IT IS probably true that the Mooring and Maneuvering Diagram in the hands of an expert, is the best all-round means of solving the various problems of course and speed involved in certain types of maneuvers. In close order formations and in anchoring where allowance must be made for variations of speed, tactical qualities of a ship, or scope of chain, there is no substitute for the Maneuvering Board.
It is desirable that all line officers be adept in the use of the Maneuvering Board. Unfortunately they are not. It is doubtful if one officer in five, without some preliminary study, can give a prompt and accurate solution of a mooring board problem. About the only officers on familiar terms with the Mooring Board are those with peculiar slants of mind who like to play with it, navigators and ex-navigators, and officers taking their examinations. The course and speed triangles are peculiarly elusive to those who have not learned to know them well.
Then too, the importance of familiarity with mooring board problems is increasing. Some years ago maneuvers involving change of bearing, of distance, or of both, were generally required only of ships in relatively close formations—ships which at least were within sight of the guide. In fleet dispositions of today outlying vessels may be called upon to change bearing or distance, or both, when many miles from the guide, and in changes of fleet course different mooring board problems may present themselves to each of a hundred ships of the screens. Any scheme that tends to simplify these problems, or to direct attention to the importance of the Mooring Board—particularly in fleet maneuvers, may well be of interest.
It was with this end in view that there were evolved the celluloid course angle cards now used to some extent in the fleet and at the War College. They have been described in the Naval Institute. It is sufficient for the purpose of this article to say that these cards are used with the mooring board diagram, the center of the card being placed at the plotted “present position,” an arm pointed at the “new position,” and the various combinations of course and speed to reach the new position read off directly from a scale. They are simple—relatively fool-proof, and do away with the troublesome “similar triangles.” They require no parallel rulers. But they do require the mooring board diagram and a clear place to lay it, not always convenient on a destroyer or a submarine.
Partly from curiosity and partly from a desire to make these problems simpler for smaller ships or for inexperienced officers, the writer for some time has been trying to eliminate the mooring boad, pencil and rulers from these problems and construct a small diagram or table which may be printed in a signal or other appropriate book or pasted up in a convenient position on the bridge. The results of this work is given in this article.
In the first place it is well to state the problem in its simplest terms. Referring to diagram No. 1, a ship at P1 wants to proceed to P2. What are the various combinations of course and speed—relative to those of the guide—which will put her there?
On the mooring board the direction of line P'-P2 is transferred to QRS, with GQ representing to scale the speed of the guide. Then GR, GS, etc., represent speeds of the maneuvering ship and the angles between these lines and the course of the guide GQ, are the corresponding changes of course for the maneuvering ships. In other words, these are two of the many combinations of course and speed which, if taken by that ship, will cause its plotted position to move along the line P'-P2. This is all there is to the problem.
This brief explanation suggests that there are three inter-related variables here. (1) the direction of the position line P'-P2 relative to the guide’s course, (2) the change of course, and (3) the ratio of the lines representing speed of guide (GQ) to speed of ship, GR, GS, etc. It has been found that the relation of these three variables can be reduced to a simple diagram (See Diagram No. 2), in which the figures at the top represent inclination of position line to course of guide, those at the left side the change of course, and the curves the speed ratio—ship’s speed to speed of the guide. If we know the inclination of the position line we can enter the top of the table, drop down to a speed ratio line and moving horizontally to the left pick out the corresponding change of course.
Having this diagram the next problem is to develop a ready means of finding the inclination of the position line. Of course we can plot our positions and measure this inclination but we want to avoid plotting anything.
In these problems the present and new positions are generally given us by bearing and distance from the guide. In some cases the bearing of the new position may be given as representing an increase or decrease in bearing. For our purposes we must use only relative bearings.
