VM 161 U58 1902 BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Hcttrg M. Sage 1S91 r—r Cornell University Library VM161 .U58 1902 The oscillations of ships 3 1924 030 901 726 olin Cornell University Library The original of tiiis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030901726 THE OSCILLATIONS OF SHIPS Compiled and Edited by the DEPARTMENT OF MARINE ENGINEERING AND NAVAL CONSTRUCTION, UNITED STATES NAVAL ACADEMY, FOR THE "^ INSTRUCTION OF THE MIDSHIPMEN Published by the U. S. NAVAL INSTITUTE, ANNAPOLIS, MD. 1902 THE FRIEDENWALD CO, BALTIMORE, MD. PREFACE The course in Naval Construction given at the Xaval Academy is planned to include in one term's work (1) a theoretical study of ship calculations and conditions of equilibrium and stability, and (2) a more extensive study of the details of practical ship- building with special reference to the construction of various types of naval vessels, shipyard processes, modes of fitting and connecting various parts, etc. Any study of the first division which does not include an investigation of the statical and dynamical elements and conditions contributing to the behavior of ships at sea among waves, of which a theoretical knowledge at least is an essential part of the proper education of a naval officer, is lacking in completeness. The time allotted for the course in Naval Construction is so limited, considering the ground to be covered, that, with the most economical arrangement of lessons, it has been necessary to confine that part dealing with the oscil- lations of ships to four lessons at most. "While this subject is most admirably treated in White's " Man- ual of Naval Architecture," which has been used as a text-book in this institution, as a whole it is not well adapted to purposes of instruction under the limitation of time imposed. Previously, certain parts only have been selected from this work to meet the requirements; but this has proved very unsatisfactory, being not only inconvenient but necessarily wasteful of time due to the impossibility of selecting paragraphs giving brief explanations of principles and results of experience, rather than exhausfive discussions of scientific investigations and conclusions, without 4 Peeface taking much more matter than is really wanted and imposing on the student lessons of almost prohibitive length in order to cover the ground. Hence this little book in which the material has been condensed as much as possible and arranged to better suit the requirements mentioned. These notes have been compiled and arranged by Lieutenant E. C. Moody, U. S. F., instructor in Naval Construction and Engineering. John K. Bahton, Lieutenant-Commander, U. S. N. Head of Department Marine Engineering and Naval Construction. U. S. Naval Academy, Annapolis, Md., October, 1902. CONTENTS CHAPTER I. Unresistkd Rolling in Still Water. PAGE. 1. Introductory T 3. Parallelism between motions of a ship and a simple pendulum 8 3. Length of simple pendulum with which oscillations synchronize. ... 8 4. xVngular velocity, accelerating force, and moment of inertia 9 .5. Position of the instantaneous axis of rotation 10 6. The agents producing rolling 11 7. Isochronous oscillations 12 8. Determination of the period 13 9. Changes in period produced by changes in distribution of weights. . . 13 CHAPTER II. Influence of Resistance on Rolling in Still Water. 10. Character of fluid resistance 15 11. Estimation of the resistance 1.5 12. Rolling experiments — how made 16 13. Observations made during rolling experiments 17 14. Curves of extinction 17 1.5. Rolling experiments with models 18 16. Periods of various classes of war vessels 19 17 Dynamical stability corresponding to decrease of range 19 18. The value of total resistance 31 19. Surface disturbance 21 30. Value of bilge-keels 22 21. Number and arrangement of bilge-keels 33 33. Effect of emersion of bilge-keels 33 33. Size of bilge-keels 34 34. Effect of " water chambers " 34 3.5. Thornycrof t's automatic steadying apparatus 35 36. Influence of free water in the hold 36 CHAPTER III. Deep-Sea Waves. 37. Fundamental conditions of trochoidal theory 37 35. Construction of trochoidal profiles 38 29. Orbital motion of particles and advance of wave form 30 30. Internal structure of wave 31 6 Contents PAOE. 31. Decrease of downward disturbance 33 32. Height of centres of tracing circles above still-water level 34 33. Displacement of vertical columns of particles 34 34. Variation of direction and intensity of iluid pressure 34 3.5. Maximum slope of wave 36 36. Extremes of intensity of fluid pressure 36 37. Motions of a loaded pole 37 38. Formulae for dimensions and speed of waves 37 39. Observations of lengths of waves 38 40. Observations of heights of waves 40 41. Cause of unintentional exaggeration in the estimate of wave heights. 41 CHAPTER IV. This Oscillations op Ships Among Waves. 42. Character of oscillations in a seaway 43 43. "Virtual upright " and "effective wave slope " 44 44. Virtual weight and buoyancy 44 45. Effect of waves in producing rolling 46 46. Importance of ratio of period of rolling to half-period of wave .... 46 47. Fundamental assumptions for mathematical investigation of unre- sisted rolling 48 48. Rolling amongst waves of synchronizing periods 49 49. Variation of period of rolling due to increase of arc of oscillation. 50 50. "Permanent " oscillations 51 51. " Phases " of oscillation 53 52. Practical conclusions on unresisted rolling 54 53. Effect of fluid resistance on rolling among waves 55 54. Influence of resistance in the critical case of synchronism 55 55. Influence of resistance in the case of "permanent" oscillations. . 56 56. Steadying influence of bilge-keels 56 57. Pitching and 'scending .57 58. " Heavine " oscillations 58 59. Observations of rolling and pitching of ships ,59 60. Errors of pendulum observations in still-water rolling 60 61. Errors of pendulum observations when rolling among waves 61 62. Observations of rolling by use of battens 62 ERRATA. Page 12, line 15, second is to read in. Page 13, line 4, 554a/ ^ /o reafZ .554a/ Pcu/e 29, Zme 14, /S, to read Sc,. Page .30, line 17, silri'^e out first in. Pa^e 33, ?t«e 27, f to read f . -P«^e 34, line 29, insert a behveen in awtZ wave. luuucis in ffrat wauer, lu wiiaL reBpeciB me uuiii;! asiuns amvcu at in the first case are modified by the operation of resistance; and third to see in what manner the oscillations are affected by the presence of waves. The principal oscillations of ships take place in the transverse and fore-and-aft directions; in the former case the motion is called rolling, and in the latter pitching or 'scending according as the bow of the ship moves downward and the stern upward, or vice versa. Of these oscillations rolling is much the more im- portant, and will alone be dealt with for the present. The extent of the rolling motion is measured by the inclina- tions which the ship reaches from the upright, and in swinging from her extreme inclination on one side of the vertical to her next extreme inclination on the opposite side, that is, a single swing from starboard to port, or vice versa, she is said to perform an oscillation. The arc of oscillation is the sum of the angles on either side of the vertical swept through by the ship in an oscil- lation, and her period is the time occupied in performing one oscillation. THE OSCILLATIONS OF SHIPS CHAPTER I. Unkesisted Rolling in Still Watek. 1. Introductory. — The main object in studying this branch of Naval Architecture is to ascertain what elements govern or afEect the rolling motions of a vessel in a seaway, so that her probable safety at sea may be secured in her design. But in approaching this question, although it is evident at once that a ship will ex- perience resistance when rolling through the water, it is conve- nient to consider first, the purely hypothetical case of a vessel rolling unresistedly in still water, where the only two forces acting upon her are the equal ones due to her weight and buoyancy; second, to ascertain by means of experiments on actual ships or models in still water, in what respects the conclusions arrived at in the first case are modified by the operation of resistance; and third to see in what manner the oscillations are affected by the presence of waves. The principal oscillations of ships take place in the transverse and fore-and-aft directions; in the former case the motion is called rolling, and in the latter pitching or 'scending according as the bow of the ship moves downward and the stern upward, or vice versa. Of these oscillations rolling is much the more im- portant, and will alone be dealt with for the present. The extent of the rolling motion is measured by the inclina- tions which the ship reaches from the upright, and in swinging from her extreme inclination on one side of the vertical to her next extreme inclination on the opposite side, that is, a single swing from starboard to port, or vice versa, she is said to perform an oscillation. The arc of oscillation is the sum of the angles on either side of the vertical swept through by the ship in an oscil- lation, and her period is the time occupied in performing one oscillation. S The Oscillations of Ships 2. Parallelism between motions of a ship and a simple pendu- lum. — There is an obvious parallelism between the motion of a ship %et rolling in still water and that of a simple pendulum swinging in a resisting medium. Apart from the influence of re- sistance, both ship and pendulum would continue to swing from the initial angle of inclination on one side of the vertical to an equal inclination on the other side; and the rate of extinction of the oscillations in both depends on the resistance, the magnitude of which depends on several causes to be mentioned hereafter. 3. Length of simple pendulum with which oscillations syn- chronize. — Supposing the rolling of a ship in still water to be unresisted, it may be asked, what is the length of the simple pendulum v\-ith which her oscillations keep time, or synchronize? It has been sometimes assumed that a comparison between a ship held in an inclined position and a pendulum of which the length is equal to the distance between the centre of gravity and the metacentre held at an equal inclination will remain good when the ship and pendulum are oscillating. In fact, it has been supposed that the whole of the weight may be concentrated at the centre of gravity while the metacentre is the point of sus- pension for the ship in motion as well as for the ship at rest; but this is obviously an error. If it were true, the stifEest ships, having the greatest metacentric heights, should be the slowest swinging ships; but all experience shows the direct opposite to be true. The error of this assumption may be simply illustrated by means of a bar pendulum of uniform section siispended at one end, as in Fig. 1, having its centre of gravity at the middle of its length. To hold this pendulum at rest at any inclination to the vertical must require a force exactly equal to that required to hold at the same inclination a simple pendulum of half the length of, and of equal weight to, the bar pendulum. This simple pendulum, considered as having all its weight concentrated at one point (the '•' bob "), which becomes the " centre of oscilla- tion," and supposed to be hung from the point of suspension by a weightless rod, if set ^winging would be found to move much The Oscillations op Ships faster than the bar pendulum; because, where in the bar pendu- lum its mass is not concentrated flt any particular point (though at rest its weight acts through its centre of gravity), but is distributed uniformly along its whole length, the centre of oscillation" does not coincide with its centre of gravity • but lies in a point two-thirds of the length of the bar distant from the point of suspen- sion equal to the radius of giTation of the bar about its end. 4. Angular velocity, accelerating force, and moment of inertia. — Consider the bar as made up of a number of heavy particles, and take each separately. For example, take a particle of weight w at a distance x from the ^^s- 1- point of suspension (axis of rotation); the moment of the acceler- ating force upon it is given by the expression — 'LO X 3/" Moment = X rate of change of angular velocity. At any instant the rate of change of angular velocity is the same for all particles in the bar-pendulum, whatever may be their dis- tances from the point of suspension; whence it follows that for the whole of the particles in the bar-pendulum — ■ Moment of accelerating ] forces at any instant | — X weight of bar X ^^ X rate of ^ change of angular velocity. To determine fc^, we have only to sum up all the products wy^x^ for every particle and divide the sum by the total weight of the bar. Or, using - as the sign of summation — 2' (jya;'") "^ weight of bar " Turning to the case of a rigid body like a ship, oscillating about a longitudinal axis which may be assumed to pass through the centre of gravity it is necessary to proceed similarly, finding V- for the ship when turning about the assumed axis. If the 10 The Oscillations of Ships whole weight were concentrated at the distance /.; from the axis of rotation, the moment of the accelerating forces and the mo- ment of inertia would then he the same as the aggregate moment of the accelerating forces acting upon each particle of lading and structure in its proper place. It will he obvious from this ex- planation why we cannot assume that a ship in motion resembles a simple pendulum suspended by the metacentre and having all the accelerating forces acting through the centre of gravity. These accelerating forces developed during motion constitute, in fact, a new feature in the problem, not requiring consideration when there is no motion. For a position of rest it is only neces- sary to determine the statical moment, but for motion there is the further necessity of considering the moment of inertia as well as the statical moment. 