Considering that the most critical time to get a cross-wind component assessment is for a landing, the problem is to assess the wind at the intended landing runway from a TAF and use that assessment to decide whether or not you need to plan for an alternate.

A TAF is hardly going to give you an accurate wind speed and direction for the few seconds between crossing the fence and touching down on the runway.

The other alternative is to check the wind soc from 1500ft over the top and guess the speed and angle to the runway. Are we really going to fool around with decimal points on a scale when the starting values are wild approximations?

Can't see the forest for the trees. The whole idea of a P chart is to allow a pilot to make a reasonably safe assessment of take-off or landing criteria with a generous margin of error.

The wind side scale on the left hand side of the computer has two scales for angle ?

John will be able to clarify this more, but from my understanding, the outer scale is the sine angle, used for xwind component, whereas the smaller scale is the cosine angle used for headwind/tailwind.

I knew I should have put some graphics in the previous post's discussion. I can't recall now who the instructor was this far removed but, when I were but a wee boy, one of the then ancient ground instructor's favourite war cries was "draw a piccy". I'll come back later with some amplified descriptions. Hopefully, we can sort out some confusion evident in previous comments.

Bob makes some good points, which I probably should offer comment to ...

Are we really going to fool around with decimal points on a scale when the starting values are wild approximations?

Absolutely not. The point of my comment was that using the CR for the calculation is of more than adequate accuracy. Its value lies in its being quick and easy to do the sums. A bit like weight and balance CG calculations. The starting empty weight CG values are going to be very lucky if they are accurate to several mm. Not much point in real world calculations running answers to decimal points of mm.

The whole idea of a P chart is to allow a pilot to make a reasonably safe assessment of take-off or landing criteria

Generally, takeoff and landing charts will be pretty accurate. This is especially so for the major heavies where the OEMs invest lots and lots of dollars for marketing commercial reasons. I say "generally" because, sometimes, we see short cuts taken in light aircraft data which make a bit of a mockery of the whole thing, For instance, sometimes you will see landing charts based on one approach speed (based on MLW stall speed) because they could and it saved a few dollars in the flight test program. Surprise, surprise, the landing distances increase with reducing weight due to the approach speed's carrying an increasing stall factor as the weight reduces. A bit silly, don't you think ?

The baseline flight test data, with a bit of care, can be very precise. Cinetheodolite data, while a pain to analyse, provides very accurate baseline distances. The mathematics data expansion take a lot more effort to establish relevant fudge factors for a particular aircraft and the extent to which the applicant has the dollars to spend will have a marked bearing on how much effort is directed to this problem in the data workup.

with a generous margin of error.

Generally, if the charts have been worked up sensibly, the emphasis is on getting the smallest margins achievable consistent with the dollar outlay. The charts, themselves, only have three elements of known "error"; the surface rolling coefficients of friction assumed may not match the field on the day, the wind carpets have the usual 50% headwind and 150% tailwind fudge factors included, and there is an overall fudge factor for some of the charts.

Please don't treat your takeoff and landing charts as having much fat unless you know some of the history which went into them or have done some field test work on your aircraft and have quantitatively determined that there is more than a minor conservatism involved. Having produced quite a few such charts in years gone by, I know how much effort I used to put into minimising unnecessary conservatism. The customer was paying for good numbers to aid the marketing program so he was never enthusiastic about numbers which were bigger than absolutely necessary.

The important thing to concentrate on is to use the charts conservatively when you are flying, sufficiently so that, should you get into an argument with CASA or the legal system (the latter usually following a fatal mishap) you have a strong argument allowing you to say "not my fault, your Honour". I've been tangled up with several post-fatal accident witch hunts and, in each case, my being a sensibly conservative engineer has let me out the side door before things got heated. Every time I read court transcripts which relate to a non-conservative pilot's, maintenance engineer's, or professional engineer's getting their fingers burnt, I have to shudder and drop a tear. Keep in mind that, on occasion, the system can hang a competent and largely innocent party. As a pilot, you CANNOT afford to be cavalier in your compliance with the rules and general legal requirements.

