PART II.
MANEUVERING.
CHAPTER I.
IDEAS ON NAVAL KINEMATICS.
29. Preliminaries.—The motive that induces us to cite a few fundamental ideas on naval kinematics must not be sought for in a desire for mathematical divagations, nor for the study of battle maneuvers on the basis of a prioristic hypotheses concerning the movements of the enemy. The object that we propose for ourselves is that of determining criteria of the maximum simplicity, holding it to be an axiom that, in offensive contact, it is absurd to place confidence in tables, diagrams, or instruments for geometrical constructions. Furthermore, it is well to give notice that, while not excluding such means in contact out of range, and during exercises (and in the latter only until a sufficient habit in maneuvering is acquired), the only one of them that we deem indispensable for the conning of a ship under the fire of the enemy, is that composed of a horizontal disc upon which are marked the sectors of offense of the weapons, and in the center of which is a revolving alidade furnished with a sight vane.
But precisely in order to be free from all shackles, clearness of ideas is necessary concerning the solution of the principal problems of kinematics, and toward this we shall tend, limiting ourselves to the purely indispensable.
To fix the idea, let us refer to the case of two ships opposed to each other, observing that, on the basis of the deductions of Chapters I and IV of Part I, we cannot confine ourselves to considering the rectilinear tracks.
Indeed, in long-range battle, if we suppose that our ship is in the proper tactical zone, that the enemy bears approximately in a direction of maximum utilization, and that we maintain a constant course, after a short time—except in very particular cases—the inclination to the plane of fire could not be held to answer to the tactical necessities; and hence it will be necessary to follow a new course.
In general, rather than change the course at intervals, and so disturb the fire control, there naturally comes the idea of satisfying the tactical necessities continuously rather than intermittently, by keeping the polar bearing of the enemy constant for a certain time, which can easily be done by means of the alidade of the instrument just mentioned. In this way, the track passed over is generally curvilinear; and its curvature is naturally a function of the enemy's track. As—steering thus by means of the sight vane—the ship continually changes her course, the doubt may arise that the disturbance of the firing may be continuous; we establish, then, the idea of taking also into consideration these curvilinear tracks, unless upon examination their radii of curvature prove to be very great. It is clear that if such conditions are realized, they present a real advantage by substituting continuous, but very slow, changes of course for those of notable amplitude between the successive courses of a broken right line.
Setting aside the effect upon the firing, it is intuitively seen that steering on a constant bearing may be advisable under some circumstances, because it permits the maximum simplicity of maneuvering that can conveniently be adapted in a continuous manner to the maneuvering of the enemy.
When the advisability of the continuous adaptation just now mentioned does not exist, and the problem is that of taking, with the greatest rapidity, a determined position with respect to a ship or a fleet of ships, the necessity of rectilinear tracks is evident. In fact, they permit of attaining the desired object in the mini- mum time, unless a curvilinear track may be convenient in order to diminish the uncertainties of the maneuver, and may permit of reaching the desired position in a space of time only slightly greater than the minimum.
In general, then, we may establish the necessity of rectilinear tracks:
(a) In contact out of range.
(b) In the maneuver of approach of a light, swift vessel.
(c) In the movements of friendly ships (evolutionary problems).
Bearing in mind what has previously been said, for the study of the movements of two ships—which forms the object of this chapter—we are to consider rectilinear movements and those on a constant bearing.
30. Indicator of Movement (Fig. 12).—Let us consider the simultaneous positions of two ships, A and B, of which VA and VB are the respective speeds, and AA1 and BB1, the courses. If A, and B, are the simultaneous positions of said ships after an infinitesimal timed t, let us take A1A1’ parallel and equal to BB1, but in a contrary direction. The distance A1’ B is equal to that of A1B1; moreover, the joining lines A1B1 and A1’B form the same angles with the courses of B and A. The geometrical locus of the points A1’ (of which A is the origin) is the trajectory or indicator of the relative movement of A with respect to B, called,
{figure}
FIG. 12
briefly, the indicator of movement; or, in other words, it is the track that the ship A, at a speed which is the resultant of VA and VB (the relative speed), passes over with respect to the ship B, which is, supposed to be stationary.
31. Generalities Concerning Rectilinear Movements (Fig. 12).—If the tracks of A and B are rectilinear and the speeds VA and VB are constant, the indicator of movement will be rectilinear.
In fact, from what has been said, the points of A1' are aligned with A, the resultant of the speeds being also constant.
Consequently, if A must determine the proper course for bringing itself into a given position with respect to B, it is well to observe that the course that corresponds to the hypothesis of an immovable B is the indicator of movement; the problem is then reduced to that of passing from the indicator of movement to the course of A.
If we imagine a circumference described with its center at A and with any radius AA1 which we take as a unit of measure, the course sought is obtained by joining A with the point A1 of said circumference, the parallel from which point to the course of B cuts the indicator in a point, A1’, such that we may have*
A1A1’/AA1=VB/VA;
If a and ? are simultaneous polar bearings of A and B respectively (counted from the bow), indicating the relative speed by Vr, from the triangle AA1A1’ we have
Vr=VA √1+(VB/VA)2 +2VB/VA cos (a± ?)
wherein it is necessary to use for 13 the positive or the negative sign, according as the two ships move toward the same side or toward opposite sides of the line joining them. The minimum value of Vr is VA — VB, which corresponds to parallel courses in the same direction.
If s is the segment of the indicator included between the points corresponding to the initial position and the final position, the time t necessary for completing the movement is
t=s/Vr. (1)
Furthermore, indicating the angles formed by the courses of A and B with the indicator of movement by yA and yB respectively, from the aforesaid triangle AA1A1’ we obtain
sin yA=VB/VA sin yB.
Since it must be that sin yA≤1, there results from this relation: 1st, that any change of position of a ship, A, with respect to another, B, which is steering a fixed course, is possible when VA>VB; 2d, that if A has not superior speed, there are possible only those changes of position for which sin yB≤VA/VB; 3d, for a value of yB, two supplementary values of yA satisfy the above-
*Marking off from A a segment VB/VA, parallel to the course of B and in the same direction as VB, and drawing from its extremity the parallel to the indicator of movement, the point A1 is determined by the intersection of this parallel with the circle of center A and radius 1. (Author's note.)
mentioned relation. It is easily seen, concerning the afore said geometrical construction, that when VA=VB, a relative speed, O, corresponds to one of the said values of yA; if VA< VB the relative speeds corresponding to the two values of yA are made in the same direction; and this must be taken into account in order to select the proper value of yA, since both the values of yA bring one to the desired position, but in different times; finally, if VA>VB the two values of the relative speed have contrary signs.
32. Chase Problems.—The simplest application of the preceding deductions consists in determining the proper course for over-taking another ship.
In this case, the indicator of movement is the joining line AB; therefore, during the movement, the two ships keep each other on the same polar bearings. The course sought can be determined with the geometrical construction, already mentioned, which it is superfluous to recall as it is practically done with an instrument having two alidades. The immediate determination of the proper course would require exact knowledge of the course and speed of the other ship; this not being presumable, recourse must be had to successive rectifications.
When there is no instrument available except one with a single alidade—mentioned in Section 29—the course to be steered must be estimated by eye and afterwards corrected until the polar bearing remains constant. In this way the ship A which gives chase generally follows a broken rectilinear track; we shall presently see how the rectifications just mentioned may be limited.
Conformably to observation 3 of the preceding section, the condition that the course steered should keep the polar bearing constant is necessary, but it is not sufficient to bring about a meeting; in fact, this may be verified for two courses of A which form supplementary angles with AB. When VA<VB the meeting is not possible if B holds A on a polar bearing, from the bow, greater than that of which the sine is VA/VB. Both the above-mentioned courses may VB lead to a meeting, but naturally the one which makes the greater angle with the said joining line cannot secure the object in the shorter time; and the existence of such a solution is worth remembering in practice in order to avoid errors. In the case of VA=VB, the course that makes the greater angle with the joining line and keeps the polar bearing constant, is parallel to the course of B; or, geometrically, it would lead to a meeting in an infinite time. Finally, when VA>V, the course of A, which makes the greater angle, diverges from the course of B, and the ships may be considered as departing simultaneously from the point of intersection of the courses.
This being said, we observe that if A steers on a constant polar bearing equal or nearly equal to the one necessary for reaching a meeting in the shortest time, the indicator of movement is rectilinear, or it is very nearly a straight line; and hence A arrives at the meeting in the minimum time, or in a time slightly greater. It is unnecessary, then, to establish the precise condition that the
{figure}
FIG. 13.
track of A be rectilinear; and this suggests the practical rule of not troubling oneself too much about the aforesaid rectifications of the course, but determining an approximate course, and then steering so as to keep constant the polar bearing thus obtained.