5. Position, of the instantaneous axis of rotation. — A ship roll- ing in still water does not oscillate about a fixed axis correspond- ing to the point of suspension in one end of the bar-pendulum; but if we may imagine in the swinging of the bar-pendulum that the end before taken as the point of suspension swings about a fixed axis of rotation through the original centre of oscillation, instead of the contrarj', preserving the same relative motions of these two points, we will see more clearly the similarity between the motion of the pendulum and that of a ship oscillating about a longitudinal axis passing approximately through her centre of gravity. The position of the instantaneous axis about which a ship is turning at any moment, supposing her motion to be unresisted, and the displacement to remain constant during the motion, may be determined graphically, considering only the simultaneous motions of the "centre of flotation" and the centre of gravity. Let us imagine the ship to be supported at the bow and stern by projecting trunnions having perfectly smooth surfaces and their cross-sections of the same form as the "curve of flotation" and resting upon a rigid, perfectly smooth, and plane, water surface. Then the ship rolling on these trunnions will fulfill the conditions for unresisted rolling. The point of contact of the trunnion with The Oscillations of Ships 11 the water surface -will then be the "center of flotation" and point of support. This pouit has its instantaneous motion in a horizontal direction and consequently the instantaneous centre will lie in a vertical line drawn through it. Eesistance being supposed non-existent, the only forces acting upon the floating ship are the weight and buoyancy, both acting vertically through the centre of gravity; therefore the instantaneous motion of the centre of gravity must be vertical, and the instantaneous centre will lie on a horizontal line drawn through it. Hence it follows that the iatersection of the vertical line through the centre of flotation and the horizontal line through the centre of gravity will be the instantaneous centre of motion. In warships the centre of gravity ordinarily lies near to the water-line for the upright position; and for oscillations of 12 or 15 degrees on either side of the vertical, the centre of flotation does not move far away from the middle line of the load water plane. In other words, the common case for warships of ordinary form is that the instantaneous axis passes through or very near to the centre of gravity. Although, as its name implies, the position of the instantaneous axis changes from instant to instant, it is suffi- ciently accurate to regard the ship as rolling about a fixed axis passing through the centre of gravity. 6. The agents producing rolling. — We have seen that, besides the influence of the moment of inertia about a longitudinal axis passing through the centre of gravity, the active agent in pro- ducing rolling, after a vessel has been once inclined and then set free, is the moment of statical stability. In the first place, to incline the ship requires the expenditure of worh or energy, the amount of which can be ascertained from the curve of dynamical stability. Then in the mclined position she has acquired energy which she holds by virtue of her inclination from the vertical, which becomes a constant quantity during her rolling, since by the law of conservation of energy, if no external resistances act, the energy due to position combined with that due to motion remains a constant quantity. At the position of extreme inclination there is no motion, and consequently her total energy is the dynamical stability in that 12 The Oscillations of Ships position. Upon reaching the upright she has no energy by virtue of position, all the dynamical stability having been converted into energy of motion. Finally, on coming to rest on the oppo- site side of the vertical all the energy she possesses is that due to position, so that the dynamical stability there must be equal to what it was in the first case when at rest at the extreme in- clination on the other side of the vertical. That is to say, the ship must reach, on the opposite side of the vertical, an angle equal to that from which she first started. If, therefore, a vessel rolled unresistedly in still water, she would continue to oscillate from side to side without any diminution in the angle of roll from the perpendicular. 7. Isochronous oscillations. — Assuming that the front part of the ship's curve of stability is a straight line (an assumption which is approximately true is most ships up to 12 or lo degrees), since then the value of the righting force varies directly with the angle of inclination, it follows mathematically that the period is the same for large as for small arcs of inclination within this limit of degrees on each side of the vertical; that is, she will swing through a total arc of, say 20°, in the same time as through one of, say 4° This fact is expressed by saying that the ship is isochronous within the limits of roll mentioned above. For larger angles of oscillation such ships would probably have a somewhat longer period than for the small oscillations, and it is possible to approximate to this increase. But as yet direct ex- periment has not been applied to determine the actual periods when high-sided ships swing to 20 or 30 degrees on either side of the vertical. Vessels of low freeboard or exceptional form may not be isochronous through arcs of oscillation so large as those named for ordinary vessels. For unresisted rolling the theo- retical condition may be very simply stated: Within the limits of inclination to the vertical, for which the statical righting moment varies directly as the angle of inclination, the rolling of a vessel will be isochronous. In other words, if the curve of stability is practically a straight line for a certain distance out from the origin, the rolling will be isochronous within the limits of inclination fixed by that distance. The Oscillations of Ships 13 8. Determination of the Period. — If the period be denoted by T, the metacentric Might by m, and the radius of gyration about a longitudinal axis through her centre of gravity by fc — ' gm ' m A fair approximation to the still-water, or " natural " period of oscillation for a new ship is given by this formula. The meta- centric height is determined for a warship as one of the particu- lars of the design; and the distribution of the weights is known, so that the moment of inertia can be calculated about the assumed axis of rotation passing through the center of gravity. This latter calculation is laborious, the weight of each part of the structure and lading havmg to be multiplied by the square of its distance from the axis; but with care it can be performed with close approach to accuracy. Calculations of this kind are rarely made except in connection with novel types of ships for which thorough investigations are undertaken in order to be assured of their safety and seaworthiness. This formula given for the period supposes rollinar to be un- resisted; but the influence of resistance is much more marked in the extinction of oscillations than it is in affecting the period, and this accounts for the close agreement of estimates made from the formula with the results of experiments. Dr. Froude discovered that the period of the Greyhound remained practically the same after exceedingly deep bilge-keels had been fitted, as it was without such keels. 9. Changes in period produced by changes in distribution of weights. — The preceding formula for the still-water period en- ables one to ascertain approximately the effect produced upon the period by changes in the distribution of weights on board a ship. Such changes usually affect both the metacentric height and the moment of inertia, and their effects may be summarized as follows: Period is increased by — (1) Increase in the radius of gyration; (2) Decrease in the metacentric height. 14 The Oscillations of Ships Period is decreased by — (1) Decrease in the radius of gyration; (2) Increase in the metacentric height. " Winging "' weights — that is, moving them out from the middle line towards the sides — increases the moment of inertia and tends to lengthen the period. The converse is true when weights — such as guns — are run back from the sides towards the middle line. Eaising weights also tends to decrease the moment of in- ertia, if the weights moved are kept below the centre of gravity; whereas if they are above that point, the corresponding change tends to increase the moment of inertia. But all such vertical movements of weights have an efEect on the centre of gravity, altering the metacentric height, and affecting the moment of inertia by the change in the position of the axis about which it is estimated. It is therefore necessary to consider both these changes before deciding what may be their ultimate etfect upon the period of rolling CHAPTEE II. Influence of Eesistance on Eolling in Still Water. 10. Character of fluid resistance. — An actual ship set rolling ta still water differs from the preceding case in the addition of fluid resistance to its rolling. This resistance may be subdivided into three parts: (1) Fric- tional resistance due to the rubbing of the water against the immersed portions of the vessel, and particularly experienced by the amidship parts where the form is more or less cylindrical. (2) Direct or head resistance due to the opposition offered to the passage through the water in a more or less flat-wise manner of projections such as the keel and bilge-keels and the comparative flat parts of the ship at the ends. (3) Surface disturbance or wave-mahing resistance due to the creation of waves as the vessel rolls, which cannot be propagated without the expenditure of energy, which must he supplied to the water by the ship, thus affecting her motion. 11. Estimation of the resistance. — The aggregate effect of these three parts of the fluid resistance displays itself in the gradual extinction of the oscillations when the ship rolls freely under the action of no external forces other than gravity. To estimate by direct calculation the value of the resistance for a ship of novel form, or for any ship independently of reference to rolling trials of similar ships or models, is not, in the present state of our knowledge, a trustworthy procedure. The flrst two parts of the resistance may be calculated to a degree of approximation that is sufficiently accurate for practical purposes, but the third seems beyond calculation. When the character of the bottom is known — iron or steel, copper- or zinc-sheathed, clean or dirty — it is possible to obtain the " coefficient of friction " for the known conditions ; then knowing the area of surface upon which this fric- tion operates and the approximate speed of rolling, the total fric- 16 The Oscillations of Ships tional resistance may be estimated. Similarly, when the " coeffi- cient of direct resistance" for the known speed has been de- termined by experiment it may be applied to the total area of keel, bilge-keels, etc., and a good approximation made to the total direct resistance. But the wave-making part cannot be so treated, and so it becomes necessary to make rolling experiments in still water, in order that the true value of the resistance may be de- duced from the observations. The importance of the deductions arises from the fact that fluid resistance has very much to do with controlling the maximum range of oscillation of a ship roll- ing in a seaway. This will be considered later; for the present it is sufficient to note that, if the rate of extinction of the still- water oscillations is rapid, the range of rolling at sea will be greatly reduced by the action of the resistance; but if the rate of extinction is slow, resistance will exercise comparatively little control over the behavior of the ship at sea. 12. Rolling experiments — ^how made. — The objects of rolling experiments are twofold: (1) To ascertain the period of oscilla- tion of the ship; (2) to obtain the rate of extinction of the oscil- lations when the ship is rolling freely and being gradually brought to rest by the action of resistance. Various methods may be employed to produce the desired incli- nation from the vertical, at which the rolling is left free and the observations are commenced. Small vessels have been "hove- down," and suddenly set free. Large vessels are usually rolled in stUl-water by running a large number of men back and forth across the deck, their motions being suitably timed so that the amplitude of the oscillations shall gradually increase. This method will be briefly described. The men should first be lined up along the middle line of the deck and then made to run out quickly to the side and back again, reaching the middle line again at the moment the maximum inclination of the ship occurs; they should then run up the deck ta the opposite side, stay there as long as possible and get back to the middle line by the time the ship reaches her extreme angle on that side of the vertical. This is repeated several times. The movements of the men must be carefully timed with the rolling of the ship so that, running The Oscillations of Ships IT at the same rate, and always ''uphill" from the middle line, they will have run out and back again crossing the middle line by the instant that the vessel on her return roll has reached the up- right. Obviously throughout this return roll the inclining mo- ment due to the weight of the men acts with the righting mo- ment due to statical stability and so increases the rolling motion. The arcs of oscillation will therefore be gradually increased, until a maximum is reached determined by the number of men, the number of runs, their transverse movement, and the resistance to rolling. The proper timing is usually effected by an officer stand- ing in some prominent position amidships, such as on a hatch or other obstruction in the middle of the deck, carefully watching the rolling of the ship and at the proper instant directing the men to " starboard,"' '' port," etc., and, when a sufficient angle of inclination is attained, to halt and stand steady on the middle line. 13. Observations made during rolling experiments. — After the men cease running, careful note is taken of the times occupied by the ship in performing each of several successive single rolls. For vessels of ordinary forms, and for the arcs of oscillations reached in still-water rolling, the periods noted for all the rolls are for practical purposes equal, and the motion is isochronous. Hence if n single rolls are noted in an interval of T seconds, the period is equal to T ^^ it . Careful observations are also made of the extreme angles of heel reached at the end of each oscillation, the difference between the successive values marking the rate of extinction. These observations are usually continued until the arc of oscillation has diminished to 2 or 3 degrees. Suitable automatic apparatus has been devised for recording the rolling motion of the ship in such a manner that the angle of inclination, at each instant of her motion, as well as her ex- treme angles of heel, can be traced, and the period also deter- mined; but such extensive apparatus is not necessary ordinarily. 