What is your TDC up to? the Law of Cos, the perfect torpedo shot

[starvingartist507]

You can download any of my guides from www.hoofinasia.net
Also @ hoofinasia, the working stopwatch program I detailed below

EDIT: added the link to the Math guide
EDIT: added guide to main page. Sorry the pictures suck so bad. I'm trying to get my file host back online, but for now, thats the best I can do.

Once you figure out whats going on, I promise that it only takes a minute to figure out this stuff using game data, so its not like you have to pause it for an hour (or at all) to get an accurate speed reading.

To nail down a ship, do 3 things:
1. Figure range
2. Figure speed
3. Figure AOB

When you look at a target, you note several critical pieces of information. The first, Bearing, the second Range. The third piece required for a good shot is Angle on Bow. At first, just estimate AOB, so that you can point your boat in the right direction. Once you are basically in position, knowledge of a perfect heading becomes more pressing. There are a couple ways to figure out AOB and Heading with a totally normal degree of accuracy detailed elsewhere in the forum. This small and poorly illustrated example makes no attempt to dethrone the current guides, but I am trying to provide a way to get exact measurements, which are especially helpful for long range, high speed, or submerged-only shots. This method can even be used to figure a “total forward movement” and “generic heading” for a zig-zagging target, given enough patience and observation time. I might not get into that just now, however.

Let’s start with your average situation. You managed to point your ship somewhere ahead of a hapless merchie, and are bearing down on their estimated course. For the sake of simplicity, we’ll say that you are submerged, and have a good visual contact under 2.5 NM (approx 5000 yds).

Now the method I use can be used to figure exact heading and speed information on the target, even while the sub is moving, but the math gets a lot more complicated. So until I become more adept at spatial reasoning, I usually bring my sub down to a negligible speed, preferably a dead stop. This happens 90% of the time when I am approaching a firing position anyway, since I don’t want the periscope to get noticed, I am moving silently to evade escorts, or I have a little time to kill before the target is in the kill zone. Basically, it is good sub-commanding to bring the ship into a firing position, and let the target drift its way into your sights, no matter what your solution calculating tactics.

So you have a target off to your port side. We’ll imagine that the relative bearing is 330, or 60 degrees left. The target is heading towards you, so the AOB is less than 90 and they are showing their starboard side. If you can accurately identify the target, do so asap. Once this is done, you have to do a few things at once (or as quickly as possible.) First, start the stopwatch, as soon as you do that, figure range. If it is easier for you to get range then click start, go ahead. Remember in all those sub movies you watched as a kid, when the Skipper would yell “Mark! Bearing such and such, range so and so!” Well that is what you are doing. Get a range, get a bearing, and start the timer. Now is a convenient time to “down scope” if you need to be stealthy. Keep that timer going, let some time pass, the more the merrier. I usually go for 2 mins or so a lot more if they are still very far away. Once you are happy with the amount of time that has passed, up-scope, and yell “Mark! Range such and such, bearing so and so!” Don’t piss off your neighbors. Jot down the second range, the second bearing, and the time that has passed. Pause if you want to now, since you are doing the work of 2 crew members and a machine. There’s no shame in working more slowly, but the information doesn’t necessarily get outdated unless your target veers off course or changes speed. So it could be ok to remain going at real time.

Ok, with the two bearing and range readings, you have all the information you need to figure range and speed, and with that, you can not only calc AOB to the degree, you can also predict the AOB on a certain bearing. I.E. when the target is at bearing 350 (or time to shoot) the AOB will be…

So the law of Cosine.
You knew there would be math. The law of cosine states that if you know the length of two sides a triangle, and the measure of one of the angles, you can solve for all lengths and angles. Find the difference of the two bearings. Using the above example, Bearing one was 330, and we’ll say bearing two was 345. The difference is 15 degrees. We’ll say for argument that Range 1 was 2200, and range 2 was 2000. Yarr!ds… or

a=2200
b=2000
c=distance traveled
C=bearing change
B=Angle on bow start
A= 180 - Angle on bow finish.

