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ROLLING OF

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Originally appearing in Volume V24, Page 946 of the 1911 Encyclopedia Britannica.
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ROLLING OF SHIPS] respectively. Froude gave his reasons for expecting the resistance to vary partly as the first and partly as the second power of the angular velocity. The latter part he considered would be due to surface-friction and the head resistance of keels and deadwood, and the former to the resistance caused by the creation of a small wave at each roll, which, by travelling away from the ship, would cause dissipation of energy. Froude's views have been confirmed by the accuracy with which .the expression —de=¢.A+b.e2 may be made to fit the curve of extinction of practically any ship by the judicious selection of the coefficients a and b. M. Bertin has, however, preferred an expression equivalent to —de while other French investigators have preferred an expression equivalent to de = a.0. do On substituting the value of a in equation (7) it becomes ('T 1 1— 22+13) . . . (8) a simplified form of the equation for resisted rolling when the coefficient b is neglected. For the " Revenge " the following equations represent the curves of extinction given in fig. 29: For deep draught : without bilge keels—d©=•01230+.002502 with „ „ —d®= •o65 0+•017 02 without bilge keels—d©=.015 O+.0028O2 with „ —d0 = •o84 0+•019 02 (0 in all cases being measured in degrees and not in circular measure). The large increase in the b coefficient after bilge keels had been fitted has given rise to considerable discussion. Mr R. E. Froude had experimented with a deeply submerged plane oscillating in water, and he found that at a speed of I foot per second the resistance per square foot was I.61b. Using this figure to calculate the work per swing from an extreme angle of 6°, the head-resistance of the bilge keels is found to account for about one-fourth the energy lost in a single swing due to the increased value of the b coefficient in the above formula. The energy abstracted in this particular case is thus about four times greater than the theoretical head-resistance of the bilge keels. This discrepancy has been observed in many cases, and it appears that when bilge keels are added to a ship they become effective, not merely as flat surfaces moving with the ship and experiencing direct resistances, but also by indirectly influencing the stream-line motions which would exist about the oscillating ship, if there were no bilge keels. Another cause of the difference is that the bilge keels during the early portion of the swing set into motion a large mass of water, only a small proportion of whose energy is returned to the ship towards the end of the roll. This condition is accentuated when the vessel is in motion ahead, and owing also to the increase of other resistances at high speeds, a more rapid extinction is then obtained. It appears from experiments made on H.M.S. " Revenge " and on a torpedo boat destroyer that the extinction at a given angle937 of roll is given by a linear formula —dB =a+,BV, where a and ,O are coefficients independent of the speed V.' Froude attacked the problem of resisted rolling in an inverse manner, endeavouring to ascertain " what wave-series is required to keep the given ship at a given range of steady rolling with any assigned period, including the effect of resistance.” Subsequently he treated the problem in a direct manner by the process of graphic integration,” an exact method of determining the motion of a ship, the elements of the ship's rolling in still water and the wave-series acting upon her being given.' Some interesting developments of the process were made by Sir William White in a paper read before the Inst. Nay. Arch. in 1881 on the " Rolling of Sailing Ships,” in which the action of the wind on the sails and the variation of the virtual weight of the ship on the wave are included. The effect of wind-pressure in heeling a ship is very much greater when she is at the crest of a wave than when she is at the trough, because her virtual weight is less. This must be taken into account when dealing with sailing vessels; the reduction of virtual weight, and therefore of righting moment, at the crest of a wave being very considerable, although the heeling moments due to the wind suffer no such reduction. The differential equation for rolling among waves including the effect of resistances varying as the first power of the angular velocity is— We dO+K2B+W (O—0i sin ,'rr—lt) =0, which becomes on substitution (K being expressed in terms of a) dB+T dt+T'O=T.O1sinTit' The general solution is- -at