14. Curves of extinction. — The gradual • degradation in the range of oscillation is commonly represented by means of what are termed "curves of extinction,"" as shown in Fig. 2, which are 18 The Oscillations of Ships constructed in the following manner: Eectangular axes are drawn, and on the horizontal axis are set off equal spaces, each division representing an oscillation; and since each oscillation is performed in the same period, each of these spaces also repre- sents a certain number of seconds. The vertical axis is simi- larly laid off to scale to mark the angles of inclination. At each point of the horizontal axis representing the successive oscilla- tions an ordinate is erected and the observed inclination at the end of that oscillation is marked on it to the scale used. A curve drawn through all such points is thr curve required. The A. 4-5 6 r a 9 10 NUMBER OF OSCILLATIONS. Fig. 2. 12 13 14 IS 16 difference between any two successive ordinates gives the degra- dation of roll or extinction value in that particular oscillation, and the greater the resistance the greater will be this extinction value and the steeper therefore the curve. 15. Rolling experiments with models. — Similar rolling experi- ments are made with models; and a model of reasonable size and so weighted that it may have a time of oscillation proportioned to the period of the full-sized ship will have, it has been found by experiment, the same extinction value for corresponding oscilla- tions and will come to rest after the same number of rolls, show- ing that rolling experiments on models may be substituted for those on the full-sized ship. There are many obvious advantages in such model experiments. They are more easily performed The Oscillations of Ships 19 than with a full-sized ship; they may be made before the con- stmction of the ship is begun; by them it is possible to test the influence of variations of form, or proposed changes in bilge- keels, etc.; and any critical conditions affecting the safety of a ship when damaged can be investigated. 16. Periods of various classes of war vessels. — Eolling experi- ments have now been made on most classes of warships and their natural or still-water periods have been determined. The following summary gives about the average values: For small gunboats, torpedo boats, and small craft the period for a single roll is from 2 to 3 seconds; these short periods being due to the small radii of gyration consequent upon the small dimensions, and to the necessity for securing a good metacentric height. For the larger gunboats and smaller cruisers, periods of from 3 to 4^ seconds are common, and 4 seconds is a good average. The larger unarmored cruisers average from 5 to 6 seconds, but some of the later swift cruisers have periods as high as 8 seconds, their meta- centric heights being less than those of earlier types. Among armored ships the shortest periods are found in coast-defense ships of shallow draft, great proportionate beam, and large meta- centric heights. One of our monitors was found to have a period of 2.7 seconds only, and some of the French floating batteries have periods of 3 to 4 seconds. The earlier types of coast-defense battleships have periods of 5 to 6 seconds, and the larger sea- going battleships and armored cruisers, having less metacentric height and large radii of gyration, range from 7 to 9 seconds. 17. Dynamical stability corresponding to decrease of range. — The determination of the period of a ship is a matter of simple observation; but the investigations by which the value of the resistance is deduced from curves of extinction involve mathe- matical processes. The principle upon which these investiga- tions proceed will be explained briefly. The work done on a ship to heel her to any given angle of in- clination — that is, the dijnamical stability — is given by the weight of the ship multiplied by the vertical separation of the centre of irravitv and the centre of buovancv. 20 The Oscillatioxs of Ships Eeferring to Fig. 3, dynamical stability = T1'(5'Z — SG). For moderate angles of inclination to which the "metacentric meth- od '■'■ of estimating statical stability applies, the successive posi- tions (-B and B') of the centre of buoyancy lie on an arc of a circle having J/ as the centre. Hence — BM = B'U B'Z = B'M — ZM = B'M — GM cos e BG = BM~-GM=^B'M— GM . • . B'Z — BG = GM— GM cos e^GM vers e If 3 be circular measure of angle of inclination; then for small values of 6, vers ^ = -o~ nearly. Hence — Dynamical stability) . , , _,. , ■ . , /-nr . , 6' i 11 1 h == weight of ship X GM X .. • lor small angles ) Xow, the " loss of range " per oscillation represents the " work '" done by the resistance during that oscillation, which evidently can be ascertained by calculating the dynamical sta- bility corresponding to the loss of range. Suppose that a ship starts from an .inclination 6, on one side of the vertical and reaches an inclination of 6.^ on the other side. Then using the above formula and writing W for the weight of the ship and m for G2f, we have — 6 '' Dynamical stability for inclination 6^= W X m X -^ ■ The Oscillations of Ships 6 ^ Dynamical stability for inclination 6.,= Wx m X -^ Hence dynamical stability corre- ) _ W X m sponding to decrease of range \ 2 id;' -6.;') + e,) (e, - e,) .y— X arc of oscillation 2 X decrease of range. 18. The value of total resistance. — This last expression measures, as explained above, the work done by the fluid resist- ance during a single swing of the ship. Moreover, it vpill be evident that vfhen the curve of extinction for a ship has been de- termined experimentally, if any value of 6t is assumed, all the other quantities in the expression will be known. The value of the work done by the total resistance can thus be determined and some data obtained from which to infer approximately the laws which govern that resistance. On the lines here briefly indicated extensive experiments and investigations have been carried out by both English and French scientists to determine not only the values of the total resistance of various types of vessels by also to ascertain the values of the component parts, assigning values to the frictional and keel re- sistances as well as to surface disturbance. Dr. Froude made investigations of this character which led him to the conclusion that surface disturbance contributed a])out three-fourths of the total resistance. 19. Surface disturbance. — Waves are constantly being created as the vessel rolls, and are constantly moving away, and the me- chanical work done in this way results in a reduction of the amplitude of successive oscillations. Very low waves, so low as to be almost imperceptible owing to great length in proportion to their height, would suffice to account even for this large propor- tionate effect. The importance attributed by Dr. Froude to surface disturb- ance derives considerable support from experiments made on very special forms of ships. For example, in experimenting upon s 22 The (Jscillatioxs of Shil's the model o[ the li. il. S. Dcvasiatioii, it wa^ found that when the deck-edge amidshii)s was considerably immei>ed bei'oi'e the model was set free to roll, the deck appeared to act like a very powerful bilge-piece, rapidly extinguishing the oscillations. Ex- periments made in the iM-ench navy, showed that when bilge-keels were moved high np on the sides of a vessel so that as she rolled the bilge-keels emerged from the water and entered it again abruptly their effect became much greater than when they were more deeply immersed; as one would anticipate from the in- creased surface disturbance that must exist when the bilge-keels are so high on the sides. Experience with low freeboard moni- tors furnishes further support to this view, immersion of the deck and the existence of projecting armor developing greatly in- creased resistance and assisting in preventing the accumulation of great rolling motions. 20. Value of bilge-keels. — It has already been explained that direct resistance is experienced by the comparatively flat surfaces of deadwoods, keels, bilge-keels, etc., hence it follows that re- sistance to rolling may be considerably influenced by fitting such appendages. In fact, the use of bilge-keels has become the most common means employed to increase resistance to rolling and it will be of interest to examine into their mode of operation. Direct experiment and careful observation have furnished un- Cjuestionable evidence in favor of the use of bilge-keels, showing that they will greatly increase the rapidity of the extinction of still-water oscillations and limit the rolling of ships at sea, while it is also found that the period of oscillation is changed but little as the resistance becomes increased. An approximation can be made by the following formula to the work done by a bilge-keel during the swing of a ship. Assuming the resistance to vary as the square of the angular velocity, and r to be the meaii' radius of the bilge-keel from the axis of rotation (assumed to pass through the centre of gravity) — Work done during i ,. , ., , , , 4 -- .,, r, . -, . "^ I = area of bilge-keel X r' X o't^ X ^' X a single swing J ° 6 1 when T = period, and 2 5^ arc of oscillation, and C a constant, which Dr. Fronde took as 1.6 lb. per sc^uare foot with the velocity of 1 foot per second. The Oscillatioxs of Ships 53 From the general form of this expression it is evident that the effect of bilge-keels increases with — (1) Increase in area; i'i) Decrease in the period (T) of the ship; (:i) Increase in the arc of oscillation. Also, having regard to the formula for the period given on p. 13, it will appear that the etftct of such keels increase> as the mo- ment of inertia is diminished or the metacentric height increased, both of which variations shorten the period T. 21. Number and arrangement of bilge-keels. — "Warships and some classes of merchant ships are now commonly fitted with bilge-keels. Usually one such keel is fitted on each side, near the turn of the bilge and carried as far forward and aft as may be convenient. In some cases two keels have been fitted on each side, but there are objections to this arrangement. Two shallow keels have much less power in extinguishing oscillations than a single deep keel of area equal to the combined areas of the other two; and there is a difficulty, except in large ships, in placing two keels on a side sufficiently clear of each other without the risk of cmersing the upper one during rolling. The reason for the comparative loss of power in two shallow keels is easily seen. As a bilge-keel swings to and fro with the ship it mo\cs at varying velocities, and impresses accelerating motions on masses of water with which it comes in contact, these accelerations being the equivalent of the resistance. If there be two bilge-keels on each side, the water encountered by one will probably have been set in motion by the other and consequently their combined resistance is less than the sum of the resistances which they would experience if acting singly both in undisturbed water. 22. Effect of emersion of bilge-keels. — As regards the emersion of bilge-keels it is necessary to remark that more or less violent blows or shocks are received by such keels as they enter the water again; and even when no structural weakness results, the noise and tremor are unpleasant. ■'J: Till-; ( )S( ILLATIOXS OF SlIIl'S The power of side-keels placed near the water-line is very great, hut for the reasons given they are rarely used; and in cases where an overhanging armor-shelf a few feet below the water-line acted as a side-keel, it has been found desirable to fill in under the shelf in order to diminish the shocks of the sea. 23. Size of bilge-keels. — Practical considerations usually de- termine the depths given to bilge-keels. Eelatively deep keels require to be strongly constructed and attached to the hulls, since their extinctive effect on rolling is necessarily accompanied by considerable stresses on the material. In vessels of great size the limit is fixed by the necessity for compliance with certain extreme dimensions fixed by the docks which the vessels have to enter. Bilge-keels are, of course, of least importance in ships of large size and considerable inertia, and often cause inconvenience in docking. In small vessels where the periods of oscillation and the moments of inertia are small, bilge-keels are most effective. In vessels where they can be conveniently fitted their influence cannot be other than beneficial. For a given area of bilge-keels the extinctive elfect varies with the cube of the angle of oscillation and consequently that effect increases very rapidly as the angles of swing increase. In still water large angles of oscillation do not occur, but among waves the contrary is true and it is under these circumstances that the full value of bilge-keels is illustrated. 