II. The problem just discussed is a particular case of the following: To determine the course that a ship, A, must steer, with a speed VA, in order to arrive, in the minimum time, from a distance R to a distance r from a ship, B, which steers a rectilinear course at a speed VB; in case this is impossible, to determine the direction in which A must move in order to reach the minimum distance from B.
In order to solve this problem, let us first of all demonstrate that if A' and B' are the positions of A and B when A has reached the distance r from B in the minimum time, the three points A, A' and B' are in a straight line. This may easily be demonstrated by an absurdity.
Let us suppose R>r. Let us also suppose that A' is not upon the joining line AB' (Fig. 13) and that A1 is the point of intersection of AB' with the circumference whose center is at A and whose radius is AA'.
Let A" be the point of intersection of AB' with the circumference whose center is at B' and whose radius is B'A'=r. It is clear that AA" is less than AA1; if, then, the ship A had followed the course AB' it would have arrived at a distance from B less than r, in a time equal to that occupied in passing over AA'. Into such absurdity one always falls, except in the case in which A' is on the joining line AB'; the alignment mentioned is therefore necessary. The proposition enunciated is analogously demonstrated if R<r; that is to say, if, instead of diminishing the distance, A must increase it. From this it follows that when A arrives at a distance r from B, if this is accomplished in the minimum time, B must be found exactly ahead of him if r<R, or exactly astern of him if r>R.
If the ship A, which sees B pass ahead of it at the distance r, should continue on its course, it would evidently cross the track of B a time r/VA, in which time B would have passed over a space r VB/VA. It follows from this that, in order to execute the maneuver of approach from any distance, R, to a distance r, in the minimum time, it is not necessary to steer as if one wished to reach the ship, as was pointed out in Problem I, but to maneuver in order to reach an imaginary point which is situated astern of B at the distance r VB/VA, and which moves with the speed and on the course of B.
Evidently, the greater the distance r, and the smaller the advantage in speed that A has over B, the more important it is to bear in mind the difference between the exact solution—which results from the consideration just alluded to—and the approximate solution which corresponds to the hypothesis r=0, or to the other hypothesis VB/VA = 0.
In contact out of range, in which there is a possibility of making a geometrical construction, the course sought is determined by laying off AB (Fig. 14), which may represent R, then taking a segment, BH=r VB/VA, in a direction parallel and opposite to the course of B, and finally forming (Section 31) the triangle AA1A1’, in which A1A1’ is parallel to BH, and where in the ratio between the sides AA1 and A1A1’ is VA/VB. The solution may be obtained more simply by describing the arc of a circle with its center at B and with a radius r, cutting all at a point H'; which indicates the course H'B that A must follow, and in regard to which the general discussion of Section 31 is recalled to mind.
{figure}
FIG. 14.
When A and B are in sight of each other the reason for applying the approximate solution in the manner indicated under Problem I, is that one cannot steer by sight vane for the point H, because that point is not distinguishable. Nevertheless it does not seem that the exact solution may be completely neglected, since we are able to come near to it, without any complication, by observing the following rule: Maneuver initially as if it were desired to come to a meeting with B, and then, successively, at intervals, bring the sight vane by which we are steering toward B, more toward the bow, in such fashion that the said ship may be bearing ahead when the desired distance is reached.
If VA<VB, so much the more necessary is it to refer to the general problem, in as much as the problem of meeting may be an impossible one; but we may approach to the minimum distance therefrom, which, as is seen in Fig. 14, is evidently that for which H'B (and hence the course of A) is perpendicular to the indicator of movement. In order to reach this minimum distance, the cosine of the angle i between the courses must equal VA/VB.
In contact out of range it is possible that it may be foreseen that the enemy will follow a broken rectilinear track when in the vicinity of the coast.
If the track of B (Fig. 15) is BB1X, two cases may present themselves: 1st, A may arrive at the desired distance r from B, before B arrives at B1; and then the solution is the one already
{figure}
FIG. 15.
indicated. 2d, the course AY, along which A would have to move under the preceding hypothesis, intersects BB1X in some point of B1X. From B, then, we lay off on the prolongation of B1X the segment B1B’=B1B. It is evident that the direction AY', in which A must move in order to solve the problem, is obtained by supposing that the movement of B always takes place along B1X, and that B' is the initial position of B. Clearly, this course of reasoning might be extended to the case in which the broken rectilinear track has more than two segments.
When r>R, the solution of the problem enunciated is obtained in a way analogous to that required for r<R, observing that the course on which A reaches the distance r in the shortest time, cuts the line of the course of B ahead of that ship, and at a distance r VB/VA; hence unless A is exactly astern of B, the most opportune course for increasing the distance up to a certain limit as rapidly as possible, must be a diverging one.
33. Evolutionary Problems.—I. Let us suppose that B (Fig. 16) follows the course BX at a constant speed. If A finds itself,
{figure}
FIG. 16
with respect to that ship, at a distance r1 and on the polar bearing ?1, and desires to pass in the minimum time to the distance r2 and to the polar bearing ?2—that is to say, to a position P with respect to B, which is supposed to be stationary—the line AP is the indicator of movement; and hence, with reference to it—as has been set forth in Section 31—setting aside the displacements due to changes of course; we may determine the course AA' that A must follow, and the necessary time.
The problem thus set forth is the one that presents itself when we aim at transporting ourselves from contact out of range, not only to a certain distance from the enemy, but also to a deter- mined relative position.*
Moreover, what has just been said includes generically the cases realized in evolutions that are not performed in succession. If ?1=?2, AB is the indicator of movement, and the evolution reduces itself to a change of distance; if r1=r2, the evolution consists of a change of bearing; the indicator of movement is then normal to the bisector of the angle PBA; that is to say,
PBA=90° - ?/2,
? being equal to ?1-?2.
II. Let us now consider the changes of bearing. The perpendicular segment dropped from B upon AP—that is to say, AB cos ?/2 -indicates the minimum distance at which the ships will pass during the evolution; it is generally held that the distance ought not to fall below 7/10 of the normal distance, which establishes for ? the limit of 90°.
It is easy to find the formula which permits of obtaining the angle ? through which A must change course, supposing, naturally, that in the position A, the said ship has a course parallel to that of B. From the figure we get
?=180° - ?1 - PAB – yA,
in which
PAB=90° - ?/2,
and yA is given by equation (2) of Section 31; that is,
yA = arc sin (VB/VA sin yB).
Then since
sin yB = sin (180° - ?1 - PAB) = cos (?1 - ?/2),
we obtain
?=90° + ?/2 - ?1 - arc sub [VB/VA cos (?1 - ?/2)].
*If the movement of B, instead of being on a single course, can be predicted to follow a broken rectilinear track, the method to be followed is evidently analogous to that indicated in the preceding section. (Author's note.)
The change of course must be made toward B, or in the opposite direction, according as ? is positive or negative.
In order to be able to pick out the value of ? from a table, the latter might be one with two entries (that is ?1 - ?/2 and the ration of the speeds).
In order to eliminate the use of tables, recourse should be had to diagrams,* to an instrument with an alidade, or to the method by parallel courses, making the ships change course through the angle 90° + ?/2 - ?1 in a way that may result in the direction AP, B afterwards executing the reduction of speed. With this latter method the relative speed is VA—VB; and hence the time necessary for the evolution remains at that corresponding to the method above indicated (AP being always the same) in the ratio Vr/VA – VB, by virtue of equation (1) of Section 31. Hence it is evident that the evolution by the method of parallel courses can never occupy less time than that which is required by the other method, which we will call the method by oblique courses.
A general rule—which we shall suppose to be implicitly followed in the evolutions when nothing is specified to the contrary— is that the ship B at the beginning of the evolution (or before reducing the speed) makes the two changes of course (initial and final) that A makes at A and at A'; with this rule, mentioned by Admiral de Gueydon, t the displacements due to changes of course are rendered the same for A and for B.
III. Definitions.—If two ships have the same course and speed, and the one further advanced in the direction of the course is bearing from the other at an angle a from the bow, we say that it is on the line of polar bearing a.
By wheeling a line of polar bearing we mean the evolution by oblique courses which permits of rotating the line joining the two ships, at the end of which rotation the former polar bearing of the formation is re-established.
*PESCI: Sui metodi per cambiare il rilevamento fra le navi di un a formazione semplice, Rivista Marittinza, March, 1897.
vTactique Navale. Recherche des principes primordiaux et fondamentaux de toute tactique navale (1868).
