This is my first post, so hello everybody.
Quote:
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Originally Posted by Shadowmind
If you want a general purpose equation for calculating speed you can use the following:
speed(knots) = (Di * sin(theta) * 3600)/(sin(AOB) * Tm)
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Yes, you're right, that's an accurate formula to calculate the relative motion of a target.
This looks like it can't be simplified, but we can use some nice properties of sin to get an approximation for
small thetas. Since sin is 0, sin' is 1 if theta = 0 and sin is concave in the interval between 0° and 90°,
we can simply replace sin(x) by x in a small area around 0.
If theta is smaller than 30 degrees, the error factor x/sin(x) is always smaller than 1.05, so we overestimate the
real relative movement by no more than 5 percent. The only problem here is that x is theta in radians, so we must do
some ugly conversion with theta since we have theta in degrees. Now lets have a look at the formula:
speed[kts] = (3*3600/6080)[s/yd * kts] * range[yd] * pi/180 * theta[deg] / time[s] * (1/sin(aob[deg]))
Since 3*3600/6080 * pi/180 equals 0.031002559, we can simply replace this ugly factor by 0.03 resulting in
speed[kts] = (3*/100)[s/yd * kts] * range[yd] * theta[deg] / time[s] * (1/sin(aob[deg])).
Again we make a small error, this time always underestimating the target speed by about 3 percent.
Well, that's a lot easier, but we still can't calculate sin(aob) without a calculator or sin table.
So let's look at some values (aob rounded to nearest integer):
aob 1/sin(aob)
90° 1
72° 1.05
65° 1.1
60° 1.15
56° 1.2
53° 1.25
50° 1.3
46° 1.4
42° 1.5
30° 2
20° 3
15° 4
12° 5
1/sin(aob) is symmetrical to 90°, so if the aob is lets say 108°, 1/sin(aob) is 1.05.
From this table we can conclude that any aob below 50 degrees produces very inaccurate values, since we most likely
don't know the exact aob and have guessed it. At these low aobs only a few degrees change the factor by 10 or more percent.
We also see that we get really good approximations when the aob is between 70 and 110 degrees.
So to get the real relative speed of a target you can use this
procedure:
1. get range to target
2. get bearing and wait t = range/100 seconds
3. after waiting t seconds get 2. bearing
4. estimate aob
5. compute theta (2.bearing - 1.bearing, ignore minus sign)
6. look up value of 1/sin(aob) of the nearest aob in the above table, use next higher entry if theta < 15
7. multiply theta by 3
8. multiply the result by 1/sin(aob)
and you have computed the target's speed
Note:
If the target is moving too slowly, you can of course wait 2*t or 3*t etc. seconds, but then you have to divide by that
factor afterwards. Waiting 3*t seconds looks promising since the 1/3 cancels out the 3 in step 5. But watch out, theta
should not become much larger than 30 degrees. Another danger is waiting a too small time, resulting in a too small theta.
If theta is too small, you have to be really precise. For example, an error of 0.5 degrees means 10% when theta = 5°,
and in that game it's hard to find out exact bearings.
This also works if you are not using imperial measures. If your range is in meters, simply add 10% to the final speed.
(The true conversion factor is 1/0.9144=1.0936, but 1.1/0.9144 = 1.00584, so 1.1 is very exact)
Now a few questions remain:
1. It's an approximation, how accurate is it?
Due to the fixed under- and variable overestimation, this method produces values that lie within 5% of the true speed
if theta is in the interval between 0 and 40 degrees. Of course, this error estimation is only valid when your
measurements are accurate...
The ideal theta range is from 20 to 30 degrees where over- and underestimations nearly cancel out each other, resulting
in an error of under 1%.
But of course the 1/sin(aob) estimation using the table produces an additional small error, this time over- or underestimating the speed
by up to 2.5% if your aob estimation was perfect.
So if theta is smaller than 15° always use the higher of the two possible 1/sin(aob) values if the aob lies between two table entries,
or use the next higher value if the aob is in the table or very close to any of these values. This way, the underestimation of about 3%
in that interval is replaced by a small overestimation of less than 2%. And we know now, that all speed estimations are either nearly
exact (±2%) or small overestimations of up to 5% (only if theta > 30, which we want to avoid anyway).
Of course, since this is only an approximation of the target's speed, you should always fire a spread unless your target is very close
(say under 1000yd) or moving so slowly, that the error of up to 5% can be ignored.
An error of 5% means 1kts when your target is moving at 20kts, which sounds much when you want to fire at long range targets.
But actually it isn't. If your target and your torpedos travel at 30kts, and the torpedo has to travel 3000m to reach the interception point,
then your target will also travel 3000m, and 5% error means 150m. But most of the times your target will be much slower than the torpedo, and
the error will be less than one ship length if you fire at targets in a reasonable range.