24. Effect of " water chambers." — Other means have been de- vised for checking rolling, and for small oscillations some of these are more efliective than bilge-keels; but, on the whole, bilge-keels are the simplest and most effective means of limiting rolling mo- tions which can be taken in addition to those which lie in the power of the naval architect in regulating the stability or the period of oscillation. For certain classes of shi]is having large metacentric heights and comparatively short periods of oscillation, the plan of fitting a •' water chamber " has been tried. H. M. S. Edinburgh, a cen- tral-citadel turret-ship, with which very extensive experiments were carried out by the British Admiralty, may be taken as a The Oscillatioxs of Ships 2o typical case. Above the protective deck in that vessel a " water chamber" was built 16 feet long fore and aft, and 7 feet high; its breadth could be varied by bulkheads from the entire breadth of the vessel at that part, 67 feet, to either 51 feet or 43 feet. Using each breadth, the chamber was partially filled two or three times to different depths giving various quantities of water, and rolling experiments made, which proved that water-chambers ex- ercise a great extinctive effect at small angles of rolling, for which bilge-keels have little influence; that at larger angles (up to 1"2 degrees of heel) a considerable increase of depth of the bilge-keels actually fitted to the Edinburgh would have been needed to give an extinctive effect equal to that of the free water in the chamber; for still larger angles the bilge-keels gained rapidly on the water chamber; that increase in breadth was accompanied by a great increase in extinctive effect. The most effective depth of water was shown to be that which would permit the transfer of the water from side to side to keep time with the rolling motion of the vessel, the water always moving so as to retard the rolling. This motion of the water to produce the best results should be the reverse of that described for the men running in a still-water rolling e.xperiment. Instead of acting witli the moment of the righting couple, it should act against it during each return roll and thus retard the motion of the vessel. 25. Thornycroft's automatic steadying apparatus. — If a weight were moved transversely across a ship as she rolled, and its phases of motion were the converse of those described for the motion of men in a rolling experiment, then it would exercise an extinctive effect resembling that obtained when the water in a chamber ha« its maximum infiuence. In other words, if at any moment the moving weight is so placed as to act against the righting moment of statical stability, this virtual diminution of righting moment must diminish the rolling. Various attempts have been made to utilize this idea; most of them have failed because no efficient controlling apparatus has been devised which would secure the appropriate motions of the weight, more particularly in a seaway. Failure in this respect might result, of course, in increased rolling and possible danger, ilr. Thornycroft succeeded hijwever in 211 The Oscillations of Ships solving this problem and gave proof of his success by the behavior of a yacht fitted with his apparatus. His success lay on the side of the mechanical controlling gear in which ]\Ir. Thoruycroft ex- ercised great skill and ingenuity, the gear being automatic in its action, and practically incapable of error. 26. Influence of free water in the hold. — It will appear that there are essential differences between the case of free water present in large quantities in one or more compartments of the hold of a ship, and that of the comparatively small quantities of free water carried for steadying purposes in the specially con- structed closed chambers. Common experience proves that the presence of large quantities of free water in the hold of a ship affects her rolling, is always objectionable, and may be dangerous. The transference of the water from side to side when the depth is considerable may be very rapid, and it will be obvious that the period of oscillation may be sensibly affected, while heavy blows may be delivered on decks, bulkheads and other portions of the structure. Considerable damage has been done in some cases where water ballast has been carried in large tank compartments of considerable depth and where the surface has been left free either by leakage or by failure to fill the compartments. CHAPTEE III. Deep-Sea Waves. 27. Fundamental conditions of trochoidal theory. — Before con- sidering the motion of a ship at sea among waves it is important to study something of the character of deep-sea waves. Many and divers attempts have been made to construct a mathematical theory of wave-motion and thence to deduce the probable be- havior of ships at sea. It is now generally agreed that the mod- ern, or trochoidal, theory of wave-motion fairly represents the phenomena presented, and without going into any discussion of the earlier theories, it is proposed in this chapter to explain the main features of this trochoidal theory for deep-sea waves. According to this theory a single, or independent, series of waves is regarded as traversing an ocean of unlimited extent, where the depth, in proportion to the wave dimensions, is so great as to be virtually unlimited also. The bottom is supposed to be so deep down that no disturbance produced by the passage of waves can reach it, and the regular succession of waves requires the absence of boundaries to the space traversed. It is not sup- posed, however, that an ordinary seaway consists of such a regular single series of waves, though sometimes the conditions assumed are fulfilled; but more frequently two or more series of waves exist simultaneoiTsly, over-riding one another or perhaps moving in different directions, causing a confused sea. But in what fol- lows it will be understood that, unless the contrary is stated, we are dealing with the primary case of a single series of deep-sea waves. A few definitions must now be given of term? that will be fre- qixently used. The length of a wave is its measurement in feet from crest to crest or hollow to hollow; in Fig. 4, QR would be the half-length. The heif/ht of a wave is the vertical distance in feet from hollow -'^ The Oscillations of Ships to crest; thus in Fig. 4, for the trochoiclal wave, the height would be Fh, or twice the tracing arm. The period of a wave is the time in seconds its crest or hollow occupies in traversing a distance equal to the wave length; or the time which elapses between the passage past a stationary point of successive wave crests or hollows. If an observer proceeds in the direction in which the waves are advancing the period of the waves will appear to be Ibnger than it really is; travelling in a direction opposite to that of the wave advance likewise leads to an apparent period shorter than the real period. Similarly, proceeding in any oblique direction affects the apparent period; retreating obliquely from the waves increases, and advancing obliquely towards them diminislies, the apparent, as compared with the real, period. The velocity of a wave in feet per second will, of course, be the quotient of the length divided by the period, and would com- monly be determined by noting the speed of advance of the wave crest. It is important to note that, when speaking of the advance or speed of a wave, it is only the ivave form which advances and not the water composing the wave. This may be seen by watching the behavior of a piece of wood floating amongst waves; it is seen that, instead of being swept away as it must be if the particle> of water on which it is borne had a motion of advance, and as it would be on a tideway where the particles of water do move on- wards, it simply moves backward and forward about a fixed mean position. Such a motion of wave form may be illustrated by securing one end of a rope and giving a rapid up-and-down mo- tion to the other end; a wave form will travel from one end to the other, but it is evident that the particles composing the wave have not so travelled. 28. Construction of trochoidal profiles. — The trochoid is the general term applied to the curve described by any point on the radius of a circle rolling on a straight line. In Fig. 4, suppose QR to be a straight line, under which the large circle whose radius OQ is made to roll. The length QR being made equal to the semi-circumference, the rolling circle will have completed half a revolution during its motion from Q to E. Then suppose n The Oscillations of Ships -29 point P to be taken on the radius of the rolling circle; this will be called the " tracing point," and as the circle rolls P will trace a trochoid which will be the theoretical wave profile from hollow to crest. To determine a point on the trochoid, take as centre and with OP as radius describe the small circle; as the rolling circle advances, a point on its circumference (say 3) comes into contact with the corresponding point of the directrix QR, so the centre of the circles must at that instant be at S and the angle through which the large circle and the tracing arm OP have both turned is given by QOZ, or, its equal, POc; through S draw S'r.,, parallel to Oc and mak:e 8c^ equal to Oc; then Cj is a point on the trochoid. Or, the same result may be reached by drawing cc^ parallel to OS, finding its intersection (Cg) with the line S. and then making c„f,. equal to cc^. The tracing arm OP may, for wave motion, have any value not greater than the radius of the rolling circle OQ. If OP equals OQ, and the tracing point lies on the circumference of the rolling circle, the curve will be a cycloid; such a wave is on the point of breaking. The curve R^TR shows this form and it will be noticed that the crest (at R) is a sharp ridge or line, while the hollow is a very flat curve. 30 ThK ()^( 'ILLATIONS OF ShIPS 29. Orbital motion of particles and advance of wave form. — Accepting the condition that the proiile of an ocean wave is a trochoid, the motion of the particles of water in the wave re- Cjuires to be noticed, and it is here the explanation is fonnd of the rapid advance of the wave form, while individual particles have little or no advance. The trochoidal theory teaclic- that every particle revolves with uniform speed in a circular orbit (situated in a vertical plane which is perpendicular to the wave ridge), and completes a revolution during the period in which the wave ad- vances through its own length. In Fig. .5, suppose /-", P, P, etc., to be particles on the upper surface, their orbits being the equal circles sliown; then, for this position of the wave, the radii of the orbits are indicated bv OP, OP, etc. The arrow below the wave -S-— J2-^ OIRECTIOW PF ADVANCE- Tin. 5. profile indicates that it i^ advancing from right to left; the short arrows on the circular orbits show that at the wave crest the particle is moving in the same direction as the wave is advancing in, while at the hollow the particle is moving in the opposite di- rection. For these surface particles the diameter of the orbit ecjuals the height of the wave. Kow suppose all the tracing arms, OP, OP, etc., to turn through the ec|ual angles POp, POp. etc. ; then the points p, p, etc.. must be corresponding jDOsitions of particles on the surface formerly situated at P, P, etc. The curve drawn through p, p, etc.. will be a trochoid identical in form with P, P, etc., only it will have its crest and hollow further to the left; and this is a motion of advance in the wave form produced by a simple revolution of the tracing arms and particle- (P). The motion of parti ^-le- in the direction of advance is limited by the The Oscillations of Ships 31 diameter of their orbits, and they s^^'ay to and fro about the centres of the orbits. One other point respecting the orbital motion of the particles is noteworthy. This motion may be regarded at every instant as the resultant of two motions — one vertical, the other horizontal — except in four positions, viz. : (1) when the particle is on the wave crest; (2) when it is in the wave hollow; (3) when it is at mid- height on one side of its orbit; (4) when it is at the correspond- ing position on the other side. At the crest or hollow the par- ticle instantaneously move? horizontally, and has no vertical mo- tion; at mid-height it moves vertically and has no horizontal motion. Its maximum horizontal velocity will be at the crest or hollow; its maximum vertical velocity at mid-height. Hence uniform motion along the circular orbit is accompanied by ac- celerations and retardations of the component velocities in the horizontal and vertical directions. 30. Internal structure of wave. — The particles which lie upon the upper surface of the wave are situated in the level surface of the wave when at rest. The disturbance caused by the pas- sage of the wave must extend far below the surface affecting a great mass of water. But at some depth, supposing the depth of the sea is very great, the disturbance will have practically ceased; that is to say. still, undisturbed water may be conceived as underlying the water forming the wave. Eeckoning downward from the surface, the extent of the disturbance must decrease according to some law. The trochoidal theory expresses the law of decrease, and enables the whole of the internal structure of a wave to be illustrated in the manner shown in Fig. 6. On the right-hand side of the line AD the horizontal lines marked 0, 1, 2. 