Fig. 17 shows a line of bearing that executes a wheel through an amplitude ?. The ship B—which is the one situated on the side toward which the wheel is to be executed—changes to the course BI', which makes with the original course BI the angle to in the direction of the wheel; after this B reduces the speed, assuming a speed VB, whileA maintains the evolutionary speed, which we will designate by VA. In order that the distance at the end of the evolution may be the normal distance, the indicator of movement must be AP, such that PAB=90° + ?/2; so that the course that A must follow is AA'. When A has nearly arrived at
{figure}
FIG. 17.
the position A', it changes to a course parallel to BI', while B again takes up the normal speed.
Evidently this problem is but a particular case of the one before presented, and, in its turn, the problem of wheeling a column of vessels is a particular case of this one. In such particular case the choice of the pivot ship is optional; that is, the line may be made to wheel about the rear ship or about the leading ship.
In 1905 we were led to consider the wheeling of a column of vessels by reading an important article entitled, Notes on the Principles of Naval Tactics, which appeared in the September number for that year of the United Service Magazine, signed by the pseudonym Experience. In the said article mention is made f pivoting the line at one-third of its length from the leading vessel; we shall occupy ourselves with this problem hereinafter. In wheeling the column, while the pivot ship turns through the angle ? in the direction of the wheel, the other ship must change from the original direction through an angle ?=?+?, ? being the angle given by equation (3); the angle ? is positive or negative according as the change of course is made in the direction of the wheel or in the opposite direction.
Let us indicate by ?c and ?t the respective values of ? for the wheel on the rear ship and that on the leading one.
In the first case the value of ? is determined, bearing in mind that the pivot ship changes course through the angle (0, and that hence, in equation (3), it is necessary to put ?1=?; in this way we get
? = 90° - ?/2 - arc sin (VB/VA cos ?/2),
and hence
?c = 90° + ?/2 - arc sin (VB/VA cos ?/2) (4)
For the wheel with the leading vessel as a pivot it is necessary to substitute in equation (3)
?1 = 180+ ?
from which we obtain
-?t = 90° + ?/2 - arc sin (VB/VA cos ?/2)
Taking absolute values from ?c and ?t we get
?c – ?t = ? (5)
which, together with equation (4), permits of the construction of a table which will give the angle of the change of course for a given speed ratio.
From equation (5) it is clear that, in order to apply the rule of De Gueydon, the pivot ship, in wheeling on the rear vessel, must change course through ?t in the direction of the wheel, and afterwards execute a change through ?t in the opposite direction; in the case of pivoting on the leading ship, the pivot ship must change course through ?t in the direction opposite that of the wheel, and afterwards through ?c in the direction of the wheel.
IV. By equation (1) of Section 31, the time necessary for executing a change of bearing (or a wheel) of an amplitude ?, is
t=AP/Vr = 2AB sin ?/2 / Vr,
from which it is seen that the duration of the evolution is proportional to the length AB of the line or column.*
Let us now compare the rapidity of wheeling a column of vessels by the two methods indicated.
Let B be the pivot ship (Fig. 18). According as the ship A is at A2 or at A1—that is to say, ahead or astern of the pivot—in order to wheel the line through ?, we find ourselves in the case of pivoting on the rear or on the head.
A2’ and A1’ being the corresponding positions of A at the end of the evolution, let us determine what must be the ratio A1B/A2B, to the end that we may have A2A2’=A1A1’. By proportionality before mentioned, this ratio indicates the relative rapidity of the two methods, which we will indicate by virtue of the t2/t1, t2 and t1 being the times respectively employed in the case of pivoting on the rear and on the head.
Considering the triangles A1BA1’, A1BA1’, by equation (5), we have
A2A2’B=?t; X1A1’B=?c
and hence
sin ?t/A2B = sin ?/A2A2’; sin ?c/A1B = sin ?/A1A1’
* The ratio between the times necessary for the evolution by the method of parallel courses and by that of changing direction in succession is
2 sin ?/2 / VA - VB VA,
or the first of these methods is more rapid if
sin ?/2 ≤ 1 VB/VA / 2.
With the formula VB/VA - 1/2, ? ≤ 29°; for VB/VA = 8/10, ? < 12°.
Then, from what has been said in division II, the evolution by oblique courses is proper in limits wider than these. (Author's note.)
from which-as, by hypothesis, A2A2’=A1A1’- we obtain
A1B/A2B = sin ?c/sin ?t
Calculating ?c and ?t by by means of equations (4) and (5), we deduce from this formula the values of the ration A1B/A2B, which
{figure}
FIG. 18.
values, set down here below opposite ?, are determined under the hypothesis that VB/VA =1/2.
? | t2/t1 |
15° | 1.16 |
30° | 1.35 |
45° | 1.57 |
60° | 1.78 |
75° | 2.00 |
90° | 2.24 |
It results from this table that the greater rapidity is obtained by pivoting on the head; by the other method, for the supposed speed ratio, about double the time is employed if ? is between 60°and 90°, and one and a half times more if ? is between 30°and 45°.*
Let us now consider the time required for changing the direction of the line through the angle ? by changing course in succession, and compare it with that required by a wheel with a pivot at the head; setting aside the time employed in changes of course.
Indicating by t0 the time required for the change of direction by changing course in succession, we have
t0=AB/VA, (7)
while the time required for the wheel is given by equation (6); hence
t1/t0 = 2VA sin ?/2 / Vr.
In the case we are considering (pivoting on the head), in the formula of Section 31, which gives Vr it is necessary to put
a=?t; ?=180° - ?
and by equation (5) we obtain
a - ? = ?c - 180°;
therefore we have
Vr = VA √1+( VB/VA)2 - 2 VB/VA cos ?c.
There results, then,
t1/t0 = 2 sin ?/2 / √1+(VB/VA)2 - 2 VB/VA cos ?c
* It is easily seen that t2/t1 increase the more VB/VA approximates to unity. It is well to note that, in any evolution, we may not adopt in practice a ration VB/VA less than 1/2; indeed it must be remembered that the speed of the pivot ship is not instantaneously reduced, and hence, in order to have the mean speed VB during the evolution, the engine must be regulated for a lower speed. (Author's note.)
With this formula, for VB/VA = ½, there are calculated the values of t1/t0 set down in the following table, in which are also written the values of t2/t0 obtained by multiplying the values of t1/t0 by the corresponding values of t2/t1 given in the preceding table.
? | t1/t0 | t2/t0 |
15° | 0.28 | 0.32 |
30° | 0.51 | 0.69 |
45° | 0.71 | 1.11 |
60° | 086 | 1.53 |
75° | 0.99 | 1.98 |
90° | 1.09 | 2.44 |
Having regard to the rapidity and to the simplicity of the evolution, and reserving I tto ourselves to discuss the subject in relation to the movements of the enemy, this table permits us to announce the following conclusions:
1st. When ? is greater than 30°, a change of course in succession is preferable to a wheel on the rear ship.
2d. A wheel on the leading ship may be preferred to a change of course in succession when ? is within the limit of 60°.
Within the limits thus established, indicating by t a value of t1 or of t2, and t0 being the corresponding value given by equation (7), putting t/t0=m, we have
t0-t=AB/VA (1-m).
The gain in time permitted by a wheel is then directly proportional to the length of the line.
34. Determination of the Course and of the Speed of the Enemy.—I. The principal force (or the main body of the fleet), on information from the units that keep the enemy in sight and determine his course and speed with the closest approximation possible, can execute the maneuver of approach in the way indicated in Sections 32, II, and 33, I.
The preliminary problem reduces itself to the hypothesis of a ship, A, that wishes to determine the course and the speed of another ship, B.
With this object in view, the ship A keeps on a constant course and, at any interval, t, of some minutes, it repeats the measurement of the distance and the polar bearing on which B lies. Drawing the triangle AA1’B (Fig. 12), where in AB and A1’B represent two measured distances, and the angle at B is the difference between the corresponding bearings, the line AA1' is the indicator of movement; consequently, marking on the drawing the course of A and taking on it the segment AA1, equal to the distance passed over by the ship in the time t, the segment A1’A will represent, in direction, the course of B, and in length, the distance passed over by the said ship in the said time.
With these measurements, two values for the course as well as for the speed sought may thus be obtained; hence, the mean of these may be taken, and so proceeding, an approximation may be reached which is so much the closer, the greater the number of measurements made.
But we note that, under the circumstances in which the problem must be solved, the distance AB may be predicted to be in the neighborhood of 15,000 meters, and may be even greater; hence but scant confidence can be placed in the accuracy of the indications supplied by the range finders. For this reason it is well to observe that, in the determination of the elements we are seeking, we may proceed with a graphic method, based upon the measured polar bearings and the spaces passed over in the intervals, supposing the distance to be only roughly known.
For this very simple method, devised by Lieutenant P. Corndon, it is expedient: 1st, to follow a course that is not parallel to that of B; 2d, that, considering the surface of the sea to be divided into two parts by the joining line AB, the two courses be not directed toward the same side of the line.
