[Rockin Robbins]

There were many plotting tools that the US Navy had in WWII that we do not have available to us: bearing rate plot, bearing difference plot, stadimeter plot, slide rules up the wazoo, etc. Technically, with the bearing rate plot and bearing difference plot it was possible to compute relative course, but not range from passive sonar or visual information. However I have not found a single incidence where this was ever done during the war.

For those interested, the relevant parts of the Submarine Torpedo Fire Control Manual:

561. BEARING RATE PLOT

As was explained under the duties of the Navigational Plotter we usually plot true bearing and ranges to determine the target's true course. If this is done on a maneuvering board we get target relative course and speed.

Now if we substitute for actual range a relative or abstract range we will get the direction of the relative movement line of the target (or relative target course). It can be proved mathematically that:

r = square root (K / (db/dt))

It is obvious that it would not be practicable to

5-43
CONFIDENTIAL SLM 1
actually compute the various values of relative range during an approach. We can, however, assume a convenient value for K which is a constant and construct on a large maneuvering board a graph on which we can plot values of rate of change of bearing (db/dt) against true target bearing. Plate IX is a picture of such a plotting sheet in which K = 200

Plate X is a set of data which has been recorded by the Sonar Plot Recorder. The numbered lines are the actual sonar bearing of the target at 30 second intervals. The data which are plotted by the sonar plotter are the differences between bearings one minute apart and the mid-bearing from the recorder sheets For example, the first db/dt is 2 1/4 degrees plotted at true bearing 330 1/2 degrees. The mid-bearing from the recorder sheet say not be the exact mathematical mean of the two bearings but it is close enough for practical application. These values represent the rate of change of bearing db/dt, for a dt of one minute and the average true target bearing during the period for which the rate has been computed. These values are shown plotted on Plate IX and indicate that the relative course of the target is 120 degrees T. Since Plate IX is also a maneuvering board we may lay out the own course and speed vector of 000 degrees T, 3 knots, and transfer the relative course line to the end of the vector and for a target speed of 15 knots obtain a true course for the target of 110 degrees T.

5-44
CONFIDENTIAL SLM 1
This method of obtaining the target's course is obviously laborious and subject to sonar errors and arithmetical errors of the plot recorder. In obtaining the target course by this method there are several points which should be remembered:

a. Even though each change of course by the target will produce a new relative motion line they will not be connected in such a manner as to produce a plotted track of the target.

b. Since the errors in the present JT sonar can be as much as plus or minus 1 to 2 degrees the value of db/dt which are plotted will vary considerably and the best that can be expected is a relative motion line obtained by "fairing" in the plotted points.

c. The inherent errors of the sonar system render it impracticable to plot values of db/dt of less than 2 degrees/mm.

d. Although a dt of one minute is used in the example smaller values may be used to obtain more plotted points without changing the graph itself or the answer desired. Experienced plotters will ordinarily use a value of dt = 30 seconds. This value of dt, however, cannot be changed during a problem.

e. The method will not work if continuous bearings are not available. Bearings obtained at intermittent intervals are of no practical value.

f. One erroneous bearing will cause two plotted points to be in error.

5-45
CONFIDENTIAL SLM 1

g. When the target changes course an immediate change will occur in the rate of change of bearing (db/dt) and tile the change of course is in progress it will change at a varying rate which will continue until the target is steadied on its new course.

562. BEARING DIFFERENCE PLOT:

The Bearing Rate Plot as has been shown plots a varying amount of bearing change occurring over a fixed time interval. The Bearing Difference Plot plots a specified amount of bearing change occurring over a varying time interval. In the Bearing Difference Plot as in the Bearing Rate Plot the result obtained is the relative course of the target. (a) Plate XI is a Bearing Difference plotting sheet using a scale factor of twenty. The mathematical proof of this method is long and involved and will not be discussed. The formula used in the construction of a bearing difference plotting sheet is Tan B = X Tan A in which B is the angle between the Y axis and any radial line; A is the bearing difference angle and X is the scale factor. In Plate XI the tangent of angle B = 20 X tan 0.5 degrees or

Tan B = 20 X .0087
Tan B = .174
B = 99 degrees

In like manner radial lines representing each 1/4 degree of bearing difference are laid out up to 5 degrees.

5-46
CONFIDENTIAL SLM 1
Beyond this varying difference angles are used as desired. The time scale used has no effect upon the solution obtained. A scale which will facilitate accurate plotting should be selected. It can be seen that increasing the scale factor will decrease the value of the minimum value of bearing difference which can be plotted for a given size of plotting sheet. When the rate of change of bearing of the target is large it will be found that a plotting sheet made up for a scale factor of ten or fifteen will give better results. Plate XII is a plotting sheet made up using a scale factor of ten.