3, etc.. show the positions in still water of a series of particles which during the wave transit assume the trochoidal forms num- bered respectively U. 1, •?. 3. etc., to the left of AD. For still water every unit of area in the same horizontal plane has to sus- tain the same pressure; hence a horizontal plane is termed a surface or subsurface of '" equal pressure '" when the water is at rest. o-> The Oscillations of Ships A> the wave passes, the trochoidal surface corresponding to that horizontal plane will continue to be a subsurface of equal / / / / X" "i F C \ \ \ \ \ \ 1 a 2) c d e i 1 1 \ ^ ^ -r e \ t 1 — e^-^ p r"**^/ -^ £/ 1 Q \ , / JUm L nU-^ -- ^ ^ 1 ^ --- ?■ -' / ^.r n ra ^ ^ - # . r ^ W T " tsf ^ 1 ^ "" A " V - 4 -T J -+- y s : s I J i- \ 1 1 6 ts ■Y 1 1 ~ \^ ■?; £ t ■ ~ — . — J " / ? 8 " f 9 9 Z*' 10 // II 11 r ^_ B Tiu-. 6. pressure; and the particles lying between any two planes (say 6 and 7) in still water will, in the wave, be found lying between the corresponding trochoidal surfaces (G and 7). In Fig. 6, it The Oscillations of Ships 33 will be noticed that the level of the still-water surface (0) is sup- posed changed to a cycloidal wave (0), the construction of which has already been explained; this is the limiting height the wave could reach without breaking. The half-length of the wave {AB) being called L, the radius {CD) of the orbits of the surface ]iar- ticles will be given by the equation — CD =z R=^,ot:^ L (nearly). 31. Decrease of downward disturbance. — AU the trochoidal sur- faces have the same length as the cycloidal surface, and they are generated by the motion of a rolling circle of radius K ; but their tracing arms — measuring half the heights from hollow to crest — rapidly decrease with the depth, the trochoids becoming flatter and flatter in consequence. The crests and hollows of all the subsurfaces are vertically below the crest and hollow of the upper wave profiie. The heights of these subsurfaces diminish in a geometrical progression as the depth increases in arithmetrical progression; and the following approximate rule is very nearly correct. The orbits and velocities of the particles of water are diminished by one-half, for each additional depth below the mid- height of the surface wave equal to one-ninth of a wave length. For example — Depths in fractions of a wave length below ) the mid-height of the surface wave j "> 9' ¥' 9' -g- «^'•'-■ Proportionate velocities and diameters . . . 1, J, :i, a, tVj ^te. Take an ocean storm-wave 600 feet long and 40 feet high from hollow to crest: at a depth of 200 feet below the surface (f of length), the subsurface trochoid would have a height of about 5 feet; at a depth of 400 feet (f of length), the height of the trochoid — measuring the diameter of the orbits of the particles there — would be about 7 or 8 inches only; and the curvature would be practically iasensible on the length of 600 feet. This rule is sufficient for practical purposes, and we need not give the exact exponential formula expressing the variation in the radii of the orbits with the depths. ■ '4 The Oscillatidxs (if Siiii's 32. Height of centres of tracing circles above still-water level. — It will be noticed also in Fig. 6, that the centres of the tracing circles corresponding to any trochoidal surface lie above the still- water level of the corresponding horizontal plane. Take the horizontal plane (1), for instance. The height of the centre of the tracing circle for the corresponding trochoid (1) is marked E. EF being the radius: and the point E is some distance above the level of the horizontal line (1). Suppose r to be the radius of the orbits for the trochoid under consideration, and R the I'adius of the rolling circle: then the centre (E) of the tracing circle (/. r. the mid-height of the trochoid) will lie above the level line (1) by a distance equal to r'-'^2E. Xow 7? is known where the length of wave is known : also r is given for any depth by the above approximate rule. Consequently we have the means of drawing the series of trochoidal subsurfaces for any wave. 33. Displacement of vertical columns of particles. — ( 'olunins of particles which are vertical in still water become curved during the wave passage. In Fig. 6, a series of such vertical lines is drawn; during the wave transit these lines assume the posi- tions shown by the heavy lines curving towards the wave crest at their upper ends, but still continuing to enclose between any two the same particles as were enclosed by the two correspond- ing lines in still water. The rectangular spaces endured bv tliese vertical lines and the level -lines are clianged during the motion into rhomboidal-sliaped figures, but remain imclianged in area. 34. Variation of direction and intensity of fluid pressure. — We now come to the consideration of the direction and intensity of the resultant fluid pressure in a wave which must dilfer greatly from those for still water. Each particle in wave moving at uni- fni'iu speed in a circular orljit will be subject to the action of centrifugal force as well as the force of gravity : and the resultant of lliese two forces must be found in order to determini' the direc- tion and intensity of the pressure on that particle. This may be done by a simple diagram of forces as shown in Fig. T for a surface particle. Let EED be the orbit of the particle; .1 its The 0>cillatioxs of Ships 35 P-- centre; and B the position of the particle in it? orbit at any time. Join the centre of the orbit .1 with B; then the centrifugal force acts along the radius AB and the length AB may be taken to represent its magnitude. Through A draw .IC vertically, and make it equal to the radius (i?) of the rolling circle; then it is known that AC will represent the force of gravity on the same scale as AB represents centrifugal force. Join BC, and it will represent in mag- nitude and direction the resultant of the two forces acting on the particle. Now it is an established property of a fluid that its free surface will place itself at right angles to the resultant force impressed upon it. resultant pressure shown by BC must be normal to that part of the trochoidal surface PQ where the particle B is situated. Similarly, for the position B^, GB^ will represent the resultant force; P-^Q^ drawn perpendicularly to CB^ being a tangent to the trochoid at B.^. Conversely, for an}' point on any trochoidal stirface in a wave, the direction of the fluid pressure must lie along the normal to the surface at that point. At the wave hollow the fluid pressure acts along a vertical line: and as its point of application proceeds along the curve its direction becomes more and more inclined to the ver- tical until it reaches a maximum inclination at the point of in- flection of the trochoid: thence onwards towards the crest the in- clination of the normal pressure is constantly decreasing until at the crest it is once more vertical. If a small raft floats on the wave (as shown in Fiir. S). it will at every instant plncc its mast in the Sll/ff'^C£ Of sriiL tv^rs/f 30 The i_)~cillatioxs of bHiPS t' the direction of the normal to the trochoid may be compared with those of a pendulum, performing an oscillation from an angle equal to the maximum inclination of the normal on one side of the vertical to an equal angle on the other side, and completing a single swing during a period equal to half the wave period. 35. Maximum slope of wave. — The maximum slope of the wave to the horizon occurs at a point somewhat nearer the crest than the hollow, but no great error is assumed in supposing it to be at mid-height in ocean waves of common occurrence where the radius of the tracing arm (or half-height of the wave) is about one- twentieth of the length. For this maximum slope, we have — radius of tracing circle Sine of an£le= — ^. 7 — rp ■ — i — radius oi rolling circle half-height of wave half-length of wave -^- tt height of wave length of wave The angle of slope for waves of ordinary steepness will be small and the circular measure -of the angle may be substituted for the -ine, so we may write — ^^ . , ^ height of wave Maximum slope m degrees = 180 X • length of wave Take, as an example, a wave 360 feet long and 14 feet high — Maximum slope = 180 X tbtj = '^°- The variation in the direction of the normal is in this ease simi- lar to an oscillation of a pendulum swinging 7 degrees on either side of the vertical once in every half-period of the wave. 36. Extremes of intensity of fluid pressure. — It is also necessary to notice that in wave water the intensity as well as the direction of the fluid pressure varies from point to point. Since lines such a= BC in Fig. 7 represent the jjressure in magnitude as well as direction, we can at once compare the extremes of the variation The Oscillatiox* of Siiip> 3'7 in intensity. In the upper half of the orbit of a particle centri- fugal force acts against gravity and reduces the weight of the particle, which reduction reaches a maximum at the wave crest, when the resultant is represented by CE = R — r. In the lower half of the orbit gravity and centrifugal force act together, pro- ducing a virtual increase in the weight of each particle, the maxi- mum increase being at the wave hollow where the resultant is represented by CD ^: R -\- r. 37. Motions of a loaded pole. — Instead of the raft in Fig. 8, if the motions of a loaded pole on end (such as SS-,} be traced, it will be found that it tends to follow the original vertical lines and to roll always toward the crest as they do. Here again the motion partakes of the nature of an oscillation of fixed range performed in half the wave period, the pole being upright at the hollow and crest. A ship differs from both the raft and the pole, for she has lateral and vertical extension into the subsurfaces of the wave, and cannot be considered to follow either the motion of the sur- face particles like the raft or of an original vertical line of par- ticles like the pole. This will be discusged in the next chapter. 38. Formulae for dimensions and speed of waves. — The tro- choidal theory connects the periods and speeds of waves with their lengths alone and fixes the limiting ratio of height to length in a cycloidal wave. The principal formulae for lengths, speeds, and periods for trochoidal waves are as follows: — I. Length of wave (in feet) = a T'^ ~~ = 5.123 X square of period (in sees.) = 5:i X square of period (nearly). II. Speed of wave ) (in feet per see.) }- 5.123 X period^ V3.123 X length = 2:^ \'length (nearly). III. Speed of wave (in knots per hour) :=>3 X period (roughly). lY Period (in seconds) = 7 !|11SL^ = * yiength (nearly). oS The O^CILLATIOXri OF SjIII'S Y. Orbital velocity of ) , , . , ' p h = speed ot wave particles on surtaee J 3.1416 X height of wave ^, heiojht of wave . , X 1 Zu -f = 7* X -/, - --^r= . = (nearly j. length of wave V length of wave ^ "^ To illustrate these formulae, take a wave 400 feet long and 15 feet high. For it we obtain — Period = |- V4'J0 = S^^ seconds. Speed = I y 400 = 4.5 feet per :^econd. = 3 X 8|- ^265 knots per hour. Orbital veloeitv of 1 15 - , . , ,. ^. ," y = 7 1 X — =^ = a- ±eet per second, surface particles j ^ v^4uo It will be remarked that the orbital velocity of the particles is very small when compared with the speed of advance; and this is always the ca>e. In Formula V, if we ^ub-titute, as an aver- age ratio for ocean waves of large size, Height = 5-VX length, the expression becomes — Orbital velocity of ) 1 v length ^.rf ace particles \='iX "^li^ = «-3- V length. Comparing this with Formula II for rpeed of advance it wiU be seen that the latter will be between six and seven times the or- bital velocity. As a mathematical theory, that for troehoidal waves is com- plete and satisfactory, under the conditions upon which it is based. Sea-water is not a perfect fluid such as the theory contem- plates; in it there exists a certain amount of viscosity, and the particles must experience resistance in changing their relative positions. But there is every reason to believe that the theory closely approximates to the phenomena of deep-sea waves. 39. Observations of lengths of waves. — From a scientific point of view, and as a test of the troehoidal theory, the observations made when a -hip falls in with a series of approximately regular wave= is most valuable. More frequently observations have to The Oscillatioxs of Ships 39 be conducted in a confused sea, successive waves differing from one another in lengths, heights, and periods, and occasional waves occurring of exceptional size as compared with their neighbors; but supposing a single series of waves to be encountered, the lengths and periods of successive waves can be easily determined if the speed of the ship and her course relatively to the line of advance of the waves are known. The method adopted will be briefly described. Two observers (.4 and B, Fig. 9) are stationed as far apart as possible at a known longitudinal distance from one another. At oiReenot/ of --(M+Vcos.Qg/t,- V,t. *<' Fig. 9. each station a pair of battens is erected so as to define, when used as sights, a pair of parallel lines at right angles to the ship's keel. The observers at both stations note the instant of time when a wave crest crosses their lines of sight; they also note how long an interval elapses before the next wave crosses their lines. Com- paring their records, they determine (1) the time (say t seconds) occupied by the wave crest in passing over the length (L feet) between their stations: (2) the time (say t^ seconds) elapsing be- tween the passage of the first and second crest across either line of sight: this time is termed the "apparent period" of the waves. 