This being said, here, in a word, is the method: We mark off on a straight line which represents the track of A (Fig. 19) the points A, A1, A2,…corresponding to the space passed over in the interval established between the successive measurements, and then we draw from these points the straight lines AB, A1B1, A2B2,. . . forming with the above-mentioned line the measured polar bearings.
With a graduated ruler, estimating by eye the direction followed by B with respect to A, we seek, by trials, to find the position that the ruler must assume in order that the segments BB1, B1B2… may be equal among themselves. Such a position of the rule is the line of the course of B, and the scale of the drawing supplies his speed.*
Having a fairly good knowledge of the distance, the two methods above indicated may be combined; in other words, AB and A1B1 being marked down, taking into account the first two distances measured, we place the ruler in the direction of BB1. When the next measurement permits of marking down A2B2, we determine a more approximate position of the rule, so that BB, may be equal to B1B2, and so on.
II. When the fighting forces that have to execute the movements corresponding to the preceding sections are in sight of the enemy, it is obvious that they may not delay beginning those movements in order to determine his course and speed; but it is well to take an approximate course, keeping it constant for some minutes, in order to attempt such a determination by the first of the methods just pointed out, and afterwards be governed by the established rules.
{figure}
FIG. 19.
*Evidently if the successive joining lines were all to intersect in one and the same point at a distance finite or infinite (parallelism), the position of the rule would be indeterminate. It is to be noted that theoretically this indetermination exists in any case because the curve traced through the intersections of the joining lines AB, A1B1, A2B2,....is a parabola; consequently an error in the initial distance AB produces errors in the speed and in the course. However, when the angle between the courses is within opportune limits, the above-mentioned errors are very small, even when the distance is very roughly known; for example, when a distance of about 15,000 meters is estimated with an approximation of ± 2000 meters.
If the polar bearing were exactly measured, for the application of the method, it would suffice to limit the said measurements to three; but on account of the inevitable errors it is well to take at least five or six bearings, well distanced, for example, at intervals of five minutes. Concerning the probability of so doing it is well to bear in mind that in contact out of range, when the naval forces are not in sight of each other, the course and the speed will not be frequently changed. (Author's note.)
35. Fundamental Tactical Relation.—In order to establish criteria of practical utility, the study of the offensive contact must be made in the most general form, considering the supposed adversary free at any moment to keep our ship on the bearing that he may deem most advisable.*
The basis for the study of the types of maneuvering that we shall develop in the following chapters must be sought in a general relation that shall bind together the elements of movement of the two ships; that is to say, the respective speeds, the polar bearings, and the variation of the distance. To such a relation we shall give the name of fundamental tactical relation. Two ships, A and B, follow any two tracks. At the moment we are considering, they have the respective speeds VA and VB and they have each other on the bearings a and ? from the bow
{figure}
FIG. 20.
The indicator of movement is a curve AX (Fig. 20), of which the tangent at A, which is the direction of the relative speed Vr, forms with AB an angle which we indicate by ?.
If m is a point on the indicator infinitely near to A, describing the arc mn of a circle with its center at B, the triangle Amn may be considered a right triangle, right angled at n; hence
An=Am cos ?
Letting dr and dt be respectively the differentials of the distance and of the time, since dr is negative when a diminution of the distance is produced, by substituting for An and Am their values, we have
dr=-Vr cos ?dt.
*Suppose, for example, that the enemy follows a constant course, or some other track of geometrically determined form, it is well understood how a proper measure might easily be decided upon; but such a deduction would be of small importance, because it would be based upon too particular a hypothesis. (Author's note.)
Projecting Vr and its components, VA and VB, upon AB, as the projection of the resultant is equal to the algebraic sum of the projections of the components, we obtain
Vr cos ? = VA cos a + VB cos ?.
Substituting in the preceeding equation, there results
dr/dt= -(VA cos a + VB cos ?),
which is the relation sought. Putting
?=180° - ?,
the fundamental relation may be written
dr/dt= VB cos ? - VA cos a,
in which it is to be remembered that ? is counted from the stern and a is counted from the bow.
36. Constant Bearings.-If the two ships A and B do not alter their speed, and steer on constant polar bearing, dr/d, or the distance in a time sufficiently short to enable us to consider a and ? as practically constant during that time. Let us see what may logically be held to be the variability of the elements of movement.
Trusting to our dexterity in firing, we may propose to ourselves to cause the distance to vary rapidly; that is to say, to render dr/dt the maximum. It is well to observe, however, that we control our polar bearing and our speed; that is to say, only one of the
*It is to be noted that when the speeds and bearings are constant, Vr and ? become constant; the indicator of movement of one ship with respect to the other thus cuts the straight lines drawn from its initial point at a constant angle; it is, then, an equiangular or logarithmic spiral, the pole of which is in the said initial position. The spiral naturally becomes a circle when the distance is kept constant, and is reduced to a straight line when the conditions of movement cause the ships to arrive simultaneously at the point of intersection of their attacks, or when the two ships may be considered as setting out simultaneously from the said point. (Author's note.)
terms of the second member of the fundamental relation. Well and good; by keeping the enemy on a variable polar bearing we shall not be able to oblige him to do likewise, and we shall renounce important benefits. In fact, a constant bearing may opportunely be chosen (Part I, Chapter I); moreover, it facilitates the control and the execution of the firing. The control is advantaged since it must necessarily be based upon the hypothesis that the variation of the distance in the interval between two successive salvos is approximately constant, and certainly the fact that we do not change the gun pointing in direction is good for the execution of the firing.
It is very probable that the enemy also may maneuver well; that is, he may be inspired by the same ideas. In studying the types of maneuvering we shall not exclude the possibility of his withdrawal; but meanwhile it may be held to be a sufficiently general hypothesis that a and ? do not change for a certain time.
With regard to the speed, it is evidently important to vary it as little as possible.
37. Curvature of the Ship's Track.—It is now expedient to ascertain whether the movement on a constant polar bearing may in any case produce sensible disturbance to the firing.
The ship A, when the enemy B has him bearing at an angle ? —to port, for example—produces, by virtue of the fundamental relation, the same variation of distance whether he keeps B bearing at the sight vane angle a to starboard or to port; because, in that way, the values of a to be introduced into the fundamental relation are equal and with contrary signs; that is to say, cos a does not change. However, according as A keeps B at an angle a, presenting to the fire of the enemy the starboard or the port side, it changes, not only the track of A, but also that of B; although the side that the latter presents may be the same in both cases.
Let us indicate by ?'A and ?’B the radii of curvature of the tracks of A and B at a given instant, in case the ships present to the fire sides of opposite names; while ?"A and ?” B, indicate the radii in the case in which the ships present sides of the same name.
As, by hypothesis, the bearings are kept constant, the angle through which A and B change course in a given time is the same; and is equal to the angle through which the line joining the adversaries rotates. Let d?" be the said angle for the movement in the time dt, with sides of opposite names, while we indicate its value in the movement with sides of the same name by d?". Denoting that dS A and dS B respectively the differentials of the arcs of the tracks A and B, there results
dS A = ?'Ad?"= ?'Ad?"= ?'Adt
dS B = ?'Bd?"= ?'Bd?"= VBdt (8)
and hence
?'A/?'B = ?”A/?’B = VA/VB. (8)
We may then affirm that, at any instant, the ratio of the radii of curvature of the tracks is constant and equal to the ratio of the speeds.
Indicating by ?' the radius ?'A or ?’B and by ?" the corresponding ?” A and ?" B, there results from (8)
?’/?” = d?”/d?’
This being the case, let B1 (Fig. 21) be the position of B after the interval of time dt, and let A' and A" be the corresponding positions at which A arrives according as he selects the movement with the side of opposite name or with the side of the same name.
Prolonging AB and A'B, to their intersection at M', and calling M" the point of intersection of AB and A"B1, we have
AM'A'=d?’, A"M"A=d?";
but since by hypothesis
A"AB=A'AB=a,
we have also
AM"A'=d?”.
{figure}
FIG. 21.
The angle d?" being thus an exterior angle of the triangle A'M"M', we have
d?”>d?’;
and hence, by equation (9),
?"<?’.
It is thus established that, if a ship fights presenting to fire the side of the same name as that of the enemy, the radius of curvature of its track is at any moment less than it would be if, with the same sight-vane angle, the ship should present the side of the contrary name. The rotation of the joining line is more rapid.*
With the object of completely fixing the ideas concerning the subject we are discussing, it is sufficient to have recourse to the relation that gives the value of d?/dt, which is determined in a manner analogous to that of the fundamental relation, as follows: In order to get the angular displacement in the time dt of the line joining the two adversaries, we observe that the length of the arc corresponding to the angle—that is, rd?—is equal to the difference of the displacements normal to the joining line in case the sides presented are of opposite names, and to their sum incase the sides presented are of the same name; we have then,
d?/dt = V A sin a ± V B sin ? / r, (10)
wherein the negative sign corresponds to ?’ and the position sign to ?”.v
We observe that there results from (8)
? = V/V A sin a ± V B sin ? r,
*Evidently we may also reach this deduction when referring to the relative speed. When A and B present to fire sides of the same name, the relative speed is greater than in the other case, and is nearer to the direction normal to the joining line AB. (Author's note.)
v The values of ?' deduced from this relation are positive when the ship A sees the line joining it with the enemy inclined in the direction in which the ship is moving; they are negative in the opposite case, which is realized when
VA sin a < VB sin ?
