn and the ETI would be 180 minutes, whereas the ground speed on the return leg would be 100 kn and the ETI 90 minutes for a total flight time of 270 minutes. 4.4.2 Estimating heading and ground speed Plotting the wind vector triangle is the most accurate method for ascertaining heading and ground speed, but there are two other methods that are quite accurate enough for light aircraft cross-country navigation – use of the 1-in-60 rule and the use of tables. But first a quick look at trigonometrical relationships. Trigonometrical relationships - sine and cosine The trigonometrical relationships of the two wind components — crosswind (that component of the wind velocity that acts at right angles to the track) and headwind/tailwind (that component of the wind velocity that acts inline with the track) — is shown in a modified wind triangle (Figure 5). The sine of an angle = opposite side/hypotenuse, while the cosine of an angle = adjacent side/hypotenuse. In this example the wind angle is 30° relative to the required track of 045° true and the wind speed is 20 knots; the hypotenuse represents the wind velocity vector, the side opposite to the wind angle is drawn from the start of the wind vector so that it forms a right angle with the track so representing the crosswind component of the wind velocity while the side adjacent to the angle represents the headwind component. Reading from the abridged trigonometric table below, sine 30° is 0.5 and cosine 30° is 0.866 — near enough to 0.9, thus the crosswind is 0.5×20=10 kn and the headwind is 0.9×20=18 kn. Abridged trigonometrical table Relationship between an angle within a right angle triangle and the sides: Tangent of an angle=opposite side/adjacent side Sine of an angle=opposite side/hypotenuse Cosine of an angle=adjacent side/hypotenuse Angle Sine Cosine Tangent Angle Sine Cosine Tangent 1° 0.017 0.999 0.017 50° 0.766 0.643 1.192 5° 0.087 0.996 0.087 55° 0.819 0.574 1.428 10° 0.173 0.985 0.176 60° 0.866 0.500 1.732 15° 0.259 0.966 0.268 65° 0.910 0.423 2.145 20° 0.342 0.939 0.364 70° 0.939 0.342 2.747 30° 0.500 0.866 0.577 75° 0.966 0.259 3.732 40° 0.643 0.766 0.839 80° 0.985 0.173 5.672 45° 0.707 0.707 1.000 90° 1.000 0 infinity In wind triangle plots we assume that the forecast wind velocity is accurate and constant, the aircraft's magnetic compass is accurate, and the pilot will maintain a constant heading in flight. However, there will be considerable variability in each (for example read the boundary layer turbulence paragraphs in the microscale meteorology module), so there is no reason to try for absolute accuracy in the initial calculation of heading, ground speed and ETI. So, rather than plotting the wind triangle we can introduce a few shortcuts to the process by using some simple mental arithmetic to estimate the crosswind and headwind/tailwind components of the wind velocity relative to the required track. Even so, it is wise to become familiar with plotting the wind triangle; the experience makes it much easier to mentally envisage the relationship between the vectors thus avoiding flying entirely in the wrong direction — which is remarkably easy to do. The 1-in-60 rule The 1-in-60 rule provides a rule of thumb based on the reasonably accurate assumption that the sine of any angle, up to about 45°, is equal to 0.01666 times (or 1/60) the number of degrees; e.g. sine 30° is 0.01666 × 30=0.5 or 30/60 = 0.5. The sine is the ratio — in any roughly right-angle triangle — of the side opposite the angle to the hypotenuse (the longest side). Thus the 1-in-60 rule is very handy in the mental arithmetic of flight theory and basic navigation, as the angles involved in en route corrections are usually much less than 45°. For angles up to 15° or 20° the tangent (opposite side/adjacent side) is practically the same value as the sine. For angles between 50° and 75° the sine is about 1/70 times the number of degrees, and for angles between 75° and 90° the sine approaches unity. Using 1-in-60 to estimate WCA The two/three-step technique described below approximates the sine/cosine relationships