One point regarding crosswind limits is that, generally, the published "limit" is not limiting (you may have seen references to "demonstrated crosswind") and will be either the minimum required under the certification rules or that resulting from the highest crosswinds the flight test program could find (the serious OEMs will spend considerable effort and dollars chasing winds for crosswind testing). For these aircraft, you may not get a fright if you operate in higher winds. However, be very aware than some aircraft are seriously limiting at the published crosswind maximum figure. I recall one Type, years ago - when it was introduced into Australia it had a pretty low crosswind limit. Importer wanted that increased and I got the task to fix the problem. I did some preliminary flight testing and frightened myself only a few knots above the published speed. When we went out to do the formal flight testing, the pilot was the then CASA Chief Test pilot (who was one of the best stick and rudder guys I have ever flown with). We eventually ended up with several knots increase above the original POH limit and that new limit was a SERIOUS limitation. So, unless you know what the history is with an aircraft, please do be very cautious when considering busting any POH limits. This is from a consideration of breaking things, quite apart from the potential for legal liability.

How fortunate we are to have someone like John to contribute so appropriately to the forum conversations. Your input is of enormous value John and much appreciated. It's not often we have the chance to communicate with someone who was so profoundly involved in the process of creating the P charts. Pay attention folks!

"So, if you put the "TAS" mark on the inner scale against the wind speed on the outer scale, then if you read around clockwise to the angle that the wind is off from the runway, then the outer scale answer gives you windspeed x sin (wind angle). Similarly, if you read around anticlockwise, the outer scale gives you the value for windspeed x cos (wind angle).

Sure, it all sounds a bit complicated but, once you have done a few examples, it becomes pretty straightforward.

Now, if you look at the usual graphical wind component solution, as Bob posed at post #3,

(a) the headwind/tailwind component is just windspeed x cos (wind angle)

(b) the crosswind component is just windspeed x sin (wind angle)

Easy peasy - that's what we had set up on the navigation computer. Looking at the graphic Bob posted, the wind is 30 degrees off at 40 knots and we can read off a headwind of about 35 knots and a crosswind of about 20 knots.

If we set up the solution on the CR - set "TAS" to 40, then, if we read

(a) clockwise to 30 degrees, we read the crosswind as 20 knots (actual value is 20 knots), and

(b) anticlockwise to 30 degrees (on the black scale), we read the headwind as 34.7 knots (actual value is 34.64)"

Another question on using the CR to calculate wind component , if the wind angle is greater than 45°, how would you find that answer using the method described? I ask this as previously I stated that CASA are now giving wind directions that are 50° or more off the runway heading. We could use the wind triangle drawing method, I'm interested to hear your thoughts, again much appreciated.

First, some basic trig(onometry) background so we all are on the same page.

Trig derives from very old triangle work dating back 2500 years or so. In the triangle shown, the basic trig ratios (and these are about all that are of interest to the typical pilot) are noted.





In the CR graphics, shown later, if the “TAS” inner scale mark is aligned with the “10” outer scale mark, then the slide rule scales are set up to provide a simple sine or cosine look-up table. The CR computer provides values for sin(e) and cos(ine) in the range 1 degree to 90 degrees which covers our range of interest nicely. Indeed, it is unusual to see a situation where we need to go to as small an angle as 1 degree.

The following table gives you some values for sine and cosine over a range of angles. You can look up any other values to check that the CR does, in fact, tell you the story. As an aside, you will often find a small “error” between the calculated mathematic value and what you can reasonably read from the CR. This is just a fact of life with slide rules. They are limited by their design to only a few decimal places.
We notice that sin(A) = cos(90-A) straight away. So, for instance, sin(1) = 0.017452 and cos(90-1) = cos(89) = 0.017452.





The first problem we might note is that the value for sine or cosine passes the “10” point periodically as the number of zeroes following the decimal point changes. The main difficulty is keeping track of where the decimal point is located. When using a slide rule you always need to keep in mind that the slide rule is interested ONLY in significant figures (the numbers) – the user has to figure (or remember) where the decimal point is located.