(Author’s note.)
which confirms the preceding results, and demonstrates further that the radius of curvature varies proportionally to the distance. This relation is the one sought and upon which we may base conclusions.
When we say that the changes of course disturb the firing we have reference to the changes with the helm hard over, or with a radius of 400 or 500 meters; this is the limit to which it is necessary to compare the radii of curvature of the tracks.
Referring to the case in which the adversaries present sides of the same name—that is, to the case in which the radii are shorter —it is necessary to establish whether they are great enough not to affect the fire control.
Let us, then, consider the values of ?", laying down the observation that, while the values of ? are different in the case of sides of opposite names and in that of sides of the same name, their ratio, by virtue of (8'), is constant; that is, the relative conditions of the adversaries do not change, and, for the values that the speed ratio may have in practice, these conditions may be held not to differ sensibly.
We will therefore consider the values of ?"B, supposing B to be the slower ship; these are given by (10), making therein V=VB, and taking the positive sign in the denominator. Let us refer to a ratio VA/VB=½ and to a single, r=500 meters; because, VB for the other values of the distance, it is sufficient to take into account the proportionality between p and r. Given the ordinary amplitude of the sectors of maximum offense 45° forward of and 45° abaft the beam), we may limit ourselves to varying a and ? between 45° and 90°, because the values of ?"B that are realized in the other cases enter into those that are deduced with the preceding data between 45° and 90°, because the values of ?"B that are realized in the other cases enter into those that are deduced with the preceding data.
VALUES OF ?"B (METERS) | |||
a/? | 45° | 60° | 90° |
45° | 3210 | 2860 | 2620 |
60° | 2910 | 2620 | 2420 |
90° | 2700 | 2450 | 2270 |
values of ?" are such that steering by sight vane may be adopted, even against an enemy who keeps his course unchanged.
In closing the distance the values of ?" diminish in a way to cause disturbance to the firing, while the values of ?' are always very large; in fact, it is enough to bear in mind that the minimum ?' is 5060 meters, which in practice (under the logical hypothesis that B fires with a maximum number of guns) corresponds to 0=45°and a=90°, for r=250°. Hence, at close range we may steer by sight vane, presenting to fire the side of the name opposite that of the enemy; when presenting the side of the same name we must steer rectilinear courses.
CHAPTER II.
MANEUVERS OF Two SHIPS OPPOSED TO EACH OTHER.
38. Importance of the Study of the Naval Duel.—In this chapter we propose to study the maneuvers of two ships opposed to each other, excluding the case of the, attack upon a battleship by a torpedo boat, which will be discussed hereafter.
First of all, it is well to reflect that there is but scant probability of single combats, considering that the more important a ship is, the less rational will be its isolation. Thus, in order to increase the zone explored by a fleet, we might be induced to extend the battleships also in chain, as well as the cruisers. This might seem to be advisable if our ships possessed a speed superior to that of the enemy's similar ships, so as to be able to effect concentration in good time. But such advisability must be excluded, because the conditions under which the meeting with the enemy will take place can never before seen. Without need of entering into discussions of a strategic character, as an elementary measure of precaution we may establish the general rule that the battleships should cruise together, and that the armored cruisers should be able to rejoin the main body with facility. Only on account of the material and moral superiority that one has in the pursuit which succeeds a victorious battle, can it be permissible to abandon, at least in part, those precautions that are indispensable in the presence of an enemy in full efficiency.
Duels, then, are to be predicted between the lighter ships; nevertheless, it is not expedient to limit ourselves to examining the hypothesis that may be formulated in this respect. The study of combat between two ships furnished with vertical armor, powerfully armed and protected (battleships or armored cruisers), if it is of improbable practical application, is of importance to us because it serves as a starting point for the study of squadron combat. Indeed, in maneuvering a fleet, the ideal at which it is necessary to aim is that of securing an advantageous position with respect to the enemy, minimizing for each ship the hindrances that derive from its association with the others; or, it is necessary that the maneuvers of each ship, with respect to that of the enemy upon which it is directing its fire, should approximate, as far as possible, to the maneuvers that it would make if it were alone.
Thus there results the advisability of studying the naval duel in a general way. We shall suppose that the maneuvering is not hampered by the coast or by other causes, and afterwards we shall allude to some special cases.
39. Distance Kept within Limits and Constant Distance.—It is clear that the maneuvering of our ships in long-range battle must generally satisfy the conditions of keeping the enemy bearing in a sector of maximum offense. Subordinately to this, in whatever way the enemy may maneuver, the maneuvering may be intended either to preserve the distance that one has at the actual moment, or to change it.
Let us now consider the first of these hypotheses. We have recognized (Part I, Chapter IV) the impossibility of assigning a strictly determined value to the distance of maximum utilization; we likewise know (Part 1, Chapter 1) that the directions of maximum utilization are to be considered as elements of the highest importance for tactical maneuvering. Hence, it would seem logical to establish that the maneuvering should be developed by keeping the enemy in directions of maximum utilization alternately forward of and abaft the beam; in this way the conservation of the distance that is deemed favorable for our ship should be understood in the sense of causing it to vary, keeping its variation, however, with in the limits of the zone of maximum utilization. This form of maneuvering, which is called maneuvering with limited distance, cannot be established in an absolute way. In fact, it evidently leads to changes of course that disturb the firing when one passes from the bearing forward of the beam to that abaft the beam or vice versa; and, furthermore, one is obliged to present the beam at such moments; further still, one of the two phases, either that of the bearing forward of the beam or that of the bearing abaft the beam, will be very short, as may easily be seen by calculating (with the fundamental relation) how rapidly the distance varies between two ships that keep each other bearing in the same general direction from the beam.
From this arises the idea of keeping constant some value of the distance included in the zone of maximum utilization, on a suitable polar bearing, whenever this may be possible.*
Evidently, the first condition necessary in order that the distance may be constant, is that of keeping the enemy bearing abaft the beam when he has us bearing forward of the beam, or vice versa.
The necessary relation between the speeds and the polar bearings in this form of movement is that which, in the fundamental relation, makes dr/dt = 0 that is to say:
VA cos a = VB cos ?,
*The French admiral, Fournier, in his book called "La flotte necessaire" (1896), was the first to study the combat with limited distance, putting the question in the following terms: The task of the vessel that wishes to draw profit from an advantage in speed is to maneuver, presenting his side in such a way as always to keep his adversary at the most effective range of his guns, without allowing him to approach within a distance arbitrarily fixed as the limit of safety. He studies the maneuver of the swifter ship on the basis of the following theorem: If two ships having speeds VA and VB (VA>VB) start at a distance r0, and the swifter vessel follows a logarithmic spiral with its pole in the initial position of the slower ship, and inclined to the radii vectors in such a way as to keep the pole bearing at an angle whose cosine is— VB/VA, while the slower ship steers on a radius vector, the distance between the two ships varies until it return store, when the swifter ship passes ahead of the other. Fournier based himself on the hypothesis that the slower party, with the intention of diminishing the distance, might follow a rectilinear course, or keep his bow constantly on the enemy. In consequence of this, and of the fact that the faster ship, which passes over arcs of a logarithmic spiral conformably to the theorem aforesaid, has not the enemy constantly bearing in a sector of maximum offense, the maneuver with limited distance in the way proposed by Fournier is not acceptable. (See our study entitled "La velocita nella tattica navale," in Rivista Marittitna of January, 1900). Nevertheless, Fournier's book efficaciously contributed to the progress of Tactics by initiating the study of long-range battle. Following Fournier came Commander (now Admiral) Baggio-Ducarne, who, studying the application of Fournier's criteria (Rivista Marittima, April, 1897), adjudged to Admiral Saint-Bon the merit of having, in Ms, perceived and demonstrated, in a tactical exercise, the advantage that a ship, swifter and more powerful than another, may draw from long-range battle. Comandante Ronca first pointed out the convenience of keeping the distance constant (Rivista Marittima, June, 1897). (Author's note.)
which we will call the equation of constant distance. It results from this that, theoretically, a ship, A, can maneuver at a constant distance from another ship, B, in two ways: 1st, at a constant speed, pre-establishing VA and determining the polar bearing a on which he must keep the enemy by means of the equation
cos a=VB/VA cos ?,
so that a is constant if B keeps ? and VB constant; 2d, on a constant bearing, pre-establishing the polar bearing in which the enemy must be kept, and assuming the speed
VA=VB cos ? / cos a,
so that VA is constant if B keeps ? and VB constant.
