WHEN the problem at sea is the interception of one vessel by another by radiocompass, it is not necessary to use those complicated methods which are ordinarily involved in tracking problems. The problem is not that of determining the course, speed, and position of the target prior to sight contact, but merely to set a collision course which must bring the searcher into sight contact with the target. It is true, of course, that the target can readily be intercepted after its course, speed, and position have been determined, but such determination is not necessary for the interception of the target.
In what follows it is assumed that the target is steaming on a steady course at a constant speed. The problem is considered as solved either when the searcher arrives within visual distance of the target, or when the searcher is brought to a course upon which the bearings of the target remain constant. Two methods will be given which differ only in the manner of their beginning.
First Method
In Fig. 1, the searcher of A takes a radiocompass bearing of the target and after steaming for any length of time, t, at speed, S, takes another bearing from B. True courses and bearings and all distances must be plotted carefully since the solution is entirely graphic.
If the target is drawing aft, the two bearings will converge, theoretically, to a point. But since the course and speed of the target can in no way be determined by these two bearings, it is necessary to make some initial assumptions upon which to proceed. There is little or nothing to indicate offhand what the course of the target may be, but a rough guess as to the limits of its speed may be possible considering the type of the target ship or the circumstances connected with the search. Only certain types of war vessels or fast liners would be likely to make 25 knots or more, and it is not likely that many vessels would make less than 10 knots. To fix limits to the speed of the target is to limit its course and its distance from the searcher. If, then, we consider that its slowest speed would be 10 knots, we may find T, where a line VT of length 10t will fit between the bearing lines. If the target is making 10 knots it cannot be nearer to the searcher than T.
Suppose that the identity of the target is known, and that the known maximum speed of the ship is 20 knots. With T as a center, and a radius equal to 2Ot, we may strike an arc cutting the other bearing line at H and K. The lines HT and KT, then, are the limiting courses that the target can make if it is at T. In the absence of any better information we may commence operations upon these impromptu assumptions, relying upon later bearings to yield corrective data.
These limiting courses and speeds, together with the known bearing of the target from B, will enable us, by means of the mooring board, to find the corresponding collision courses between which to limit our choice of a new course from B. In the choice of the first new course it does not matter greatly which is chosen as long as the new course is between the limits obtained from the mooring board. On the principle that it will take longest to intercept a target that is headed away from the searcher it is well to take the course to intercept the target on its most unfavorable course, such as HT in Fig. 1. Such a course keeps the searcher in a favorable position to deal with other target positions without great delay.
Having chosen his new course, the searcher obtains another bearing at C. If this bearing is different from that taken at B, the difference will indicate whether the target is drawing ahead or drawing aft. If it is drawing ahead, the target must have been nearer to the searcher than T and must
have been making a speed somewhere near its maximum. If it is drawing aft, its course and speed are not readily estimated at this time, but the change in bearing can be offset by applying a correction to the next collision course.
With the new bearing taken from C, consult the mooring board for the new collision course. Apply a correction equal to twice the difference between the bearings at B and C. This correction is applied in the same direction as the last change of bearing; that is, if the bearing of the target from C was to the right of the bearing from B, apply the correction to the right of the collision course calculated on the mooring board. This corrected collision course is the new course to be steered from C. The same process is repeated at each later bearing until the searcher is steadied on a course that gives no change of bearing.
The searcher need not hesitate to make radical changes of course for it is better to swing on either side of the real collision course than it is to allow the target to draw steadily ahead or aft. In all cases it is necessary for the searcher to study the plot and the changes of bearing in order to guide his selection of new courses to be steered. After a few bearings have been taken, an inspection of the bearing lines and their intercepts may enable the searcher to learn within narrow limits what the course and speed of the target actually are. With a little experience almost any problem of interception can be solved with six good bearings.
It will be noted that it is not necessary for the bearings to be taken at equal time intervals, nor is it necessary for the searcher to maintain a constant speed. But he must exercise care in providing the correct data for his mooring board calculations.
Second Method
In this method, three bearings are taken on the initial course. These three bearings enable the searcher, through the medium of the mooring board, to confine his initial assumptions of the target course and speed within narrower limits.