Now looking at Diagram No. 1 again we see that the inclination of P1-P2 to the course of the guide equals angle P2-P1-B, which in turn equals angle BP1C (the present bearing) plus angle CP1P2 (inclination of position line to present bearing). In other words if we can find a ready means of obtaining angle CP1P2 we have merely to add it to the present bearing to get the desired inclination of the position line to the course of the guide. But we have found that our Diagram No. 2 is a graphic means of solving triangles. Comparing triangles P1P2G and QGS we find that angle P1GP2 (the change of bearing) corresponds to the angle QGS (the change of course), that the ratio of P2G to P1G (distance ratio) corresponds to the ratio of SG to QG (speed ratio) and that the angle CP1P2 (inclination of position line to present bearing) corresponds to angle DQS (inclination of line QS to course of guide).
Using Diagram No. 2 we may, then, enter the left column with the change of bearing, move to the right to intersect the proper distance ratio curve and thence up to the top of the diagram where we find the corresponding value of angle CP1P2.
We have therefore the rule that the inclination of the position line to the course of the guide is the present relative bearing plus or minus a certain angle according as the change of bearing is toward or away from the guide. Given the change of bearing and distance ratio this “certain angle” is obtainable from Diagram No. 2.
Having obtained, as above described, the inclination of the position line to the course of the guide we can obtain by again using Diagram No. 2, the various course and speed combinations which will take the maneuvering ship to its new position.
It now remains to combine Diagram No. 2 and a diagram representing the present bearings so as to make one diagram for each of the two cases, (1) increasing bearing and (2) decreasing bearing. The complete diagram for increasing bearing is given as Diagram No. 3. For simplicity when reduced in size some lines are omitted. The diagram for decreasing bearing is not given, but it is similar to Diagram No. 3 except that the triangular diagrams at the right are inverted and the numbering on the “present bearing” scale is reversed.
In Diagram No. 3, the upper left triangle (Diagram No. 2) gives the inclination of the position line to the present bearing. The diagonal parallel lines are marked for “present bearing” and diagramatically add the present bearing and the inclination of position line as obtained from the upper triangle. These two angles combined give the inclination of the position line to the course of guide. The triangles at the right are each a Diagram No. 2. They use the angle of inclination in combination with the speed ratio curves and give as an answer the corresponding change of course (relative to the guide) and the direction of that change—whether away from or toward the guide.
The construction of this diagram as above explained may appear somewhat involved (it seemed so in working it out) but the rules for its use are simple. To use this diagram:
- Enter upper left side of table with “increase of bearing” and move horizontally to right until intersecting “distance ratio” curve corresponds to ratio of new distance to present distance.
- From this intersection drop down vertically to the diagonal line corresponding to “present bearing.”
- From this intersection move horizontally to the right to intersect speed ratio curves in a right hand triangle.
- From any one of these intersections move down or up vertically and at the edge of that triangle read the change of course (from course of guide) and direction of turn.
The heavy dotted line on Table No. 1 shows the path to be followed for the following problems:
- 40°, 5 miles on starboard bow of guide, to take station 15 miles on that beam. (Here increase bearing is 50° and distance ratio 3.)
Answer: Maintain speed of guide and turn out 36°; or take twice speed of guide and turn out 80°, etc.
- 100°, 6 miles to port of guide, to take station 170° to starboard of guide, distant 12 miles. (Here increase of bearing is 90° and distance ratio 2.)
Answer: Take two-thirds speed of guide and turn toward guide 27° or 87°; or maintain speed of guide and turn in 108°, etc.
The above rules and examples cover the most common use of this diagram. It should be noted, however, that this diagram gives the relation of the five variables:
Change of bearing.
Distance ratio.
Present bearing.
Speed ratio.
Change of course.
It is therefore possible to obtain any one of these items when the other four are given or assumed. Some of these cases are perhaps merely interesting but they explain the flexibility of the table.
Suppose a destroyer 60° on starboard bow, 5,000 yards from the guide, changes course to right 60° and takes twice the speed of the guide. The table shows she will have increased her bearing 26° when at double her former distance.