So if you fire a spread you can cancel out that error. If theta was greater than 30°, you're overestimating too much and should aim at the stern
or slightly behind the target if it is far away. Otherwise aim at the mid of the target.
I've tried this method several times in a quick mission in SH3/GWX and my 4 torpedos fired as a 5° spread arrived always 100 to 300 meters
behind a destroyer moving at about 25kts 2.5km away from me, whose speed I estimated with theta = 5.5° and an aob of around 70° as 20 kts.
I wanted to know what was wrong, and did it again with autoaiming. And, what a surprise, my estimated aob was always way too high. The real aob was
around 50°, so I have used 1.05 for 1/sin(aob) instead of the right value 1.3, resulting in an error of about 20%.
But the error induced by the aob estimation isn't so important when engaging slow targets at small ranges.
In the SH3/GWX torpedo exercise mission I engaged 2 targets at speeds of 8 and 7 kts and ranges of 1500 and 1000 meters. The first, a small
freighter, with 1 torpedo at an aob of about 80° which hit about 20 meters behind the mid, at which I had aimed. The other was a tanker which
I engaged at an aob of 120° with a 4 fish 5° spread, of which 2 hit and 2 passed behind, one only by a few meters.
2. Relative movement? So how much does my own movement affect the measured speed?
Well, your movement subtracted from the true target movement is the relative movement, when you have all movements as vectors.
So if you are moving in the same direction as the target, you have to add your speed to the calculated target speed (in case the bearing moves
towards your stern, multiply the calculated speed by -1 before adding your speed). If you are moving in the opposite direction, subtract your
speed from the calculated value. In cases where these vectors form a isosceles triangle, typically with your small speed between the two long
target speeds, your speed doesn't influence the relative speed. So you should try to move nearly perpendicular to the target's course, but
slightly in the same direction as the target. The course difference should be 60° if you expect it to be as fast as you are, and over 80° if
it is 4 times as fast or even faster.
Of course there is an exact formula to calculate the exact speed under all conditions, but it is too complicated to explain now,
this post is already long enough... 
I will post something about it later.
For the moment just stay as slow as possible while estimating speed.
Well, that was much. And it sounds really complicated.
But it isn't, what we can easily see at an
example:
(the following measurements have been made in the SH3/GWX mission "operation cerberus" using autoaiming)
The first sighting of the lead destroyer was at 3600m, bearing 340°. The target speed is 25kts.
3600/100 = 36, so we have to wait t = 36 seconds.
After 36s the target is at 345°, aob 46°.
theta = 345°-340° = 5°
1/sin(aob) = 1.4, since theta < 15 use 1.5
theta*3 = 15
15*1.5 = 22.5
since range was in meters, multiply by 1.1: 22.5 * 1.1 = 24.75
so we have computed a target speed of 24.75 kts, 1% lower than the true target speed
Lets wait another 36 seconds, so we have waited a total of 2*t = 72 seconds
Bearing is now 351.5°, aob 53°.
theta = 11.5°
1/sin(aob) = 1.25, theta < 15, use 1.3
11.5 * 3 = 34.5
34.5 * 1.3 = 44.85
meters, 44.85 * 1.1 = 49.335
since we waited 2*t, divide by 2: 49.335 / 2 = 24.6675
so we have computed 24.6675 kts, less than 2% too low
And now a third estimation after 3*36 = 108 seconds
Bearing = 0°, aob = 61°
theta = 20°
1/sin(aob) = 1.15 (this time theta > 15, so we don't increase it)
20 * 3: skip it, canceled out by / 3
20 * 1.15 = 23
23 * 1.1 = 25.3
so the computed speed is 25.3 kts, an overestimation of about 1%
And it is indeed very simple. Depending on the aob and the resulting 1/sin(aob) value, you can do these multiplications in a few seconds
in your head, or on a sheet of paper.
The numbers sometimes become very ugly, but you can round them to the nearest quarter knot or so to simplify the multiplications.
It is impossible to enter the speed more precisely, anyway.
Of course these estimations are not so exact if you misjudge the aob by 10° or so. The error may easily reach 10% or more, and you will miss targets
that are far away or very fast, as my example above shows (the first from question 1, it's the same target as in the last example).
So if you are very good at estimating aob, you can precisely estimate the target's speed.
If you tend to estimate the aob right, but sometimes over- and sometimes underestimate it by 10° or so,
you can also compute the speed after t, 2*t and 3*t seconds and then calculate the average of these 3 speeds to even out aob estimation errors.
But if you are not that good, like I am, you can still use this estimation when you want to shoot from the hip at a target of opportunity,
especially when the range is under 1000m and the aob is nearly 90°, then the formula is simply the
rule of thumb: theta*3. (or even simpler: theta, if you waited 3*t seconds. maybe +10% if the target isn't perfectly aligned.)