z e =Ae'rsin (T, • +R) +A10, sin (• r—#1} . (9) where 2 4a2 A (1 T-) 2 ' T—T-1 and 131=tan—1r (2aTTT2) and A and are arbitrary. The first term represents a free oscillation of the ship, which in time dies out, leaving a forced oscillation in the period of the waves. From observations on rolling, however, it is found that, owing to the departure from exact uniformity in the waves encountered, a ship seldom, if ever, completely forsakes her own natural period of rolls; for each slight alteration in the wave period Ti introduces afresh terms involving the free oscillations of the ship. In the synchronizing conditions where T=T1, the forced oscillation is represented by 0=—Z-Olcos'r—It., the amplitude being limited entirely by the resistance; the phase is 2 before that of the wave slope. The vessel is then upright in mid-height, and inclined to its maximum angle on the crest and in the hollow of the wave. The maximum amplitude 0 is given by 2.01=a.0. Since the right-hand term represents the decrement of roll due to resistance, the left-hand side must represent the increment of roll due to the wave in this synchronizing steady motion. If this latter relation be assumed to hold when the resistance to motion is represented by the more general decremental equation, then the maximum amplitude 0 is given by 201=a.0+b.02. In 1894 and 1895 M. Bertin, at the Institution of Naval Architects, extended this relation to cases in which Ti is not equal to T, obtaining at the same time not simply the angles of steady rolling for these cases, but the maximum angles passed through on the way to the steady condition; to these maximum angles he gave the name of " apogee " rolls. In 1896, at the Institution of Naval Architects, Mr R. E. Froude investigated the probable maximum amplitude of roll under the influence of a non-synchronous and non-harmonic swell. He imagined three identical ships, A, B and C, the first rolling in still water, and the two others placed in the same swell assumed recurrent in period 2T1, but not necessarily harmonic. Assuming resistance to vary as de, then denoting the vessels by suffixes, the effective wave slope by Ol, and constants by K, K' and K", d20A +KdBA dt2 at +K'eA = 0; d2Ba dOB dt +K dt ~K'en=K"el; d20e dOc dt2+K-T +KOc=K"Ol. ' See papers on this subject read before the Institution of Naval Architects in 1900 by Professor Bryan and in 1905 and 1909 by Mr A. W. Johns. 2 See Trans. Inst. Naval Arch., 1875. A miro ro , r .S ig a 8 E , c 'IM1111111•1111.111111al II ENEEMI.Mm ~, c 6P• M• K• 7P' /O' B' Q• P• ' D' Angle of roll (8) from vertical. -at O=AeT sin For light draught: 938 If at any instant den dec den it follows that d2BA d~ec ddes ate°ate-dt2' whence the above three relations hold at the successive instants and consequently for all time. Hence the rolling of C differs from that of B in having the free oscillations of A in still water superposed upon it. If, therefore, it is possible to obtain any one motion in the swell, any other motion due to a different phase relation between ship and wave slope can be at once determined. A convenient motion in the swell to form a basis for obtaining other motions is the forced oscillation proper to the swell, i.e. the particular oscillation that is recurrent in the period of the swell. The amplitude of roll at any instant is therefore the sum of the amplitudes due to the forced oscillation and to an arbitrary free oscillation in still water. If the latter component be regarded as perfectly arbitrary there is no limit to the angle of roll obtained by postulating suitable initial conditions; to determine the practical limitation of rolling, however, it may reasonably be assumed that at or near the commencement of the motion there will be a brief period of no roll, and that the maximum angle of roll obtained will occur at no great interval of time after this period. At the instant when there is no roll, the forced and free oscillations are equal in magnitude and opposite in phase, and the period of maximum (termed the " criterion ") amplitude O, will occur as soon as the two components are in phase; the time interval between the two conditions isnT, where n= Ti It is assumed also that during the above interval—(1) the effect of the swell was sensibly the same as that of a simple harmonic wave, A being the amplitude of the forced oscillation (and of the initial free oscillation) ; (2) the extinction equation of the free oscillation —=aO-hbO2 can be replaced by the simple form —do=EO, where E=a+be,, approximately; this has been implied by the absence of terms containing ((Ti 2 in the differential equation above. The amplitude of the free oscillation during the maximum roll is, from equation (8) Ae-' ; whence Oe=A(1+e-'e). Also, from equation (9), the forced oscillation is given by z z \\/11 01=A\ (1—T2+/ . 