According to the trig law: The distance of the third leg (which is the distance that the ship traveled) is:


Distance Travled^2= First Distance^2 + Second Distance2^2 – 2*Distance1*Distance2*Cos(angle change)

Using our data:
1. add squares of both distances (8,840,000)
Distance 1 was 2200, squared to 4,840,000
Distance 2 was 2000 squared to 4,000,000
2. Figure the Cosine of the angle. (.9659 )
(Remember the difference of the two angles was 15 Degrees. The cosine of 15 is .9659)
3. Double this, multiply by both distances. (8500140)
(2 times .9659 times 2000 times 2200)
4. Subtract the first total (step 1) from the second total (step 3) 339,860
(8,840,000-8,500,140)
5. The square root of this, (582.9751) is the distance that our merchie traveled between the two marked times.
If I waited 2 minutes (120 seconds), this gives me a base speed of 4.852812 yards per second (582.9751 divided by time). From there,
6. multiply by 1.777 to get a base speed of 8.63288 knots.
Any yards/second times 1.777 = knots.

That’s what your stopwatch button is supposed to calculate, but it doesn’t. An exact speed is usually enough to make a perfect shot, with a roughly accurate AOB.

To figure an exact AOB using 2 ranges and 2 bearings:
You know the angle of one leg, 15 degrees, and the opposite distance (582.9751 from the above work, step 5)
You also know the distance of all sides of the triangle. Dusting off your trigonometry book again. Recall that:

So! Sin of bearing change, divided by distance travled, equals Sine of Angle on Bow, divided by the range at Mark 2. Or!

• Find the sin of the bearing change. Recall from above that the bearing change was 15 deg. (0.2588)
• Multiply by the second distance to target (which was 2000) (517.6380902)
• Divide by the distance traveled (found using the above method) (.889792)
• The angle on bow is the inverse sin of this number (so the angle on bow was 62.6136)

The greatest part about knowing the exact angle on bow at an exact bearing is that, all future AOB will be related to that. So when our mystery ship was on bearing 330, and had an AOB 62.61; I am absolutely positive that at Bearing 345, it will have an AOB of 77.61. Why? Because all these angles and ranges just graph out a big triangle and all the angles of a triangle have to add up to 180. So trust me:

If the target moves 5 degrees, and you don’t move at all, the angle on bow increases by that amount. Simple as that.

Don’t try to solve for the final AOB, since you are actually solving the “inside” angle of the triangle (see below)…
So if you pop up the scope and figure an AOB of 44 Deg at bearing 50, you can be darn sure that when that target is at bearing 10, that it will have an AOB of 84 Deg. In this way, if you get a sure range and speed, you can calculate and predict AOB with a fantastic amount of certainty. So now you can fix the AOB for any time, and find the target speed with pinpoint accuracy. Those fish should be hitting right on every time.

This can also be used to predict range at a future bearing. Just re-solve with new information, invent any old situation you want. But for the most part, the only thing you need to get a good shot is AOB, Speed, Range. And Range is easy enough to equate, and this will give you the rest, down to pinpoint accuracy.

Remember this method is reliant on accurate range findings, as the manual states. So use the sonar station to back up your calculations. If you use the Sonar to re-check your range, remember as of 1.2 sonar was still bugged, so go the station directly, don’t trust what the sonarman told you, because he’ll just say whatever you have entered on the TDC. Sit down, and ping the target yourself for best results.

To figure an exact heading, knowing their AOB and bearing
This is very easily estimated using the charts, but if you for some reason need to know. As I sometimes do, here’s how I figure it out. Its quick and dirty math.
I just remember, Port = negative, starboard = positive (with straight ahead being the origin), and the enemy ship is opposite.

Take the first AOB that you just calculated, and the first bearing that you noted at Mark 1.
1. Your heading minus their relative bearing if they are on the port side (left), or plus their relative bearing if they are on the starboard (right)
If from the above example, they were at bearing 330, this means 30 port. So take our heading (270) and subtract 30. 240 is the Absolute Bearing
2. 180 + absolute = (heading if AOB=0). So if AOB is 30 starboard, subtract 30 from this result. If 40 port, add 40.

From above, AOB was 62.61 at bearing 330. 330 = 30 to port side. So our heading (270) minus relative bearing since we are looking to port (30) = 240 Absolute bearing, or about WSW. Then take this absolute bearing and add 180, then add or subtract their AOB. So Absolute bearing of 240, plus 180 (420) minus their AOB (which was starboard 62.61) for an exact heading of 357.39. I apologize that the chart below doesn’t perfectly match the math. I was just throwing lines together to show how they relate. But you get the idea I hope!

Happy Hunting! If you actually use this with any measure of success let me know! It seems way to complicated once I write it down…