Velocity from Target Length (Fixed-Wire Method) Using Tables

Introduction

This is meant as a companion guide for DARKNESS' Das Book table and map collection mod.

It will teach you how to measure the speed of a vessel by having it cross a fixed line. Realistically, that fixed line can be anything, from a natural feature, to an optical implement.

The latter will be the basis for this guide. The line to cross will be the fixed wire that your periscope graticule provides.

Mathematical Basis

Disclaimer: you don't need to know any of this if you want to use the method effectively. This section is for those who are interested. If you're not into maths at all, you don't miss anything for you need to know for applying the method by skipping this section.

The basis of this method is the simple relation

 velocity = distance / time .        [1]

The most simple case is when you let the target cross your 0° (or 180°) bearing, meaning the bearing rate is zero. I will refer to that as the ZeroBearing method. In this case, you introduce no lateral movement, and Eq. 1 can be used as is. Since you time how long the target takes to cross your wire from bow to aft, the distance will always be the length of the ship.

Another special case is where you're on a parallel course (Parallel Bearing), which presents the maximum bearing rate. Then, you have to modify Eq. 1 by correcting by the lateral speed of your parallel movement:

 velocity_target = (distance_target + distance_own) / time 
                 = velocity_measured + velocity_own.        [2]

This is still simple, since you know your own speed.

The most general, and therefore most complicated case is when you're on any course (General Bearing), with an arbitrary bearing rate. Then, Eq. 1 will also need to be corrected by the parallel movement, but this time it is not equal to your own movement, but by it's parallel component:

 velocity_target = [distance_target + distance_own * tan(theta)] / time 
                 = velocity_measured + velocity_own * tan(theta).        [3]

Keen eyes will see that this is an approximation. It will work best if you're approaching the target at an AOB of around 90°.

Here's a basic sketch of the three mentioned cases:

Figure 1: Diagram showing the two special cases of the fixed-wire method, as well as the general case. Not to scale.

Table and Method

The following table lets you calculate the distance from a known targets length:

Figure 2: Table for the fixed-wire method in the Forward Bearing case.


The three areas of this table are

• The top-most row (red). This contains the target length you base your velocity calculation on in meters.
  
• The left-most column (blue). This is in units of velocity. Here you will find the result of your measurement.
  
• The center (green). This is the time it took for the target to cross from bow to aft.

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For the Forward Bearing method, you first note the target length from your identification booklet.
Stay on a constant course and wait until the bow of your target crosses your 0° bearing marker. As soon as it does, start the timer.
Keep your periscope still and let the ship pass your wire without changing course.
As the aft crosses the wire, stop the timer.

Let's use a NA-1 coastal ship with its length of 78.65m as an example. Say it took 24 seconds to cross its length.

Use the table and find the target length, or the value closest to it, in the top-most row. Here, that will be the first row, with 80 meters.
Now go down this row until you find a time that matches the one your measured the closest. Here, that will be between 22-, and 26 seconds.
Follow that row to the left. The target speed in knots will be the corresponding number on the far left. In our example, since 24 is exactly half-way between 22- and 26 seconds, we measured a speed of 6.5 kn.

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For the Parallel Bearing method, the same scenario could play out like this:
You position yourself in such a way, that the target crosses any bearing (90° works best) when you start your timer, and stop again when its aft crosses that same bearing.
Since you moved along with your target, this time will be longer than you'd expect from the Forward Bearing method.
In our example, let us assume we travelled at 3.2 knots and measured 47 seconds.
Look again for that time in the center. The closest match we have is 52 seconds, which corresponds to 3 knots. Since 52 is 10% larger than 47, we know the speed we measured is 10% larger than 3 knots, namely 3.3 knots.
Now add your own speed to that. You get 3.3 knots + 3.2 knots = 6.5 knots. The same value as before.

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You can basically skip the rest of this section, since these two special cases will be 99% of the situations you'll find yourself in. If you, for any reason, don't have the luxury to position yourself advantageously, you can still use the

General Bearing method. This requires you to apply the correction using a tangent function (Eq. 3). Thankfully, we have a table for that instead:

Figure 3:
Table for the velocity correction when using the General Bearing method.


The process is exactly analogous. We're on an arbitrary path, and let the vessel cross our 20° bearing. It takes 29 seconds to do so. Use that number as if you had carried out the Forward Bearing method. According to the table from above (Fig. 2), that corresponds to a velocity of around 5.3 knots, since its between 31, and 26 seconds, but closer to 31.
Now look at Table 3. A bearing of 20° at a velocity of 3 kn would have us apply a 1.09 kn correction. Since we're a little faster than 3 knots, let's take that number up to 1.2 kn.
The end result will be 5.3 knots + 1.2 knots = 6.5 knots.

Conclusion

As you can see, all methods will yield the same result.
But going from the Forward Bearing, to the Parallel Bearing, to the General Bearing method progressively introduces more error sources:

• Your course may not be perfectly parallel
  
• Your interpolation for bigger time measurements may not be entirely accurate
  
• The granularity of the correction table makes interpolation potentially less accurate

That means you should use the Forward Bearing method when you require an accurate result, the Parallel Bearing when you're stalking on a parallel course anyway, and don't need the result to be perfect, and the General Bearing only because you can, and using tables is fun.

General tips to for an accurate measurement

• The closer you are, the less the periscope sway is going to affect the accuracy. Use this method as a last minute measurement to double check the velocity you got from tracking the target[en.wikipedia.org] (you did that, right?).
  
• No matter which method you use, starting your timer when the target is at an AOB close to 90° will always improve accuracy. That is because for non-90° AOB, you may accidentally time the crossing of the apparent length, not the actual length

I hope you learned something from this guide.

Every table used in this guide is available in-game with Das Book!
We used historical sources as inspiration, and DARKNESS' did a great job of designing an authentic booklet that seamlessly integrates into the UI.
Go and check it out!

https://steamcommunity.com/sharedfiles/filedetails/?id=3419126557

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[JazzMano]

Great guide, thank you very much !

Just a quick note for anyone not familiar with english naval terminology:

When the guide says “wire”, it’s not an actual cable or anything. It refers to the vertical line at 0° in your periscope view.
Also, “aft” just means the back of the ship. Sometimes you’ll see “stern” used instead.

[Drexack]

Thank you for bringing this up.

You have to add/substract the component of the speed you travelled parallel to the target course.
For the zero bearing case, this component happens to be zero.
For the parallel case, this component happens to be your unaugmented speed.
For a general case, you have to calculate this parallel component using trigonometry. So much is clear. In this scenario, that component happens to be the your speed times the tangent of the bearing.

It should be correct for right triangles, which isn't always the case. That's why it's an approximation that works best if you're on a nearly perpendicular course.

Of course I won't rule out that I messed up the exact relation. Feel free to correct me in this case, but I won't go back and try to derive it again myself.

The bottom line is: I know the table and the method that goes with it works. But don't take it from me, take it from the SH3 community that has used this table for decades.

[DECAFBAD]

General solution is incorrect. It will give you wrong results whenever theta != 0 or Vs!=0

Just apply it in the special case 2 (parallel course). Say you measure 3 knots, which would be the difference between the two ships, then add or your own speed to get the target speed. Here you'd be finished, but according to the general formula, you also need to multiply your speed with the tangent of the bearing angle. Why?