1,E 2 From these equations 01 can be determined if T, T,, a, b and Oc are given; conversely if Oc is known, 01 can be tentatively obtained. The following table gives the criterion angle (Oc) and the angle of steady roll (A) for the' Revenge," both without and with bilge keels, obtained on the above-mentioned assumptions:[ROLLING OF SHIPS Maximum Wave-Slope, 3 Degrees. =I.3. 7=1.2. n=1o. =I•I.=r•o. n=3'33• n=5. n=oo. e a q o g °a: 4ti o ou °c~ od °c4 ° .d u °c4 Ud es s, U~ et, , Ud n v d v <2 in m "Revenge" (deep deg. deg. deg. deg. deg. deg. deg. deg. draught), with 8.25 4.35 12.25 6.8 21.2 13.9 41.1 41.1 no bilge keels "Revenge" (deep 6.6 4.24 8-6 6.4 11.55 io•8 .14.85 14.85 draught), with bilge keels . read before the I.N.A. in 1896 and 1898, the whole motion of the ship, including pitching and rolling, is dealt with; every variation which can reasonably be conceived is taken into account in these papers. Of the various appliances adopted to reduce rolling, the most important and successful are bilge keels. Some reference has already been made to the influence they exert on the rolling of ships, as illustrated by H.M.S. " Revenge," in which there was one bilge keel on each side, 200 ft. in length and 3 ft. in depth, tapered at the extreme ends. The great value of bilge keels. in diminishing rolling was pointed out by Froude and demonstrated by him in 1872 by experiment with the " Perseus" and the " Greyhound," Methods of which were alike in every essential respect, except that the mducing former was not provided with bilge keels and the latter roman was. The general conclusion was that the rolling of the " Greyhound," was only about one-half that of the " Perseus." Bilge keels were usual in warships until, in the design of the " Royal Sovereign " class, it was decided not to fit them, owing to the large dimensions of the vessels and the difficulties in certain circumstances of docking them if provided with bilge keels. Ultimately one of the class, the " Repulse," had them fitted for purposes of comparison, and the effect on her rolling was so marked that it was resolved to fit them to all the ships of the class. Before fitting them on the " Revenge," a careful programme was drawn up of experiments to be made before and after the bilge keels were fitted; and on carrying out this programme some valuable results were obtained. The experiments were made at Spithead in smooth water, the general effect of the bilge keels was to reduce the rolling to one-third of its former amount. When, instead of. having no motion in the line ahead, the ship had a speed of 12 knots, an even greater reduction in the rolling was observed. Their effect on other qualities of ships is on the whole beneficial, and in general little, if any, reduction in speed has resulted from their use. The experience of Great Britain with regard to bilge keels has been repeated in America. Bilge keels were omitted for the same reasons as they were in the " Royal Sovereign " class; they were afterwards fitted in the U.S.S. " Oregon," experimental investigation being made both without and with them, and the general conclusion arrived at was that the rolling was diminished by two-thirds by the adoption of the bilge keels. A method for reducing. rolling of ships in a sea-way by- the use of water-chambers was devised by the writer in 1874 in connexion with the design of the " Inflexible," which was expected to be a bad roller. It consists in fitting one or more tanks across the ship of such shape that when filled to a suitable height with water the motion of the water from side to side as the vessel rolls is such as to retard the rolling. Let fig. 30 represent a series of transverse sections of a ship fitted with a water-chamber, in various positions in rolling from port to starboard; and suppose the water to move so as to be most effective in quelling rolling. Let G represent the centre of gravity of the ship including the water in the chamber, g the centre of gravity of the water in the chamber, and B the centre of buoyancy of the ship; and let the arrows over the sections indicate the direction in which the ship is rolling at the instant considered. In position No. i suppose the ship to have reached the extreme heel to port and to be on the point of commencing the return roll, then g should have reached the middle line on its way down towards the port side and the righting couple will be that due to the angle of heel, supposing the water to be a fixed weight amidships, In the position No. 2 the ship. has performed part of the roll back towards the upright; the water will have moved farther