Considering sine, if we start at a very small angle, we have a very small value of sine. So, for example, for an angle of 0.5 degree, the value of sine is about 0.008727. As we increase the size of the angle, the value of sine increases. For example, for 2 degrees, we get around sin(2) = 0.0349. When we get to an angle of about 0.573 degrees, the value of sine is near enough to 0.01. The slide rule scale puts this value at the “10” position. If we keep increasing the angle, the sine values keep increasing until we get to about 5.739 degrees where the value of sine becomes 0.1. Again, the slide rule puts this at value at the “10” position. As we keep increasing the angle, the value of sine keeps on increasing. When we get to 90 degrees, the value of sine is 1.0 which the computer, again, puts at the “10” position.

So, what to do when we are just going around and around the slide rule scale ? It’s enough to make a pilot dizzy.

Easiest ways to accommodate this is either to mix all the values, (say) with different colours to separate things (dreadfully messy) or run a spiral scale as in the CR. A sample spiral representation of the values is shown below.

If we overlay a suitable graphic on the CR, we can make reasonable sense of the thing -





Some things to keep in mind:

(a) don’t worry about the highlight-coloured scale markings for cosine values – that is just a visual aide-memoire for the pilot’s use in wind triangle calculations per the CR manual and has nothing much to do with the actual use of the slide rule, per se. The scales go right around the computer as shown in the overlay graphic in both directions. You just need to keep in mind which is sine (clockwise increasing) and which is cosine (anticlockwise increasing).

(b) due to space, there is a limit to how much detail can be etched on the plastic. If you need an angle which isn’t marked, you need to keep the (90-angle) consideration in mind so that you can switch from sine to cosine and vice versa.

(c) you need to know the decimal place story and how many zeroes follow the decimal point.

(d) the scales are wildly non-linear. For example, the distance between, say, 1 degree and 2 degrees is vastly greater than, say, the distance between 80 degrees and 90 degrees. The scale is called a "logarithmic" scale as the C/D slide rule scales are adding or subtracting lengths related to the logarithms of numbers. If you want to delve into this via a net search, it relates to using powers of some standard, or base, number to represent values and the neat things we can do with such powers representation.

So, looking at the last graphic, we can play with some numbers to get a feel for what is going on here.

If we start at the 1 degree mark, we can read off a value for sine of about 0.01748 as compared to the tabulated value of 0.01745. You just have to get used to the idea that the electronic calculator will give you better precision than the slide rule.

If we proceed clockwise (sine increasing), say, to 2 degrees, we get a value for sine of about 0.0349 compared to the tabulated value of 0.0349 – that’s not too bad.

Now if we keep going clockwise to a bit over 5.5 degrees (nothing specifically marked) we see that we are going to go past the “10” mark on the outer scale. As we go past the “10” we need to drop a zero following the decimal point. So for, say, 10 degrees, the sine value becomes 0.174 compared to the tabulated value of 0.17365 which, rounded off to three decimals, is 0.174. Again, not a bad result.

And so, you keep on increasing the angle by going clockwise and the sine increases also. Now, when you get to 45 degrees, you get tangled up with the reverse colour highlight and a cosine scale. No problem, we just invoke the (90-angle) trick to get the answer. So, for instance, if we were after sin(60), which isn’t marked, we can use cos(90-60) = cos(30) which is about 0.868 on the outer scale compared to the tabulated value for sin(60) of 0.866 which is pretty close to what we can read off the scale reasonably well.

If we keep going, say, to 90 degrees (which is the “TAS” mark), we read off a value for sine of 1.0 compared to the tabulated value of 1.0.

Now, what about cosine values ?