40 The Oscillations of Ships Suppose the ship to be advancing at a speed of V feet per second towards the waves, her course making an angle of « degrees with that course which would place her exactly end-on to the wave*. Then expressing the facts algebraically — Apparent speed of wave (feet per second) = L t Eeal speed of wave (feet per second) ^ Fi = I y ~ ^ ) cos es beneath a ship, therefore, the direction of the action of buoyaircv is con- tinually changing: when the cre-t reaches her it acts vertically upward-; about midway between crest and hollow it acts at its maximum deviation from the vertical; and in the hollow it- direc- tion is again vertical. The cuimiuii of the buoyancy in wave-water The Oscillations of Ships 45 is also constantly varying according to the position of the ship on the wave, being sometimes greater and sometimes less than her buoyancy in still water. The causes producing this variation also operate to make the "virtual weight" of the ship change in the same manner, as well as to make it also act normal to the wave surface. These differences in amount of the virtual weight and buoyancy are not however relatively large and are usually neg- lected, the virtual weight being taken as the real weight through- out, and the buoyancy also assumed to be constantly equal to this. It is thus seen that the ship at each instant is acted upon Pig. 11. by two equal and opposite forces in a direction perpendicular to the wave surface, and hence, when her masts stand at right angles to the wave surface she will at that instant be subject to no forces displacing her from that position. She is then, indeed, in the same condition as she would be if upright in still water, ajid, therefore,, in wave water, the perpendicular to the wave surface is the " virtual upright "; and if she were displaced through an angle e from that virtual upright, the moment of stability acting upon her would be equal to that acting in still water when at the angle e from the upright. In Fig. 11, let AA be the wave surface where a ship is floating at any particular instant; then if the vessel had its mast standing 46 The Oscillations op Ships in the direction BM, perpendicular to AA, there would he no moment of stahility; but in the position shown, its mast making an angle d with BM, the righting couple is W X GZ or W X GM sin e. It is thus clear that, instant by instant, the stability de- pends upon the inclination to the virtual upright^ and if this becomes equal to or greater than the angle of vanishing stability, as given by the curve of stability, the vessel will capsize as in still water. 45. Effect of waves in producing rolling. — Suppose a ship lying hroadside-on to the waves to be upright and at rest when the first wave hollow reaches her; at that instant the normal to the surface coincides with the vertical, and there is no tendency to disturb the ship. But a moment later, as the wave form passes on and brings the slope under the ship, the virtual upright to- wards which she tends to move, becomes inclined to the vertical. This inclination at once develops a righting moment tending to bring the masts into coincidence with the instantaneous position of the normal to the wave. Hence rolling motion begins, and the ship moves initially at a rate dependent upon her still-water period of oscillation. Simultaneously with her motion, the wave normal is shifting its direction at every instant, becoming more and more inclined to the vertical, rmtil near the mid-height of the wave it reaches its maximum inclination, after which it gradually returns to the upright; the rate of this motion is de- pendent upon the period of the wave. 46. Importance of ratio of period of rolling to half-period of wave. — Whether the vessel will move quickly enough to overtake the normal or not, depends upon the ratio of her still-water period to the interval occupied by the normal in reaching its maximum inclination and returning to the upright again, which it accom- plishes at the wave crest; this interval equals one-lialf the period of the wave. Hence it appears that the ratio of the period of the ship (for a simple roll) to the half-period of the wave must influence her rolling very considerably, even during the passage of a single wave, and still more is this true when a long series of waves moves past, as will be shown presently. It will also be The Oscillatioxs of Ships 4T obvious that the chief cause of the rolling of ships amongst waves is to be found in the constant changes in the direction of the fluid pressure accompan^'ing -svave motion. Two extreme cases may be taken as simple illustrations. The first is that of a little raft having a natural period indefinitely small compared with the half-period of the wave. Her motions will consequently be so quick as compared with those of the wave normal that she will be able continually to keep her mast almost coincident with the normal and her deck parallel io the wave slope. Being upright at the hollow, she will have attained one extreme of roll at about the mid-height of the wave and be upright again at the crest. The period of this single roll will be half the wave period. As successive waves in the series pass under the raft, she will acquire no greater motion, but continue oscillating through a fixed arc and with unaltered period. The arc of oscillation will be double the maximum angle of wave slope. The other extreme case is that of a small vessel having a natural period of oscillation which is very long compared to the wave period. If such a vessel were upright and at rest in the wave hollow she would be subjected to rolling tendencies similar to tliose of the raft owing to the successive inclinations of the wave normal — her instantaneous upright. But her long period would make her motion so slow as compared with that of the wave nor- mal that, instead of keeping pace with the latter, the ship would be left far behind. In fact, the half-period of the wave during which the normal completes an oscillation would be so short rela- tively to the period of the ship that, before she could have moved far, the wave normal would have passed through the maximum inclination it attains near the mid-height of the wave, and rather more than half way between hollow and crest. From that point onwards to the crest it would be moving back towards the up- right; and the effort of the ship to move towards it, and further away from the upright would consequently gradually diminish. At the crest the normal is upright and the vessel but little in- clined — and even then inclined in such direction that the varia- tions of the normal on the second or back slope of the wave will tend to restore her to the upright. Hence it is seen that the 48 Thh Oscillations of Ships Ijussage of a wave under such a ship disturbs her but little, her deck remains nearly horizontal, and she is a much steadier gun- platform than the raft-like ship. No ship actuall}' conforms to the conditions of either of these extreme eases, nor can her rolling be unresisted as here assumed. Experience proves, however, that vessels having very short periods of oscillation in still water tend to acquire a fixed range of oscil- lation when they encounter large ocean waves, keeping their decks approximately parallel to the effective wave slopes. Actual ob- servations also show that vessels having the longest periods of oscillations in still water are, as a rule, the steadiest amongst waves, keeping their decks approximately horizontal and rolling through very small arcs. 47. Fundamental assumptions for mathematical investigation of unresisted rolling. — We now come to a more particular anal3'^sis along mathematical lines of the unresisted rolling of ships among waves. The modern method of investigation makes the following assumptions in order to bring the problem within the scope of exact mathematical treatment: (1) The ship is regarded as lying broadside-on to the waves with no sail set, and without any motion of progression in the direction of the wave advance; in other words, she is supposed to be rolling passively in the trough of the sea. (2) The waves to which she is exposed are supposed to form a regular independent series, successive waves having the same di- mensions and periods. (3) The waves are supposed to be large as compared with the ship, so that at any instant she would rest in equilibrium with her masts coincident with the corresponding normal to the " effec- tive wave slope," which is commonly assumed to coincide with the upper surface of the wave. (4) The righting moment of the ship at any instant is assumed to be proportional to the angle of inclination of her masts to the corresponding normal to the effective wave slope — the virtual upright. (5) The variations of the apparent weight are supposed to be The Oscillations of Ships 49 so small, when compared with the actual weight, that they may- be neglected, except in very special cases. (6) The effects of fluid resistance are considered separately, and in the mathematical investigation the motion is supposed to be unresisted and isochronous. Upon the basis of these assumptions a rather complex dynamical equation is formed which furnishes an expression for the angle of inclination of the ship to the vertical at any instant in terms of her own natural period, the wave period, the ratio of the height to the length of the wave, and certain other known quantities. Assuming certain ratios of the period of the ship to the wave period, it is possible from the equation to deduce their compara- tive effect upon the rolling of the ship; or, assuming certain values for the steepness of the waves, to deduce the consequent rolling as time elapses and a continuoiis series of waves passes the ship. In fact, the general equation gives the means of tracing out the unresisted rolling of a ship for an unlimited time, under chosen conditions of wave form and period. A few of the more import- ant eases will be briefly discussed. 48. Rolling amongst waves of synchroiiizing periods. — First. The first ease to be considered is that in which the natural period of the ship for a single roll equals the half-period of the wave. It is a matter of common experience that if a pendulum receive suc- cessive impulses, keeping time (or " synchronizing ") with its period, even if these impulses, individually, have a very small effect, they will eventually impress a very considerable oscillation upon the pendulum. "When a similar synchronism occurs between the wave impulse and the period of the ship, the passage of each wave tends to add to the range of oscillation, and were it not that the fluid resistance puts a practical limit to the range of oscillation, she would finally capsize. Suppose a vessel to be broadside-on in the wave hollow when the extremity of her roll is reached, say to starboard, the waves advancing from starboard to port. Then the natural tendency of the ship, apart from any wave impulse, is to return to the upright in a time equal to one-half her period, which by our hypothesis will be equal to the time occupied by the passage of one-fourth of the wave length. In other words,. 50 The Oscillations of Ships the ship would be upright midway between hollow and crest of the wave about where its maximum slope occurs. Now at each instant of this return roll towards the upright the inclination of the wave normal, fixing the direction of the resultant fluid pres- sure, is such as to make the angle of inclination of the masts to it greater than their inclination to the true vertical; that is to say,, the inclination of the wave normal at each instant virtually causes an increase of the righting moment. Consequently, when the vessel reaches the upright position at the mid-height of the wave, she has by the action of the wave acquired a greater velocity than she would have had if oscillating from the same initial inclination in still water. She therefore tends to reach a greater inclination to port than that from which she started to starboard; and this tendency is increased by the variation in direction of the wave normal between the mid-height and crest — that part of the wave which is passing the ship during the period occupied by the second half of her roll, — and since the ship, during the second half of the roll, inclines her masts away from the wave crest, with the angle between her masts and the wave normal constantly less than they make with the vertical, the effect is to make the righting moment less at every instant during the second half of the roll than it would have been in still water. For unresisted rolling, it is the work done in overcoming the resistance of the righting couple which extinguishes the motion away from the vertical. On the wave, therefore, the vessel will go further to the other side of the vertical from that on which she starts than she would do in still water, for two reasons: (1) she will acquire a greater velocity before she reaches the upright; (2) she will experience a less check from the righting couple after passing the upright. 