and produces results near enough to the trig calculations. • Step 1. First find the crosswind component of the forecast wind velocity by estimating the acute angle (i.e. less than 90°) at which the wind meets the required track, divide that by 60 and multiply the result by the wind speed. However, if that relative angle exceeds 60° just use 60. For example: (a) track = 045° w/v = 075/20 kn: relative angle = 30 = 30/60 × 20 = 10 kn crosswind. or (b) track = 045° w/v = 135/20 kn: relative angle = 90 [use 60] = 60/60 × 20 = 20 kn crosswind. or (c) track = 045° w/v = 195/20 kn: relative acute angle = 30 = 30/60 × 20 = 10 kn crosswind. • Step 2. Then use the 1-in-60 rule to estimate the wind correction angle by dividing the crosswind component by the TAS and multiplying the result by 60. For example: (a) and (c) crosswind = 10 kn; TAS = 75 kn: 10/75 × 60 = 8° WCA. or (b) crosswind = 20 kn; TAS = 75 kn: 20/75 × 60 = 16° WCA. But combining steps 1 and 2 simplifies the calculation: WCA = relative angle [60 max] x wind speed / TAS Example (a) track = 045° TAS = 75 kn; w/v = 075/20 kn: relative angle = 30 WCA = 30 × 20/75 = 8° And remember that the wind correction is applied in the direction the wind is coming from so that the aircraft crabs along the required track. • Step 3. Then to estimate the ground speed, deduct the (acute) angle at which the wind meets the track from 115 (for angles up to 60°, use 105 for greater angles) and apply that as a percentage of the wind speed. For example: (a) track = 045° w/v = 075/20 kn: angle = 30; 115 – 30 = 85% of 20 = 17 knots headwind. or (b) track = 045° w/v = 135/20 kn: angle = 90; 105 – 90 = 15% of 20 = 3 knots headwind. or (c) track = 045° w/v = 195/20 kn: angle = 30; 115 – 30 = 85% of 20 = 17 knots tailwind. Subtract the result from TAS if wind is coming from ahead to abeam, otherwise add. If you like to try a quick mental calculation with the two plots in Figure 4, you will find the arithmetic will produce much the same results as the plots. You may think it wrong that if the wind is at 90° to the track the ground speed calculation will still come up with a headwind component. This is because the track and the wind velocity are relative to the ground, not to the aircraft's heading. With a wind at 90° to the required track the aircraft must take up a heading having some into-wind component, so that it crabs along the required track; try it by plotting a full wind vector triangle incorporating a wind at 90° to the required track. All the short-cut techniques described are not ultra-precise but they are quite okay for most cross-country navigation in visual meteorological conditions. You should also read the meteorology module dealing with southern hemisphere winds and particularly section 6.3. Using tables to derive ground speed and WCA The third and simplest method for estimating WCA, heading and ground speed is to use tables such as those following. Table 1 is for wind speeds up to 30 knots in 5-knot intervals, and for wind angles relative to either side of the required track between 0° and 180°. In the table you will see that headwinds have a negative adjustment and tailwinds a positive adjustment for ground speed. However if the calculated WCA exceeds about 10° the inbuilt crab problem becomes apparent and a small additional calculation to derive a more accurate ground speed has to be made (Table 2). Note that the sum of the two wind components only equates with the wind speed when the wind angle is 0°, 90° or 180°. Table 1. Wind components Head/tailwind component in knots (for ground speed) Crosswind component in knots (for WCA) WA Wind speed Wind speed 5 10 15 20 25 30 5 10 15 20 25 30 0° -5 -10 -15 -20 -25 -30 0 0 0 0 0 0 15° -5 -10 -15 -20 -25 -30 1 2 4 5 6 7 30° -4 -9 -13 -17 -21 -25 2 5 7 10 12 15 45° -3 -7 -10 -14 -17 -21 3 7 10 14 17 21 60° -2 -5 -7 -10 -13 -15 4 9 13 17 21 25 75° -1 -2 -4 -5 -6 -7 5 10 15 20 25 30 90° 0 0 0 0 0 0 5 10 15 20 25 30 105° +1 +2 +4 +5 +6 +7 5 10 15 20 25 30 120° +2 +5 +7 +10 +13 +15 4 9 13 17 21 25 135° +3 +7 +10 +14 +17 +21 3 7 10 14 17 21 150° +4 +9 +13 +17 +21 +25 2 5 7 10 12 15 165° +5 +10 +15 +20 +25 +30 1 2 4 5 6 7 180° +5 +10 +15 +20 +25 +30 0 0 0 0 0 0 5 10 15 20 25 30 5 10 15 20 25 30 ground speed* = TAS + value shown WCA = value shown / TAS × 60 *If the WCA exceeds 10° then reduce the ground speed by an additional value that is a percentage of the TAS, as shown in Table 2. You will note that the adjustment to ground speed really only becomes particularly significant at WCAs above 20° and then, in such conditions, it is probably unwise for light aircraft to be engaged in cross-country flight. Table 2. Ground speed adjustment applied if WCA exceeds 10° WCA 10° 15° 20° 25° 30° Reduction 2% 3% 6% 10% 12% Example 1. The track required is 090°, the wind velocity is 060°/15 knots and TAS is 70 knots. Then the wind angle relative to track is 30° left and, reading from Table 1, the headwind component is –13 and the crosswind component is 7. Thus the ground speed will be 70 –13 = 57 knots, the wind correction angle will be 7/70 × 60 = 6° (to the left) and the heading = 084°. Example 2. The track required is 300°, the wind velocity is 075°/15 knots and TAS is 70 knots. Then the wind angle relative to track is 135° right and, reading from Table 1, the headwind component is +10 and the crosswind component is 10. Thus the ground speed will be 70 + 10 = 80 knots, the wind correction angle will be 10/70 × 60 = 8° (to the right) and the heading = 308° Example 3. The track required is 360°, the wind velocity is 075°/20 knots and TAS is 70 knots. Then the wind angle relative to track is 75° right and, reading from Table 1, the headwind component is –5 and the crosswind component is 20. Thus the ground speed will be 70 – 5 = 65 knots, the wind correction angle will be 20/70 × 60 = 16° (to the right) and the heading = 016°. However, because the WCA exceeds 10°, Table 2 is consulted. This shows for a WCA of 16° the ground speed should be further reduced by 3% of the TAS — about 2 knots, so the adjusted ground speed is 63 knots. 4.4.3 Navigation calculators Circular slide rules There are several 'do everything' circular slide rules, or 'whiz wheels', marketed for aircraft flight planning usage. These navigational 'computers' also incorporate a wind disc for the solution of the wind triangle. They too find the wind triangle solution by breaking the wind velocity into the crosswind/headwind components, rather than plotting a full wind vector triangle. So the adjustment to derived ground speed for WCAs exceeding 10°, similar to that shown in Table 2, is also an additional step. You will find them very useful on the ground but some can be difficult to read and adjust in a light aircraft, particularly in an open cockpit. You may have enough difficulty just handling the chart, the flight plan notes and a pencil (sharpened both ends). The Jeppesen CR2 — available from the Airservices Australia online store navigation and planning accessories — for about $60 is okay and will fit into your pocket — together with a small folding rule. It can be operated with one hand for time and distance calculations. E-6B calculations 'E-6B calculations' (or just E6B) is now a generic term for all the calculations associated with tracks, TAS, wind velocities, headings, ground speeds, density altitude, time, fuel, weight and balance and so on. The term derives from one of the many model numbers of a hand-held dead reckoning instrument generally known as the Dalton computer. It was extensively used by all the Allied air forces in WW2 and consisted of a circular slide rule with either a fixed or sliding wind scale or a wind scale belt looped inside an instrument box. All subsequent navigation circular slide rules are developments of Dalton's dead reckoning computers. There are E-6B software apps for smartphones, tablet computers and other personal electronic devices, readily available for a few dollars or possibly as freeware. To find sources, google 'E6b software'. However it is my opinion that the whiz wheel does provide a better grasp of the essentials of the wind triangle. 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