Each of these methods presents grave inconveniences. Brief, considerations suffice to show that, in general, with one of the above-mentioned methods, if the maneuvering of the enemy is not rational, our maneuvering also cannot be the most opportune.
With the method at a constant speed, if B keeps A on a variable bearing, ? (as may happen, for example, when B keeps the course constant), a must also be variable. If ?=90°, a must also be 90°; corresponding to ?=0 and ?=180°, we have, respectively, cos a VB/VA and cos a=-VB/VA. Consequently, if the enemy should constantly present his beam, in applying this method our ship should act in the same manner, which is illogical. If the enemy has our ship bearing in line with the keel or in a sector of minimum offense, with the values that the speed ratio may assume in practice, our polar bearing also would generally be outside of a sector of maximum offense. Moreover, if VA>VB cos a<cos ?; and hence, supposing B to maneuver rationally and keep A in a direction of maximum utilization, the ship A, in order to keep the distance constant and develop the maximum speed, must bring the enemy to bear nearer to the beam, or, in a direction that may be a less defensive one.
With the method on a constant bearing it might be necessary to vary the speed between very wide limits; that is to say, from the value VB/cos a to the value zero corresponding to ?=90.
Neither of these forms of maneuvering can, in an absolute way, be held to be acceptable, as has already been said concerning that with the limited distance; but, instead, it will be easy to see how the three methods, considered together, permit of formulating practical rules of great simplicity.
We may reach such rules with the aid of the following observations:
I. It would be absurd to vary the speed in the way that might be required by the method on a constant bearing, but limited variations of the speed, from the maximum to a speed inferior by four or five knots, are acceptable.
II. Analogously, it may be admitted that our sight vane may be moved from a direction of maximum utilization as much as 15° or 20°, approaching nearer to the beam, or reaching an extreme limit of the sectors of maximum offense.
III. When the enemy keeps our ship bearing in a direction 'very near the beam, the variation of the speed and of the bearing within the limits above mentioned is not sufficient to keep the distance constant; maneuvering with a limited distance is then rendered necessary, and we need not be concerned about having to present the beam at intervals, because the enemy is continually in that disadvantageous condition.
IV. When the enemy, for reasons that we shall specify, can control the variation of the distance and avails himself of that faculty, it is evidently necessary to develop the maximum speed and steer with the sight vane on an extreme limit of the sectors of maximum offense, thus hindering the maneuvering of the enemy as much as possible. This being said, the rules for the practical application of the criteria alluded to are evident.
40. Maneuvering in Long-Range Battle.—Our ship is, by hypothesis, within the limits of distance that are held to be advisable. We have the usual disk provided with a sight vane. On it, PP' (Fig. 22) indicates the direction of the keel; OS and OS' are the limit directions of the sector of maximum offense SOS'.
Let us suppose that the enemy has our ship bearing forward of the beam. We steer on a constant bearing, arranging the sight vane in the direction OM, which is the after direction of maximum utilization, and we develop the maximum speed. Three cases may present themselves; that is to say, the distance may remain constant, or it may increase, or it may diminish.
In the first case we have but to continue to steer with the sight vane in the direction OM.
In the second case, that is, when the distance increases, continuing to steer with the sight vane on OM, we reduce the speed little by little, and it is possible that, within the limits established for the diminution (four or five knots), we may find a speed that will keep the distance constant. When this is not realized, even for the said inferior limit of the speed, we gradually move the sight vane nearer to the beam, without, however, going beyond
{figure}
FIG.22.
the pre-established limit OM'. If, even with this limit direction, the distance still increases, we continue to steer with the sight vane in the direction OM' and, before arriving at the superior limit of the distance, we change our course, steering at the minimum speed with the sight vane in the direction OM" symmetrical with OM'.
In the third case, that is to say, when the distance diminishes, keeping to the maximum speed, we move the sight vane gradually toward OS'. This limit being reached, if the distance continues to diminish, we may not move the sight vane further toward the stern, because that would be without the sector of maximum offense. Evidently the enemy is master of the maneuvering, and we must continue to steer with the sight vane in the aforesaid direction in order to minimize the variation of the distance. We should maneuver in a perfectly analogous manner if the enemy had our ship bearing abaft the beam.
41. Rotation of the Line Joining the Adversaries.—It results from the foregoing that, with two adversaries that maneuver rationally, besides being able to presume that they may steer maintaining constant polar bearings, the typical case to be con-
{figure}
FIG. 23
sidered is that wherein, for a certain time, the distance remains about constant.
It is easily demonstrated that, if two ships steer keeping constant polar bearings, and if the distance also remains constant, the tracks followed by the two adversaries are concentric circumferences. In fact, by equation (II) of Chapter I (Section 37) the radii of curvature of the tracks described by the ships A and B in the case we are now considering are constant, and hence the tracks are circumferences; moreover, the normals to the tracks always intersect in a point 0 (Fig. 23), by which we have
OA/OB = cos ?/cos a,
and hence, from the equation of constant distance, there results
OA/OB=VA/VB,
and O is then the common center.
Fig. 24 shows the circumferences passed over by ships sup-
posed to be initially at A and B; the centers O’ and O” of the circumferences that the ships follow in presenting sides of the same name or sides of opposite names are naturally found on the perpendicular to the initial course of the ship B, which is supposed to have A always bearing to port. Conformably to
{figure}
FIG. 24.
what has been demonstrated in a general way in Section 37, the radii of the circumferences with their center in O” are shorter than those of the circumferences that have their center in O’; and hence, in the movement with sides of the same name, the rotation of the line joining the adversaries is more rapid; which is important in relation to what is said in Section 21 concerning the natural elements.*
*The demonstration that the tracks of A and B in the case under consideration are concentric circumferences can also be made without having recourse to the aforesaid formula (11). In fact, if the center of curvature of the tracks of A and B were not coincident and fixed, they would be, for example, at O’A and O’B on the respective normals to the tracks; and by virtue of equation (8') of Chapter I, and of the equation of constant distance, the line O’AO’B should at all times be parallel to the joining line AB. But, considering the positions A' and B' of the two ships after the time dt, O’AO’B should also be parallel to A'B', and hence we should fall into an absurdity. In order to avoid this we must admit the coincidence above mentioned. (Author's note.)
The expediency of presenting to the enemy the proper side in order to keep or to secure an advantageous position with respect to the sun, or to the coast, or to the strategic objectives, is clear; therefore it is important to fix the ideas concerning the rapidity of rotation of the joining line.
The speeds, the polar bearings and the distance being constant, it results from equation (10) of Chapter I that the speed of rotation of the line joining the adversaries is constant. With the said formula we may calculate the time required for the joining line to rotate through a given angle.
We note that, if the speeds of A and B are equal, or, if a=?, the point 0' is at infinity, and the tracks of the two ships are parallel straight lines; while the circumferences with their center at O” are reduced to one only, since ?A=?B. This being the case it is clear that, applying the type of maneuvering indicated in the preceding section, conditions near to these will be realized when the directions of maximum utilization of the ships are not inclined in a sensibly different way. We may conclude from this that, in general, when the adversaries present to each other sides of opposite names, the rotation of their joining line may be held to be very slow.
When, on the other hand, the two adversaries present to each other sides of the same name, the rotation of the joining line is very rapid; thus, supposing A and B to be distant from each other 9000 meters, that they have each other bearing at 45° from the beam, and that VA=VB=15 knots, the time necessary to rotate the joining line through 90° is only 23 minutes. However, for the case in which, by presenting to the fire the side of the same name as that of the enemy, we may expose ourselves to having a rotation of the joining line in a direction contrary to that desired, let us see how the rotation can be obtained by presenting the side of the opposite name.
It is clear that, in order to have the maximum component of the speed normal to the joining line, it is necessary to maintain the maximum speed instead of reducing it according to the criterion given in the preceding section; and it is expedient to have the enemy bearing as near as possible to the beam, or, in one of the pre-established limit directions OM', OM", of Fig. 22. If with the sight vane in one of these directions the distance changes, we continue to steer in this way up to the limit established for the distance, and then steer in the symmetrical direction with respect to the beam. So doing, by virtue of the aforementioned formula (10), the ship A sees the line joining it with the enemy incline itself in the direction in which the ship is moving, if there is realized the condition
VA sin a > VB sin ?.
or a little more than half the time above mentioned. Hence, for rotations of great amplitude, avoiding the inconveniences of keeping the enemy bearing too near the beam and gaining in celerity, it would be advisable to produce the complement in 360° of the required rotation.