49. Variation of period of rolling due to increase of arc of oscillation. — More or less close approximation to this critical con- dition will give rise to more of less heavy rolling; but it is a note- worthy fact that, even where the natural period of the ship for small oscillations equals the half-period of the wave, and may thus induce heavy rolling, the synchronism will almost always be dis- turbed as the magnitude of the oscillations increases; the period of the ship will be somewhat lengthened, and thus the further The Oscillations of Ships 51 increments of oscillation may be made to fall within certain limits lying within the range of stability of the ship. The angles of swing of ships are rarely sO' large as to make the increase of period great proportionately, but yet the increase may be sufficient to add sensibly to the safety of the ship when she is exposed to the action of waves having a period double her own period for small oscillations; although it is by the action of fluid resistance that the capsizing of a ship so circumstanced is chiefly prevented. The effects of synchronism, which m theory is the worst condi- tion for a ship to be subjected to, may often be produced by a ship steaming on a course oblique to the direction of wave ad- vance when the apparent period of the wave becomes double the ship's natural period of oscillation; and here it is worthy of notice that, so long as the ship is manageable, the officer in command can to a great extent influence her behavior by the judicious change of the course and speed to produce an apparent period of the wave such that the ratio of the periods of the ship and wave is made conducive to better behavior. 50. " Permanent " oscillations. — Second. A second deduction from the general equation. for unresisted rolling in waves is found in the "permanent" oscillations of ships. If a ship has been exposed for a long time to the action of a single series of waves, she may acquire a certain maximum range of oscillation, and per- form her oscillations, not in her own natural period, but in the possibly different wave period. This case differs from the pre- ceding one in that the period of the ship for still-water oscillations does not agree with the half-period of the wave; but, notwithstand- ing, the oscillations among waves keep pace with the wave, their period being "forced" into coincidence with the half -period of the wave. At the wave hollow and crest a ship so circumstanced is upright; she will reach her maximum inclination to the vertical when the maximum slope of the wave is passing under her (about the mid-height of the wave); and the passage of a long series of waves will not increase the range of her oscillations, which are "permanent" both in range and period — Whence their name. The maximum inclination then attained depends, according to theory, upon two conditions: (1) the maximum slope of the wave; (3) the 52 The Oscillations of Ships ratio of the natural ]3eriod of oscillation of the ship to one-half the wave period. Let a =; maximum angle made with the horizon hy the wave profile; e = maximum angle made with the vertical by the masts of the ship; r^= natural period of still-water oscillations of the ship; 2 7", = period of wave. If fluid resistance is neglected, and the conditions above stated are fulfilled, mathematical investigation for this extreme case leads to the following equation: — 6 = ., 1 = "■'^^^" Three cases may be taken in order to illustrate the application of this equation. I. Suppose r = Ti : then 6 becomes infinity; that is to say, we have once more the critical case of synchroni-sm previously discussed. II. Suppose T less than T^ : so that — ^ is a proper fraction less J i" than unity; then 6 and a always have the same sign, which indi- cates that the masts of the ship lean away from the wave crest at all positions, except when the vessel is upright at hollow and crest. The closer the approach to equality between T and T^, the greater the value of 6; which is equivalent to the statement previously made, that approximate synchronism leads to heavy rolling. The smaller T becomes relatively to T^, the smaller does 6 become, its minimum value being a. when T is indefinitely small relatively to Tj. This is the case, before mentioned, of the raft which keeps its mast coincident with the wave normal. III. Suppose T greater than T^: then e and « are always of opposite signs, and, except at hoUow and crest, the masts of the ship always lean towards the wave crest. The nearer to unity is the ratio of T and T^ the greater is d ; illustrating as before the accumulation of motion when there is approximate synchronism. The greater T become^ relatively to T^, the less 6 becomes; in The Oscillations of Ships 53 other ■words, as explained above, a ship of very long period keeps virtually upright as the wave passes. A simple illustration may be given of the influence which changes in the relative periods of ships and waves have upon roll- ing. Let AB, Fig. 12, represent a pendulum with a very heavy bob, having a period equal to the half-period of the wave. To its lower end let a second simple pendulum, jBC,be suspended, its weight being inconsiderable as compared with the wave pendulum AB; then, if AB is set in motion, its inertia will be so great that, notwithstanding the suspension of BC, it will go on oscillating at a constant range — say, equal to the maximum ^ slope of the wave — on each side of the vertical. First, suppose BC to be equal in length and peri- od to ABj then, if the compound pendulum be set in motion, and AB moves through a small range, it will be found that BC. by the property of sjmchronizing impulses, is made to oscillate through very large angles. Second, if BC i.- made very long, and of long period, as compared with AB, it will be found that BC continues to hang nearly vertical while AB swings, just as the ship of comparatively long period remains up- right, or nearly so, on the wave. Third, if B(' is made very short and of small period, when AB is set moving BC will always form almost a continuation of AB, just as the quick moving ship keeps her masts almost parallel to the wave normal. 51. " Phases " of oscillation. — Third. A third deduction from the general equation of imresisted rolling is that, except when the conditions of synchronism or permanent oscillation are obtained, the rolling of a ship will pass tlirough phases. At regular stated intervals equal inclinations to the vertical will recur, and the range of the oscillations included in any series will gradually grow from the minimum to the maximum, after attaining which it will once more decrease. The time occupied in the completion of a phase depends upon the ratio of the natural period of the ship to the wave period. If T = ship's period for a single roll, T^ ^half- period of wave, and the ratio T to T^ be expressed in the form 54 The Oscillations of Ships where both numerator and denominator are the lowest whole num- bers that will express the ratio: then — Time occupied in the completion of a phase = 2qy, T seconds. For example, let it be assumed that waves having a period of 9 seconds act on a ship having a period (for single roll) of 7^ seconds. „, ^ 7J 15 5 JO Time for completion of phase = 2 X 3 X 7^ = 45 seconds. Although the mathematical conditions for these "phases" of oscillation are not altogether fulfilled in practice, the causes actually operating on the ship — such as the differences in form of successive waves and the influence of fluid resistance — commonly produce great differences in the successive arcs of oscillation. It is important, therefore, in making observations of rolling, to con- tinue each set over a considerable period. 52. Practical conclusions on unresisted rolling. — In concluding these remarks on the case of unresisted rolling among waves, and bearing in mind that the actual behavior of ships at sea is greatly influenced by fluid resistance, we may briefly summarize our con- clusions as follows: (1) Heavy rolling is likely to result from equality or approximate equality of the period of a ship and the half-period of the waves, even when the waves are very long com- pared to their height. (2) The best possible means, apart from increase in the fluid resistance, of securing steadiness in a seaway, is to give to a ship the longest possible natural period for her still-water oscillations. (3) Changes of course and speed of the ship relatively to the waves affect the relation between the periods and may either destroy or produce the critical condition of syn- chronism. (4) Vessels having very quick periods, say 3 seconds or less for a single roll, fare better among ordinary large storm waves than those having periods of 4 to 6 seconds (which approxi- mate very closely to the half-period of large storm waves), since the tendency in these very quick moving vessels is to acquire a fiied oscillation, keeping their decks approximately parallel to the effective wave slope; though such a vessel would not be a The Oscillations of Ships 55 steady gun platform. (5) For small sea-going vessels, for which the still-water periods are made short by the smallness of their moment of inertia, and the necessity for retaining a sufficient amount of stiffness, the effective wave slope is very nearly the upper surface of the wave, and their range of oscillation among large waves is practically determined by the wave-slope; but amongst smaller waves, approaching the condition of synchronism, these small vessels are worse off than very broad vessels of equally short periods because the effective wave slope for the broad ves- sels is so much flatter. 53. Effect of fluid resistance on rolling among waves. — The de- ductions from the hypothetical case of unresisted rolling among waves can be regarded only as oi a qualitative and not of a quanti- tative character. Although the character of the motion is well described by the deduction from the hypothetical case, its extent is not thus determined. The effect of resistance, and, in the case of isochronous rolling, change of period with increased arcs of oscillation, must be considered when exact measures of the range of oscillation are required as for the determination of the safety of ships. The problem therefore resolves itself iato one of correcting the deductions from the hypothetical cases by the con- sideration of the effect of fluid resistance, and by allowing for departures from isochronism as rolling becomes heavier. In accordance with the principles explained ia Chapter 11, it is possible to ascertain the amount of resistance of a ship correspond- ing to any arc of oscillation. If a vessel rolls through a certain arc amongst waves, it appears reasonable to suppose that the effect of fluid resistance will be practically the same as that experienced by the ship when rolling through an equal arc in still water. The intrusion of the vessel into the wave, as previously remarked, must somewhat modify the internal molecular forces, and she must sustain certain reactions, but for practical purposes these may be disregarded. Eesistance is always a retardiag force; in still water it tends to extinguish oscillation; amongst waves it tends to limit the maximum range attained by the oscillating ships. 54. Influence of resistance in the critical case of synchronism. — This may be well seen in the critical case of synchronism; where 56 The Oscillations of Ships a ship rolling unresistedly would have a definite addition made to her inclination by the passage of each wave. The wave impulse may be measured by the added oscillation; the dynamical stability corresponding to the increased range expressing the energy of the wave impulse. At first the oscillations are of such moderate ex- tent that the angular velocity is small, and the wave impulse more than overcomes the efi:eqt of the resistance, and the rolling be- comes heavier. As it becomes heavier, so does the angular ve- locity increase, and with it the resistance. At length, therefore, the resistance will have increased so much as to balance the in- crease of dynamical stability corresponding to the wave impulse; then the growth of oscillation ceases. As successive waves pass the ship after this result is attained, they each deliver their im- pulse as before, but their action is •absorbed in counteracting the tendency of the resistance to retard and degrade the oscillations. 55. Influence of resistance in the case of "permanent" oscilla- tions. — When a ship is rolling " permanently "" amongst waves, her oscillations having a fixed range and period, a similar balance will probably have been established between the wave impulse and the resistance; and here also the actual limit of range will fall below the theoretical limit given by the formula for unresisted perma- nent rolling on page 53. The influence of resistance may, in this case, be viewed as similar to that of a reduction in the steepness of the waves; this diminished slope taking the place of what has been termed the " efEective slope " for unresisted rolling. 