Cosine (0) = 1.0 so we can start from "10" if we are increasing angle and proceed anticlockwise as cosine increases. If we need to use the sin(90-angle) trick, the smallest sine angle engraved is 1 degree so we can't go higher than cos(89) - which probably isn't going to cause us any problems. So, for a couple of examples, cos(40) is about 0.768 and cos(89), which we read at sin(1), is about 0.01748. Just take your time and think steadily about what you need to do to locate the specific angle values you are looking for.

Say we are after cos(60). There is no position marked for that value so we need to use cos(60) = sin(90-60) = sin(30) for which we would read off about 0.5 compared to the tabulated value of 0.5 for cos(60). And so on it goes.

Now, some of you may find it a bit difficult to get on top of this, initially, so please, do ask questions. If you don’t want to ask on the forum, then flick either me or Bob a message and we can answer the question off-line. Bob is every bit up to speed with this stuff as I am so it doesn’t matter to whom you address your question.

Wind Calculations

Now, as the scales are just slide rule C/D scales which are used for multiplication and division on the other side of the computer, we can do such operations on the wind side scales as well. So, if the “TAS” inner scale (really “10”) is aligned with any value, say a speed, then the outer scale value against any other inner scale angle location (going clockwise) will give you the value of [speed x sin(angle)] which is your crosswind. Similarly, going anticlockwise (and noting the angle or (90-angle) requirement once you are past the 45 degree position) the value will be [speed x cos(90-angle)] which is the head or tail wind. The only difficulty is that you have to keep track of what the decimal point(s) is(are) doing.

So, say I have a 15 knot wind. I need to align the “TAS” inner mark against 15 on the outer scale, as in the next graphic. Now I can read off any crosswind value I need for any given angle off the runway.





For example, if the wind is 30 degrees off the runway, I would move clockwise from the “TAS” until I get to 30 degrees and read off a crosswind of 7.5 knots (round off either to 7 or 8 knots), which you can compare to the tables in Bob’s post #3 (7 knots - which is the "correct" value rounded off).

Another example, if the wind is 60 degrees off the runway, I would read off a crosswind of 13 knots (at the cos(30) position, using (90-60)=30). Bob’s table in post #3 gives (surprise, surprise) 13 knots. And so you can continue with whatever speeds and angles might be of interest.

If you are looking for a head or tail wind, you are interested in the cosine value for the multiplication rather than the sine.

For example, using a 30 knot wind and 60 degrees off the runway, we need to run around to the cos(60) position. This isn’t marked so we use cos(60) = sin(90-60) = sin(30) which gives us about 15 knots headwind. Bob’s table in post #3 suggests (wait for it) 15 knots.



Once you get on top of it with a bit of practice ... easy peasy.

The Questions

The wind side scale on the left hand side of the computer has two scales for angle ?

See the story on spiral scales.

Another question on using the CR to calculate wind component , if the wind angle is greater than 45°, how would you find that answer using the method described?

Just use the relationship that sin(angle) = cos(90-angle) to figure which of the cosine angles marked is the one you need (or vice versa).

So, if you need to work out the crosswind for 70 degrees off the runway, use sin(70) = cos(90-70) = cos(20) and read the value against the cos(20) mark on the inner scale. So, for a 30 knot wind, this would give you about 28.2 knots compared to Bob’s table at post #3 which suggests 28 knots.

Is the surprise factor fading a bit by now ?

It is well worth getting on top of this technique as you can then use it for any calculations involving basic trig work, eg 1:60 for starters.

Is it essential that you be able to do it ? Of course not.

Is it useful to know ? You bet.

Looking through the posts, the angles dont go higher than 90. You can have bigger angles in a triangle. What happens with bigger angles ?

That usually doesn't present a problem. However, the value of sine and cosine oscillate between -1 and +1, albeit out of phase by 90 degrees. When you get to either -1 or +1, the value starts to increase or reduce, respectively, as in the linked graphic.

Nice picture here en.wikipedia.org/wiki/File:Sine_cosine_plot.svg. The angle information is for radian measure rather than degrees. Pi radians is the same as 180 degrees. Don't worry too much about the different units and, as pilots, we rarely have any need to consider angles less than 0 degrees or greater than 90 degrees..