down the incline, so that g will be some distance from the middle line on the port side as shown, and therefore G will also have moved out from the middle line on the port side; hence the righting couple will be less than what would correspond to the angle of heel if the water were a fixed weight amidships. In position No. the ship has just reached the upright and will be moving with the maximum angular velocity; the water will have moved still farther down the incline, and g will be at a greater distance from the middle line on the port side, and therefore G will have moved farther out from the middle line, whereas B will have returned to the middle line; so that the weight of the ship and the upward pressure of, the water will form a couple tending to retard the ship's rotation, although she is for the moment in the upright position. In the position No. 4 the ship is heeling over to starboard and the centre of gravity of the water is returning towards the middle line; but it and G are still on the Among the conclusions reached by Mr R. E. Froude in the case of a ship rolling in a uniform swell were: However non-uniform initially, the rolling ultimately falls into the uniform forced oscillation; it does so the sooner, ceteris paribus, the higher the resistance, and with the fewer " cycles" or alterations of amplitude of rolling, the more nearly synchronous the swell with the ship. The amplitude of the ultimate uniform rolling is an approximate mean of the alternate maxima and minima of the precedent non-uniform rolling. If the rolling starts from zero, the maximum amplitude falls short of twice the ultimate uniform amplitude, the more so the higher the resistance and the more synchronous the swell; and in a synchronous swell the maximum ampli- tude cannot exceed the ultimate uniform amplitude, unless it does 1 port side, and the righting couple is therefore greater than that so initially. corresponding to the angle of heel of the ship and a fixed centre of In two papers by Captain and Professor Kriloff of St Petersburg, gravity amidships. In the position No. 5 the ship has momentarily come to rest at the end of the starboard roll, the centre of gravity of the water should have again reached the middle line, and the righting couple should be neither increased nor diminished by the water-chamber, except in so far as it affects the displacement and the vertical position of the centre of gravity. The same process is repeated on the ship's roll back from starboard to port. Thus the water-chamber reduces the angle of roll of the ship chiefly by modifying the righting couple acting upon her throughout the rolling; it increases the righting couple which opposes the motion as the ship heels over, thereby reducing the amount of the heel, and on the return roll it lessens the righting couple and causes the ship to move more slowly than she otherwise would, so that she acquires less angular momentum on reaching the upright, and therefore tends to roll less deeply the other way. Two water-chambers were originally contemplated in the old Inflexible, but the space occupied by one of these was required for other purposes, and only one, the smaller of the two, which was 51 ft. long (across the ship), and 14 ft. wide (fore and aft), was final, fitted. This was shown to reduce the rolling by about 25 %. Several ships have since been fitted with this device., In addition to trials at sea to ascertain the diminution of roll by this means, still-water rolling experiments were carried out in the " Edinburgh " and compared with the results obtained with a model water-chamber on a linear scale of 2O 5 loaded so that its period and stability corresponded to those of the ship. A close agreement was observed between the behaviour of the model and the ship; and this enabled the experiments to be carried out over a larger range of conditions than would have been practicable with the ship alone. The model was supported on knife edges and connected to a paddle partially immersed in the water of a tank; this was adjusted to represent to scale the natural extinction of roll in the ship without 20 ' 15 10 Angle of Roil in degrees the water-chamber. The length of the chamber (in the ship) was 16 ft.; and widths of 43 ft., 514 ft. and 67 ft. were successively given to it. The displacement of the ship was about 7500 tons; the period to seconds; and the metacentric height 7.52 ft. On experimenting with different depths of water, it was found that the maxi-mum extinctive effect at all angles of roll was obtained with the depth at which the period of motion of the water from side to side of the tank is equal to the period of the ship. The best depths were found to be 2.3 ft. and 3.35 ft. with