We may then establish:
1st. In order to maintain an advantageous position with respect to the sun, to the coast, or to the strategic objectives, we must present to fire the side of a name opposite that of the enemy, applying the general type of maneuvering indicated in the pre- ceding section.
2d. We must present to fire the side of the same name as that of the enemy, applying the general type of maneuvering, when in that way the rotation of the joining line takes place in the desired direction, or when the amplitude of the desired rotation is very great.
3d. If in presenting the side of the same name as that of the enemy the rotation takes place in the direction contrary to that desired, if, furthermore, the desired rotation is of small amplitude and we have a speed superior to that of the enemy, it is well to present the, side of the opposite name, maneuvering by the rules just pointed out. When, however, we have not sufficient speed for obtaining the object established, it is necessary to maneuver as has been said in the preceding case.
It results from the foregoing that it may be easier to maintain an advantageous position than to acquire it. In general we can have but little faith in being able to apply the second of the preceding deductions, for it is presumable that the enemy, when he sees our ship present the side of the same name as his own, may change the maneuver by exposing his other side if he deems it to his interest to avoid a rapid rotation of the joining line. Still, the contrary might happen when, for example, the enemy has commenced firing before we have done so, and does not wish to change the side, in order to avoid the disadvantage to the fire control that would be produced.
Evidently it would be a fine game if the enemy were obliged to present a determined side, as would be realized if his offensive field were unsymmetrical with respect to the longitudinal axis.
42. Change of Distance.—I. It is clear that in order to obtain a change of distance without sacrificing offensive power, a ship A must develop the maximum speed and keep the enemy bearing at the forward limit of a sector of maximum offense if he desires to diminish the distance, or at the after limit if he desires to increase it. The enemy B, in order to maneuver by the same standard, hindering the change, must keep A bearing respectively at the after limit, or at the forward limit of a sector of maximum offense.
By the fundamental relation, the ship A may impose a change of the distance when
VA cos a>VB cos ?;
a and ? corresponding in this case to the limits of the sectors of maximum offense.
II. Let us now suppose that the two adversaries are at the limit of offensive contact. The object of the ship A is to engage in a decisive combat with the ship B, which, not being able to prevent the approach of A, proposes to limit the action to the maximum distances. It is easy to see that if the ship B is powerfully armed on the line of the keel (has powerful head and stern fire), it may be interested in keeping the stern always toward the enemy. In fact, if A, steering by sight vane, keeps B at the forward limit of a sector of maximum offense, calling this bearing a—counted, as usual, from the bow—by the fundamental relation (wherein we must, in this case, put ?=0) it results that the necessary condition to the end that the distance may not increase, is
VA/VB≤ 1/cos a.
Consequently, the minimum speed necessary for A, in order to develop against the enemy the maximum power without falling out of range, is that corresponding to the greatest amplitude of the sectors of maximum offense; or it is determined by the relation
VA/VB= 1/cos 30° = 1.15.
Putting ?= VA-VB, we have
VA/VB =1+ ?/VB
or
?=0.15VB.
Hence it results that the superiority in speed required by A is so much the greater, the greater is VB; or, while a greater speed per hour of 1.5 knots is necessary for the object mentioned against a ship of 10 knots speed, when V=20 knots, there is required an advantage of 3 knots.
This said, the following observation should be made:
1st. The aforesaid advantage in speed simply permits the ship A to keep B under the fire of his guns, but does not admit of his diminishing the distance.
2d. As has been pointed out in Part I, Chapter I, some ships can develop a strong intensity of fire in the direction of the keel— much greater than that of which they are capable in the sectors of minimum offense; moreover, it is well to remember (Section 3) that, at the maximum fighting distances, a ship presenting itself end on to the enemy's fire diminishes in that way the enemy's percentage of effective hits.
3d. It is true that, being removed by a small angle from the direction of the keel, one enters a sector of minimum offense; but this may also happen for the limit direction of the sector of maximum offense on which A is steering; in other words, if A desires to provide against the inexactness of steering by sight vane, rather than at 300 from the bow, it ought to keep the enemy at about 350; but then, for V=20 knots, there would be required an advantage in speed of 4.4 knots, rather than of 3 knots. It seems, then, that we may affirm that, even supposing the sectors of maximum offense to have the maximum amplitude, without a great advantage in speed, not presumable in practice, the situation is not very decidedly favorable for A. On the other hand it appears to be clear that B may not be interested, in this case, in conforming to the general rule, which is that of keeping the enemy bearing in a sector of maximum offense.
If the ship A, then, has not a speed which greatly exceeds that of B, his desire to succeed in the intention to diminish the distance constrains him to keep his bow toward the enemy; which obliges him to endure, for a very considerable time, a disadvantageous situation, when the strength of his fire in line with the keel is inferior to that of the enemy. In fact, it is sufficient to note that, in order to diminish the distance by at least 2000 meters—which it is important for A to do in order to engage in a quickly efficacious action—the necessary time is that which would be required for passing over the said space at a speed VB; or, it is more than 20 minutes if the difference in speed is about 3 knots.
When, however, the ship A, besides being the swifter, has also a more powerful fire in line with the keel, it is clear that B will be obliged to bring it to bear in a sector of maximum offense; we then come back to the case already discussed at the beginning of this section.
In the general case of two ships opposed to each other on the open sea, we may then conclude:
1st. That, for declining a decisive tactical action, powerful stern fire may constitute a compensation for inferior speed.
2d. That a speed superior by two or three knots, even if associated with the maximum amplitude of the sectors of maximum offense, cannot be held to be sufficient for imposing the tactical action; powerful fire ahead is more important for securing that object.
43. Capacity for Tactical Initiative.—A ship has complete liberty of tactical initiative with respect to another if it can
V cos (a-?)=(V+kV) cos a;
hence we have
k=cos (a-?)/cos a – 1.
If we were considering the after limit direction of the sector of maximum offense it would be necessary to put 180°—a in place of a, and 180°—a+? instead of a—?; hence the formula found for k is a general one.
In the three hypotheses (Section 2) of sectors of maximum offense extended to 30°, 45°, and 60° from the beam, the values of k for ?=5° are respectively 0.14, 0.08, 0.04. For ?=10°the values of k just given are doubled; that is, k may be held to be proportional to ix. For the ordinary case in which the two ships have sectors of maximum offense with amplitude near to 45° forward of and abaft the beam, it is then to be remembered that if a ship has, with respect to another, the limits of the sectors of maximum offense nearer to the longitudinal axis, every 50 of difference of this kind, or every 10° of advantage in the total amplitude of the sectors, is equivalent, in the particular regard now under discussion, to an increase of speed of 8/10 of a knot for every 10 knots.
IV. Generally the maneuvering of a ship in the duel is developed by keeping the adversary in one of the four fractions of the sectors of maximum offense; as, SOM", S'OM' of Fig. 22. included between the limit directions of said sectors and the directions nearest the beam to which corresponds a sufficient defensive capacity; the wider these partial sectors are, the greater is the liberty of maneuvering. From this comes the superiority that is derived from greater thicknesses of armor than those of the enemy, or from guns with greater penetrating power.
V. To the end that a ship may be able to impose the rotation of the line joining it with the enemy, it must be able to obtain, with respect to the enemy, a greater component of its speed normally to the said joining line. This depends upon the relative speed conditions and upon the angles that, for each ship, the directions OM", OM' of Fig. 22 form with the beam. In this particular respect an advantage in speed is hence equivalent to a greater thickness of armor or to a greater penetrating power of the guns.
From the preceding observations it results that the importance of the various elements of the capacity for tactical initiative cannot be considered in a one-sided manner.
It is incontrovertible, then, that the said capacity increases with the amplitude of the sectors of maximum offense; but with this amplitude (Section 3), the ratio between the potentiality in line with the keel and that of the said sectors diminishes; nor, on the other hand, can an increase of the said amplitude render an advantage in speed less desirable. This deduction is to be borne in mind when considering the types of ships.
44. Maneuvering at Close Range.—Let us examine the manner in which maneuvering at close range may be developed; that is to say, within the limits of the fourth tactical zone distinguished in Section 24, when the employment of the torpedo is possible, and hence, from what we demonstrated in Section 9, it is preferable to have the enemy bearing abaft the beam.
It must be borne in mind (Section 3) that the probability of being hit by the guns at close range, while it remains constant if the direction in which the enemy bears is not removed more than 45° from the beam, rapidly increases beyond that limit; therefore it is not advisable to have the enemy bear more than 45° from the beam, or, it is not advisable to utilize a greater amplitude of the sectors of maximum offense.