The procedure for using this plot is as follows:

1. Start the stop watch on any even degree of bearing.

2. Label the Y axis as the reciprocal of this bearing.

3. Note the stop watch time and plot a point each time the bearing changes 1/4, 1/2, 3/4, etc., from that which obtained when the watch s started.

4. When enough points have been plotted to establish a line, measure the slope of this line. In Plate XI this is done by transferring the line to the origin and reading value of the slope on the vertical or horizontal scales around the maneuvering board plot. In Plate XII it is done by placing an overlay scale on the line as shown and reading the value directly.

5-47
CONFIDENTIAL SLM 1

5. Apply this slope to the value of the Y axis determined in step 2. If the true bearing is increasing the slope is negative. If the true bearing is decreasing the slope is positive.

6. This is then the true direction of the target relative motion.

7. On the mooring board combine this value with own course and speed and known or estimated target speed to determine true target course. Each time the target changes course it is necessary to repeat again steps one through six.

(b) Plate XIII is a set of data plotted on Plate XI and labeled one (1). In this example the true bearing of the target was 000 degree T increasing. The "Y" axis is then 180 degrees T and the slope is 8 degrees. This gives a value of 172 degrees for the true direction of relative target course. This is then plotted on the maneuvering board to give a true target course of 171 degrees T. Note that the target changed course at 6 minutes and that points plotted at 06-15 and 06-17 indicate a definite change in the slope of the line. All changes of target course will be indicated in this manner.

(c) Plate XIV is another set of data in which there is a much larger rate of change of target bearing. This is shown plotted on Plate XI and labeled two (2). Note that a slope has been determined in 1 minute and 42 seconds. In the example the initial true bearing of the target was 010 degrees T increasing.

5-48
CONFIDENTIAL SLM 1
The "Y" axis is then 190 degrees T and the slope is 50 degrees. This gives a value of 140 degrees T for the true direction of relative target course. This data is then plotted on the maneuvering board to give a true target course of 132 degrees T.

It can be readily seen that it is inconvenient to plot in the lower left hand corner of the plotting sheet. This can be prevented by using a plotting sheet made with a smaller scale factor. A second method is to double the time scale. The line labeled three (3) in figure XI shows the data of line two (2) plotted with the time scale doubled. Note, and this is important, that changing the time scale does not change the slope of the line.

It is sometimes desirable to plot bearing differences against equal increments of time instead of the method used in step 3 of the procedure. This does not vary the result and allows more points to be plotted with very low bearing rates.

(d) comparison of the Bearing Rate Plot and the Bearing Difference Plot as shown in the examples brings out the following points which should be noted in selecting the method to be used.

1. The greatest advantage of the Bearing Difference Plot over the Bearing Rate Plot is that bearing inaccuracies inherent with the sonar equipment are absorbed resulting in smaller and smaller percentage errors as the plot progresses.

5-49
CONFIDENTIAL SLM 1
The Bearing Rate Plot uses bearing rates computed between one minute observations. The effect of the error of the sonar equipment for any given bearing rate will remain the same. In the Bearing Difference Plot this is not the case as the plotted values are always taken from a reference bearing. As the problem progresses the bearing difference becomes larger and larger and the percentage error becomes smaller and smaller.

Due to the inherent errors of our present sonar equipment the Bearing Rate Plot is not usable with a bearing rate less than 2 degrees/minute, The Bearing Difference Plot may be used at rates less than 1 degrees/minute.

As our sonar equipment is improved both plots will of course become more effective.

2. In the Bearing Rate Plot no data are available before an elapsed time of 1 minute. In the second example of the Bearing Difference Plot twenty-eight (28) points were plotted and a solution obtained in an elapse time of 1 minute, 30 seconds.

3. In the Bearing Rate Plot one bad bearing affects two plotted points. In the Bearing Difference Plot a bad bearing affects only one point.

4. In the Bearing Rate Plot a recorder and complicated arithmetic computations are required to obtain data. These are of course subject to error. In the Bearing Difference Plot no recorder is required and the data

5-50
CONFIDENTIAL SLM 1
is obtained directly from the sonar bearing repeaters by observation.

5. In the Bearing Difference Plot where the angle on the bow of the target becomes larger than 30 degrees the slope of the line becomes very critical and a slight error in picking off the slope will introduce a fairly large course error. However, as the angle on the bow increases the target course becomes less and less critical in the fire control solution as the optimum torpedo track is approached.

Have at it and have fun!