56. Steadying influence of bilge-keels. — The conclusion to be drawn then is that any means of increasing the fluid resistance to the rolling of a ship tends to limit her maximum oscillations among waves. Circumstances may and do arise in the designing of warships which make it difficult, if not impossible, to associate requisite qualities with the long still-water period which theory and observation show to be favorable to steadiness; and in such cases bilge-keels are frequently fitted, for, as explained in Chapter II, in their use is found one of the most convenient and efEective methods of increasing the resistance to rolling, and their employ- ment is most efEective in small ships of short period. The Oscillations of Ships 5? The form of the immersed part of a ship and the condition ol her bottom also very considerably affect the aggregate resistance, both of these conditions being included in the determination of the co-efficient of resistance to rolling; but the form of a ship is planned by the naval architect mainly with reference to its sta- bility, carrying power, and propulsion, the consideration of its resistance to rolling being a subordinate feature, and best effected by leaving the under-water form of the ship herself unaltered, and simply adding bilge-keels in cases where the size and inertia of the ship are such as to make them useful, and when the condi- tions of the service of the ship, the sizes of the docks she has to enter, or other circumstances permit their use. 57. Pitching and 'Scending. — The longitudinal oscillations of pitching and 'scending experienced by ships among waves must now be briefly considered. In still water these longitudinal oscil- lations do not occur under the conditions of actual service; and it is difficult, even for experimental purposes, to establish such oscillations, because of the great longitudinal stability of ships. On this account we have little definite information respecting still-water periods for pitching, or the " co-efficients of resistance " for longitudinal oscillations. The formula for the period of unresisted pitching may be ex- pressed in the same form as that given on page 13 for the period of unresisted rolling, where m is, in this case, the longitudinal metacentric height, and k is the radius of gyration taken with reference to the transverse axis through the centre of gravity. The effect of the great longitudinal height more than counter- balances the effect of the increased moment of inertia for longi- tudinal oscillations, whence it follows that the period for pitching is usually considerably less than that for rolling, lying in many cases between one-half and two-thirds of the period for rolling. The existence of waves supplies a disturbing force capable of setting up longitudinal oscillations, leading to disturbances of the conditions of equilibrium as they exist in still -water, and to the creation of accelerating forces due to the excess or defect of buoy- ancy. 58 The Oscillation's of Ships The chief causes influencing the pitching and 'scending of ships amongst waves will be seen to bear a close analogy to what has been said in this regard about rolling, and will not require further explanation in detail. They are: (1) the relative lengths of waves and ships; (2) the relation between the natural period (for longi- tudinal oscillations) of the ship and the apparent period of the waves, this apparent period being influenced by the course and speed of the ship as previously explained; (3) the form of the wave profile, i. e. its steepness; (4) the form of the ship, especially near the bow and stern, in the neighborhood of the still-water load-line, this form being iniiuential in determining the amounts of the excesses or defects of buoyancy corresponding to the de- parture of the wave profile from coincidence with that line; (5) the longitudinal distribution of weights, determining the moment of inertia. In addition, fluid resistance exercises a most import- ant influence in limiting the range of the oscillations. This re- sistance is governed by the form of the ship and particularly by that of the extremities, where parts lying above the stUl-water load-line are immersed more or less as the ship pitches and 'scends and therefore contributes to the resistance. 58. Heaving Oscillations. — In addition to the transverse and longitudinal oscillations above described, a ship floating amongst waves has impressed upon her more or less considerable vertical oscillations, " heaving " up and down as the waves pass her. Tak- ing the extreme case of the small raft in Fig. 8, page 35, it will be seen that her centre of gravity performs vertical oscillations of which the amplitude equals the wave height, and the period is the wave period. This is termed " passive heaving."" A ship of large dimensions relatively to the waves obviously would not per- form such large vertical oscillations. The extent of these vertical movements would depend upon her position relative to the waves, her course and speed as afEecting the apparent period of the waves, the form of the effective wave slope, and the under-water form of the ship. Since, as we have seen above, the vertical and lateral extension of a ship into the wave structure has some influence on the effec- tive wave slope, the movement of the centre of gravitv of a ship The Oscillations of Ships 59 in passive heaving is considered as similar to tliat of a particle in the wave lying on the trochoidal surface passing through the centre of buoyancy, when broadside-on to the waves. When end- on to the waves, it has been proposed to take the mean of the vertical motions in the efEective wave slope which she covers; but when dealing with longitudinal oscillations this is not considered as of much practical value. Speaking generally, it may be said that the vertical oscillations of a ship result from the operation of excesses or defects of buoy- ancy due primarily to the passage of the waves and their troch- oidal forms, but actuated in many cases by the transverse and longitudinal oscillations. Heaving motions are favorable to sea- worthiness and safety, since they make it less probable that waves will break on board. A ship so small as to practically accompany the vertical component of the wave-motion — riding " like a duck " — makes good weather. "When ships are beam-on to the sea and roll but little, heaving motions of considerable extent may take place, and amongst large waves these motions may be nearly as great as those of surface particles. When ships are end-on to waves, although passive heaving may have a small amplitude, the effect of pitching and 'scending accompanying the vertical move- ment of the centre of gravity of the ship may be considerably in- creased having an important bearing on the longitudinal bending moments. It may be remarked, before leaving this discussion of the mo- tions to which a ship may be subjected that in the actual behavior of ships at sea all of these different kinds of oscillations may be occurring simultaneously, and may mutually influence one an- other. Careful observations alone can decide upon their absolute values and ordinarily rolling motions alone are considered worthy of observation. 59. Observations of rolling and pitching of ships. — Without at- tempting any extended discussion of the numerous methods of observing rolling motions of ships at sea, some of which involve the use of elaborate instruments, a few brief remarks will be made upon two methods employing such simple apparatus as is usually found on boaxd ship or may be easily rigged. (■'> The (Js( illations ok Ships These methods are (1) t)ie use of pendulums in various forms of clinometers, these pendulums having periods of oscillation which are very short as compared with the periods of the ships; and (2) the use of sighting battens arranged so that the angle of inclination of the shii^ to the visible horizon may be directly ob- served. 60. Errors of pendulum observations in still-water rolling. — The use of the pendulum, however^ is not well adapted to obser- vations of rolling, and it is the purpose of these remarks to point out its defects, since they are often not appreciated Ijy him whiisc attention has not been called to them. When a ship is held at a steady angle of heel (as shown by Fig. 3, p. 20) a pendulum -u^pended in her will hang vertically, no matter where its point of suspension may be placed, and will indicate the angle of heel correctly. The only force then acting on the penditlum is its weight, i. e. the force of gravity, the line of action being vertical. But when instead of being steadily inclined, the ship is made to oscillate in still water, she will tttm about an axis passing through or very near to the centre of gravity; hence every point not lying in tlie axis of rotation will be subjected to angular acceleration. Supposing the point of suspension of the pendulum to be either above or below the axis of rotation, it mil be subjected to these accelerating forces as well as to the force of gravity, and at each instant, instead of placing itself vertically, the penduhnn will assume a position determined by the resultant of gravity and the accelerating force. If the period of the pendulum used is short as compared with the period of the ship, the position towards which it tends to move -will probably be very nearly reached at each instant. The case is. in fact, similar to that represented in Fig. 12, p. 53. If the length of the upper pendulum, AB. is sup- posed to represent the distance from the axis of rotation of the ship to the point of suspension of the pendulum which is intended to denote her inclinations, the clinometer pendulum itself may be represented by BC. As AB sways from side to side the point B is subjected to angular accelerations, and these must be com- pounded with gravity in order to determine the position which BC will assume; for obviously BG will no longer hang vertically. The Oscillations of Ships 61 The angular accelerating force reaches its maximum when the extremity of an oscillation is reached, consequently it is at that position that the clinometer pendulum will depart furthest from the vertical position. In Fig. 12, suppose TAB to mark the ex- treme angle of inclination received by the ship, and let AB be produced to D : then, to an observer on board, the angle CBD will represent the excess of the apparent inclination of the ship to the vertical above the true inclination. It will be seen that the linear acceleration of the point of sus- pension B depends upon its distance from the axis of rotation A in Fig. 12. If B coincides with the axis of rotation, it is sub- jected to no accelerating forces, and a quick-moving pendulum hung very near to the height of the centre of gravity of a ship rolling in still water will, therefore, hang vertically, or nearly so, during the motion, indicating with very close approximation the true angles of inclination. Hence this valuable practical rule: when a ship is rolling in still water, if a pendulum is used to note the angles of inclination, it should be hung at or near the height of the centre of gravity of the ship; for if hung above that position it will indicate greater angles, and if hung below will indicate less angles than are really rolled through; the error of the indications increasing with the distance of the point of suspension from the axis of rotation and the rapidity of the rolling motion of the ship. 61. Errors of pendulum observations when rollingr among waves. — In the more complicated case of a ship oscillating amongst waves, the errors of pendulum observations are frequently still more exaggerated. The centre of gravity is then, as explained above, subject to the action of horizontal and vertical accelerating forces. If the pendulum were hung at the centre of gravity (G) of the ship, shown on a wave in Fig. 11, p. 45, it would therefore no longer maintain a truly vertical position during the oscilla- tions, but would assume at each instant a position determined by the resultant of the accelerating forces impressed upon it and of gravity. The direction of this resultant has been shown to coin- cide with that of the corresponding normal to the effective wave slope. Hence follows another practical rule. When a ship is rolling amongst waves, a quick-moving pendulum suspended at or 62 The Oscillations of Ships near to the height of the centre of gravity will place itself normal to the effective wave slope, and its indications will mark the suc- cessive inclinations of the masts to that normal, not their inclina- tions to the trne vertical. This distinction is a very important one. When hung at any other height than at that of the centre of gravity of a ship rolling amongst waves, the indications of a pendulum axe still less to be trusted. For a ship rolling amongst waves there is clearly no fixed axis of rotation, and the problem to be solved in discussing the possible errors of indication in a quick- moving pendulum hung at various heights in a ship is one of great difficulty. Any other devices, such as spirit-levels or mer- curial clinometers, depending for their action on the force of gravity or statical conditions, are affected by the motion of the ship as the pendulum has been shown to be affected. 62. Observations of rolling by use of battens. — The use of bat- tens affords the simplest correct means of observing the oscilla- tions of ships; they can be employed whenever the horizon can be sighted. The line of sight from the eye of an observer standing 6- n IINE OF SIGHT. ■ <3/g- JTJJ-J- W/TTEf^. _ on the deck of a ship to the distant horizon remains practically horizontal during the motion of the ship. Consequently, if a cer- tain position be chosen at which the eye of the observer will always be placed, when the ship is upright and at rest the horizontal line passing through that point is determined and marked in some way; this horizontal line can be used as a line of reference when the ship is rolling or pitching, and the angle it makes at any time The 0-f illations of >^hips. 63 with the line of sight will indicate the inclination of her masts to the vertical. In Fig. 13, the point E on the middle line marks the position of the eye of the observer, and at equal distances athwart- ships, two battens C and (!G are fixed so that when the shi)) is upright and at rest these battens are vertical, and at that time the line i^ii'i^vvill be horizontal. This maybe termed the "zero- line"'; and the points i^i^ would be marked on the battens. Sup- pose the figure to represent the case of a ship rolling among waves ; when she has reached the extreme of an oscillation to starboard, GF the angle GEF, whose tangent is given bv -- , measures the angle EF of inclination of the masts to the vertical. If the battens are placed longitudinally, instead of transversely, the angle of pitching may be similarly measured. It is a great joractical convenience to have the battens graduated to a scale of angles so that an observer can at once read off and note down the angles of inclination in degrees. This is simply effected by marking vertical distances above i^'on the battens equal to the distance EF t'nwes, the tangents of the successive angles. rf^^'fft•t^JWrLlWfW^W^■rRwii«2