breadths of 43 ft. and 512 ft. respectively, thus agreeing closely with the theoretical formula, v = sigh, for the speed of a solitary wave across the water-chamber. In these circumstances the water rushed across the tank in a breaking wave or bore, and consumed energy in its passage and through its violent impact with the sides of the tank. With other depths, the motion of the water, at moderate angles, took the form of a slope gently alternating from side to side at small angles of roll ; and the effect was practically non-extinctive. With the critical depth the growth of the resistance to rolling commenced almost at zero angle; but, with other depths, the extinction was nearly nil, until a certain angle of roll was attained, whose amount increased with the departure from the critical depth. At the larger angles of roll, the disadvantage of the departure from the critical depth was not marked. The resistance of the chamber increased considerably with the breadth; the value of the 512-ft. chamber was roughly twice and that of the 67-ft. chamber three times that of the 43-ft. chamber. In order to compare the effect of water-chambers with that of other methods of extinction, it is observed that the resistance due to the former increases slowly at large angles of roll. The effectiveness of bilge keels, on the other hand, increases rapidly as the angle of roll increases. It was found that, with 12° roll, the resistance of the water-chamber was equivalent to that of 2 ft. of additional bilge keel; but at 172° the water-chamber was relatively about half as effective. With 30 of roll, however, the water-chamber was about 9 times as 4 See paper on " A Method of Reducing the Rolling of Ships at Sea " in Trans. Inst. Nay. Archs. 1883.effective as the additional bilge keel. Fig. 31 shows the comparative rates of extinction under the various conditions? Water-chambers have been successfully employed to limit the rolling motions at sea in ships of the old " Inflexible," " Edinburgh " and " Admiral " classes, and in other warships and merchant vessels. Sir John Thornycroft devised an arrangement for overcoming the rolling motion of a ship amongst waves, consisting of a weight carried from side to side so as always to oppose the heeling couple caused by the wave slope. The weight was automatically worked by apparatus controlled by two pendulums (or their equivalent), one of which—a long period pendulum—remained vertical, and the other —a short-period pendulum—placed itself perpendicular to the effective wave slope. The gear was fitted on a yacht of about 230 tons displacement, the moving weight being 8 tons; and the net effect in this case was to reduce the rolling by about one-half. (See Trans. Inst. Nay. Archs. 1892.) An interesting application of the gyroscope to the diminution of rolling was devised by Dr O. Schlick, and fitted by him to the S.S. " See-bar." The principle of its action, the details of the gear, and a description of the trials are given in papers read before the Inst. Nay. Archs. in 1904 and 1907. Particulars of the " See-bar " were : length 116 ft., breadth 11.7 ft., draught 3.4 ft., displacement 56 tons, meta-centric height 1.64 ft., and period of double roll (gyroscope at rest) 4.14 seconds. The fly-wheel of the gyroscope was one metre in external diameter, weighed 1100 1b, and it was run at i600 revolutions per minute; its axis was initially vertical, and the casing containing the wheel was capable of revolving about a horizontal athwartship axis, the centre of gravity of the apparatus lying slightly below this axis. A brake was fitted to control the longitudinal oscillations of the casing. When the wheel was revolving and the axis held by the brake, no effect was produced upon the motion of the ship ; but when the axis was allowed to oscillate freely in the middle-line plane the period of roll was lengthened to 6 seconds, but no other extinctive effect was obtained. By suitably damping the longitudinal oscillations of the gyroscope, however, by means of the brake, a large extinctive effect upon the rolling was experienced ; and during the trials made, the apparatus stopped practically all rolling motion. The equations for the pitching motions of a vessel are identical in form with those for rolling ; and the preceding remarks are, in general, equally applicable to pitching. In a large number of ships Pliehiag the period for pitching is approximately one-half of that and for rolling; but the angles attained are considerably less. heaving. Where control over the longitudinal positions of weights is possible, e.g. in small sailing vessels, weights are removed as far as possible from the ends in order to shorten the period, the safety of short ships