Since the mooring board treats primarily of relative motion, these first three bearings are laid off from the center of the diagram, 0 on Fig. 2.
Since these three lines diverge from one point, there is one system of parallel lines that can cross them with intercepts, either equal or of a given proportion. Let us suppose for a moment that these three initial bearings have been taken at equal intervals: then since the target, by hypothesis, is steaming at constant course and speed, the direction of a line which cuts the three bearing lines with equal intercepts will be the relative course of the target. If the bearings are not taken at equal intervals of time then the line which cuts the bearings with intercepts proportional to the time intervals will be the relative course of the target. It is not necessary, except for simplicity, that these bearings shall be taken at equal time intervals ; a geometrical construction will be given later by which the relative course of the target may be obtained for any time intervals between the three bearings.
In Fig. 2, XY is the relative course of the target, being the line that crosses the bearings OA, OB, and OC so that XZ equals YZ. Also, any line parallel to XY will have equal intercepts.
From the center of the board, lay off the speed vector of the searcher, OS, and from S draw ST parallel to XY. Then ST is the locus of the outer ends of the various possible speed vectors of the target drawn from 0; that is, as long as ST is the relative course of the target, then the target must actually be on course OU, if its speed is 80 per cent of OS, or it must actually be on course OV, if its speed is 60 per cent of OS, or on OW if 40 per cent, etc. OH is the minimum speed that the target can possibly be making, since OH is drawn normal to ST.
In this way, each possible speed of the target is tied up with a particular course, and if we assume limitations of the target speed on the same basis as we did in the first method, we may readily find the corresponding limits of the target course and also the limiting collision courses. We may then proceed, point by point, as in the first method.
It will be noted that it is not necessary for the bearings to be taken at regular intervals. It is also unnecessary for the searcher to maintain a constant speed, except on the initial course ABC; but accurate data must be supplied for the mooring board calculations.
Constructive Problem
Given three lines which intersect in a point, to find the direction of a fourth line that will intersect the three lines with intercepts of a given proportion.
Suppose that RST cuts the three lines drawn from O so that ST/RS = k. The problem is, then, to construct a line parallel to RST without further mechanical reference to it. be seen in Fig. 2a that at E the searcher changed course to 9° and that his next bearing changed only 2°, showing that 9° is quite close to the proper collision course. Let us assume for purposes of demonstration that 9° was the correct collision course, and that 346° is the constant bearing. In Fig. 2, we saw that ST was the locus of the outer ends of the target speed vectors as determined by the first three bearings. Then, in Fig. 2, plot / for the searcher course 9°, speed 25 knots, and from / draw JK, representing the constant bearing, 346°. Then JK is also the locus of the outer ends of the target speed vectors, and L where ST and JK intersect, must be the outer end of the actual target speed vector. If the vector OL is plotted on Fig. 2a in the same manner that OU, OV, and OlV were, the actual track of the target will have been determined.
Even with the erroneous data assumed, Fig. 2 shows the target course to be 98°.5, speed 10.5 knots, which is not greatly in error for the actual solution—course 99°.5, speed 10 knots.
Still another method of tracking the target is suggested by the above. Suppose that after changing course at C and obtaining another bearing at D, the searcher is to continue the same course and speed, and obtain another bearing of the target at E. There will then be two sets of three bearings each, with bearing C common to both, and each of the two sets will have been taken while the searcher was on a constant course and speed.
We have seen that one set of bearings, OA, OB, and OC (Fig. 2) gives a locus line ST. The other set of bearings taken from points C, D, and E, could be used in exactly the same manner to determine another locus line. Then, the vector drawn from the center of the diagram, O, to the intersection of these two locus lines, would give the actual course and speed of the target, and the application of that vector, to scale, in Fig. 2a, would give the position of the target.
This latter method has the disadvantage of inaccuracy of plotting if the course from C is close to the collision course, because the bearings C, D, and E, will be so close together that the second relative course of the target will be difficult to determine graphically with accuracy. If, as is sometimes the case, tracking is desired rather than sight contact, course CDE could be selected approximately at right angles to the collision course. This would obviously increase the accuracy of plotting rather than making it more difficult.