As previously stated, these diagrams represent the general case in which there are five variables. In the most common maneuvers, however, there is no change of distance—only a change of bearing—or there is a change of distance (opening or closing from the guide) but no change of bearing. In the first of these cases—no change of distance—the diagram gives the correct solution using the distance ratio curve (in this case a straight line) corresponding to a distance ratio of one.
In the second case—opening or closing distance without change of bearing, the diagram gives the solution by using a zero change of bearing and the zero distance ratio line for closing distance and the infinity distance ratio line for opening distance. This means that the intersection in the distance triangle for closing is at its upper left corner and for opening distance at its upper right corner.
While it is thus seen that the two diagrams cover all cases whether simple or complex, it is possible and probably desirable to construct special diagrams for these two most common cases—as such tables can be made more simple.
Taking first the case of changing bearing without changing distance:
In this case it is apparent that the direction of the position line is at right angles to a bearing half-way between the present and new bearing—in other words, the inclination of the position line is the mean bearing plus or minus 90°. If we take diagram No. 2 as the basis for our new diagram, we may mark the “inclination of position line” side, with “mean bearings,” displacing the values 90° either way to allow for the plus or minus 90°. Now in Diagram No. 2 the “inclination” angles run from 0° at the point of the diagram to 180° at the right angle corner. To cover the case of subtracting 90° from mean bearing we should therefore mark the corresponding points on Diagram No. 4, from 90° to 270° and to cover the case of adding 90° we should mark them from minus 90° through 0° up to plus 90°. Though not apparent at first glance this is what has been done at the left side of Table No. 4. The left column covers the first case of subtracting 90°. Reading from the top down this is apparent as far as the middle of the column—180°. After that instead of going from 180° to 270°, it is desirable and practicable to reckon the bearing on the opposite side—thus 170° instead of 190°, 160° instead of 200°, etc.
The second of these columns covers the case of adding 90°, the angles in the upper half really being minus angles (angles on opposite side.) It is not necessary to consider the minus sign because it ultimately concerns only the direction, not the amount of the change of course, and in this problem we can make the simple rule that the turn be always made toward the new position.
It is apparent that for each “mean bearing” there are two sets of problems depending on whether we are increasing or decreasing the bearing. Thus, if we are to go from 90° to 170° the mean bearing is 130°. We are increasing the bearing and we must use the lower half of the diagram, whereas, if we are going from 170° to 90° we are decreasing the bearing and we must use the upper half.
There remains the question of how most conveniently to obtain the mean bearing. When the two bearings are on the same side, the mean is, of course, half their sum. When they are on opposite sides we could reckon both from the same side say 90° and 220° and take half the sum, but as the new bearing will be given as left 140° rather than right 220°, it is more convenient to take half the difference between the two bearings each reckoned on its own side. This really gives the reciprocal of the mean bearing, but for our purposes both are the same.
The rules for using Diagram No. 4 are very simple. It is merely necessary to enter the left side at the appropriate point, move to right to intersect a speed ratio curve and then vertically up or down to obtain the change of course corresponding to that speed ratio.
In the case of opening or closing distance without change of bearing we again use the fact concerning Diagram No. 2, that if we have the inclination of the position line to the course of the guide, we can obtain the proper combination of course and speed from a simple triangular diagram. Now in the case of opening distance the position line runs in the direction which the ship bears from the guide. In closing distance it runs in the opposite direction. It is necessary therefore only to make a No. 2 diagram in the form given as No. 5. Enter this diagram with the present bearing, using the right side when opening and the left side (where the scale is reversed), when closing, move horizontally to intersect a speed ratio curve and dropping down vertically, read off the change of course corresponding to that speed.
This is the simplest of all these diagrams and possibly the most useful as it can be used not only in maneuvers but in station keeping, and in opening or closing the range while maintaining the bearing.
As stated in the beginning of this article it is not believed that these diagrams can or should replace the mooring board, which must be the basis of all accurate maneuvering. On the other hand they may prove simpler for some officers. They do give a picture of the relation of the various elements entering in a maneuver which the mooring board diagram does not give and it may be worth while to have them printed in some of the tactical books or posted on the bridge.