In maneuvering at close range, then, we should generally have the enemy bearing in a direction of maximum utilization abaft the beam, and should present to fire the side of a name opposite that of the enemy; because in that way it is possible to steer by sight vane (Section 37), which permits of satisfying the tactical necessities in a continuous manner.
If the enemy has our ship bearing forward of the beam, it is well to develop the maximum speed or a speed somewhat inferior —as has been said for maneuvering at long range—with the intention of keeping the enemy at the desired distance. For some moments we may change the angle of the sight vane when this may be necessary in order to launch the torpedoes; we must not, however, forget the risks we run in presenting the beam at close range; and hence it is necessary to establish it as a rule, not to execute, within the limits to which we are referring, any passage from a bearing abaft the beam to one forward of the beam, or vice versa. In consequence of this, except in the case to which we shall now allude, the launching tubes of the forward sectors should be utilized by launching for an angled run.
It is clear that if the enemy maneuvers analogously, keeping our ship bearing abaft the beam, neither of the combatants is within the radius of action of the torpedo; the duel then remains an artillery battle exclusively.
Let us consider the situation of two adversaries that, at close range, have each other bearing forward of the beam.
The distance diminishes; the ship that, at a certain moment, in order to avoid a further diminution of the distance, brings the enemy to bear abaft the beam, while the enemy continues to keep it forward of the beam, presents the beam during the change, and hence makes—as has been said before—a perilous maneuver.
If neither of the adversaries makes such a maneuver, and if they present to fire sides of opposite names, they come to close quarters.
When the sides exposed to fire are of the same name, if one of the adversaries desires to fight at close quarters, the other cannot avoid it; let us suppose that this is not desired by either of the combatants. They keep the course constant; and, if the courses are parallel, the polar bearings are the same at any moment. Thus, passing on opposite courses, in as much as it leads the two adversaries to present the beam simultaneously, may seem an opportune form of maneuvering for two combatants, both desirous of using the torpedo. However, it is to be noted that only in appearance are the combatants in identical conditions; the more strongly armored ship acquires the advantage, owing to the fact that the adversary abandons maneuvering on the bearing which would permit him to compensate for, or at least diminish, the greater vulnerability of his armor.
Under the hypothesis that the enemy has our ship bearing forward of the beam, we may then conclude:
1st. It is not necessary to have the enemy bearing forward of the beam at close range, unless we intend to provoke a battle at close quarters, or unless we have decided to undergo it.
2d. Not desiring battle at close quarters, the maneuver of having the enemy forward of the beam may be rational when our ship has side armor of greater thickness.
45. Limit of Battle at Close Quarters.—Below a certain distance from the enemy, a ship may be assured that, if a collision takes place, it may run in to the other, instead of being itself run into.
Theoretically, when we have the greater speed, we run no risk in having abaft the beam an enemy who has us forward of the beam; but it is necessary to take unforeseen elements into account, and hence the necessity of not having the enemy abaft the beam below a certain distance, determined under the hypothesis of equal speeds and equal evolutionary qualities for the two combatants. This distance marks the limit of battle at close quarters.
Let us consider two ships, A and B (Fig. 25), that have each other bearing respectively the one astern and the other ahead, and that have the same speed, V. Let us hold their evolutionary curves to be circular. We must seek the maximum distance, AB, which will permit A to arrive at K' at the same time as B.
Let us admit that the arcs BK and MK' are passed over in the same time—which is not rigorously true, since one is described
{figure}
FIG. 25.
at the beginning, and the other at the end of an evolution. KK' must be passed over in the time that A takes in passing over the semi-circumference ANM.
Indicating by ? the radius of the circle of evolution, since A loses about one-third of his speed, we have
KK’/V =??/2/3V,
KK’=3/2??.
Since AB=2CO’=2√O’K’2+CK’2=2√?2+1/4KK’2, substituting for KK’ its value, we obtain
AB=?√4+9/4?2=5? (about).
On the basis of the values that we may attribute top, and considering that the value found for AB refers to the extreme case wherein A has the enemy bearing exactly astern, we may hold the distance of 2000 meters to be the limit of battle at close quarters.
46. Maneuvering at Close Quarters.—Battle at close quarters has lost the importance that it had in the past when it was held to be the principal form of action; for this reason we shall confine ourselves to brief considerations.
The ramming maneuver against a ship with freedom of movement is extremely hazardous. Indeed, the instant at which a ship must reach the intersection of its track with that of the enemy in order to obtain the object of ramming him, differs very little from that at which the said ship would be itself rammed. For this reason, and owing to the fact that modern ships have in the gun and in the torpedo most powerful means of fighting, it is to be admitted that neither of the adversaries is likely to maneuver with ramming as the principal objective.
In maneuvering at close quarters we must nevertheless take in to account the possibilities of ramming; that is to say, we must maneuver so as to avoid being rammed. In other words, it is necessary to maneuver defensively with respect to ramming.
The offensive maneuver for ramming would require having the bow directly toward the enemy; for this reason, when the tactics of the ram were a subject of study, it was admitted that the two combatants would have steered initially bow against bow; but, afterwards, considering that such a direct clash would have placed both the adversaries in identically disadvantageous conditions (no matter how different might be their structures), it was held by the majority of the authors that the two ships would have turned aside little by little. As a collision at a small angle is dangerous for the rammer as well as for the rammed, it was also admitted that the two ships, in order to avoid it, would both have turned a side with the same helm, thus changing the collision into a grazing, or slipping past at short distance; after which they would have executed offensive turns with intent to ram.
For the defensive maneuver with respect to ramming, it is not necessary—as it is for the offensive maneuver—to tend to keep the enemy at an angle from the bow less than that at which he keeps us, but it is sufficient that the two bearings be equal. On the basis of this consideration, the form that it seems the initial phase of battle at close quarters between two modern battleships should assume, is not exactly that which appeared probable in the tactics of the ram. Within the limit of distance indicated in the preceding section, the two ships will run to meet each other; but, instead of steering bow to bow, they will keep on their guard, following parallel courses; there will thus be, not a grazing by, nor a passing at very short distance, but a passing at a distance of some hundreds of meters, such as to permit making use of the guns and the torpedoes.
After passing by, and before the enemy gets out of the after sector of maximum offense, it is evidently logical to turn in order to keep him in that sector; therefore the two ships will turn at that time toward each other; and afterwards, for the same reason, they will change course again, thus following parallel broken straight lines, having each other bearing abaft the beam; and hence, they will draw away from each other. Given—but not conceded—that the passing by may not have had the gravest effects, if one of the ships desires to provoke a repetition of the preceding phase, it will have to turn toward the enemy in order to bring him to bear about 450 forward of the beam; the other ship will be obliged to turn also in the same way.
It is to be noted that while admitting it to be unlikely that one of the ships will maneuver, after the passing by, in order to ram, the other—unless his mobile and evolutionary qualities are greatly inferior—by turning with the helm hard over, will be in time to present his bow to the adversary. Hence we deem rational the opinion of to-day favoring the abolition of the ram.
47. Particular Cases.—In practice—as was said at the beginning of this chapter—particular circumstances will determine the form that the duel will assume.
Ordinarily one of the adversaries will be sensibly weaker than the other, and will seek to reach a movable or a fixed center of protection. Hence it is possible that the enemy may or may not be found between the center of protection and the ship that desires to decline battle.
In the second case the weaker ship will evidently have to run; and will place itself in safety if it has an advantage in speed, or if its inferiority in speed is not too great, and if the radius of action in which the said ship must operate has been conveniently established, taking into account the speed of the enemy's strongest ships.
In case, however, it is indispensable to reckon with the enemy in order directly to reach the center of protection, the weaker ship could run away if it had an advantage in speed, and seek to throw the enemy off its track; but when it is feared that in this way the ship may run into preponderating hostile forces, it will steer directly for the center of protection, thus coming into a fight at close quarters with the enemy.
This is the case in which encounters between torpedo boats very often take place, especially during blockading operations, as has been demonstrated in the Russo-Japanese war. The destroyers and torpedo boats of the defense, having come out during the night in search of the enemy's battleships, returning at daylight toward their base, may encounter on their route similar units of the enemy; and as, for the said vessels, running to seaward would signify exposing themselves to certain loss, they will be obliged to fight. The adversaries will steer for each other bow to bow, so that the fight will first be developed in line with the keel, and afterwards, by passing each other on opposite courses, at very close quarters. After this, the units of the blockading party, unless they have sustained injuries that impede their freedom of maneuvering, will invert the course, following the enemy to the vicinity of the base.
Between protected cruisers, when the weaker is the slower, and has not a potentiality of fire in line with the keel that is sufficient for maneuvering by the rule given in Section 42, or when for any reason it cannot avoid a decisive action, it will seek to escape from the bad situation by provoking battle at close range. The stronger ship will evidently be interested in keeping the enemy at a distance.