and boats being secured when the deck is maintained as nearly as possible parallel to the wave slope (v. remarks by Froude ante). The single period for heaving and dipping oscillations is equal to x ~T„ when W is the displacement in tons, and T" the tons per inch immersion. When proceeding across waves of apparent semi- Resistance. The resistance of a ship in steady motion, or the force exerted by the surrounding water on the hull, opposing its progress, is equal and opposite to the thrust of the propellers. The ship is subjected to a system of balanced forces, each of which is in some degree affected by the others. It is convenient, however, first to confine attention to the resistance of the hull, assuming the See paper entitled " The Use of Water-Chambers for Reducing the Rolling of Ships at Sea," Trans. Inst. Nay. Archs. 1885. G \ A \ as 25 5 0 2 period T1, forced heaving oscillations of semi-amplitude aTi-21 s are obtained, where T is the single period of dip, and 2a is the vertical distance between the statical positions of the ship on crest and in trough of wave. These oscillations combine with the free dipping oscillations due to the circumstances of the initial motion, the resultant motion being of interest in connexion with the longitudinal bending moments in the ship caused by the waves. (See section Strength.) Pitching or rolling is frequently the cause of dipping oscillations, and the motion is then termed uneasy; this action may be of importance in ships whose sides near the water-line have a considerable slope to the vertical, since any rolling motion is then accompanied by vertical oscillations of the centre of gravity. It may also be shown that forced dipping oscillations of considerable amplitude are obtained when the period of roll (or pitch) in such cases approximates to twice the dipping period ; the complex nature of the resistances attending the motion of the ship has, however, prevented a complete investigation being made. Interference also occurs between the rolling and pitching movements of a ship, when the centres of gravity of the wedges of immersion and emersion for moderate angles of heel are separated by a considerable distance longitudinally; and occasionally uneasy rolling of a peculiar character is caused thereby. propeller to be removed, and the ship to be towed through undisturbed water. Under these conditions the power expended in towing the vessel is termed the effective horse power, and is considerably less than the indicated horse power exerted by the propelling engines at the same speed. The relation between the effective and indicated horse powers, and the effect of the propellers on the resistance of the ship will be discussed under Propulsion, below. If a body of " fair " form, i.e. without abruptness or discontinuity in its surface, moves uniformly at a considerable depth below the surface of an incompressible and perfect fluid, it can be shown that no resistance is experienced, and the uniform motion will, caeteris paribus, continue indefinitely. The motion of the fluid is extremely small, except in the close vicinity of the body. A clearer conception of the interaction of fluid and body is obtained by impressing upon the whole system a velocity equal and opposite to that of the body, which then becomes motionless and is situated in a uniform stream of the fluid. The particles of fluid move in a series of lines termed " stream lines "; and the surface formed by all the stream lines passing through a small closed contour is termed a " stream tube." If a denote the area of a stream tube, assumed sufficiently small for the velocity v at a point within it to be sensibly uniform across a section, then, since no fluid is leaving or entering the tube, a.v =constant throughout its length. The motion of the fluid is also subject to Bernoulli's energy quotation- +2g +h= constant, p, w and h being respectively the fluid pressure, the density and the height above a fixed datum. The remaining conditions affecting the flow and determining the forms of the stream lines are purely geometrical, and depend on the form of the body. The motion in a perfect fluid flowing past bodies of a few simple mathematical forms has been investigated with success, but in the general case the forms of the stream lines can only be obtained by approximate methods. It is evident that the flow is in all cases reversible since the equations are unaltered when the sign of v is changed; on the other hand any resistance must always oppose the motion, and therefore, as stated above, there can be no resistance under these conditions. The circumstances attending the motion of a ship on the surface of the sea (or that of a stream of water Corn- flowing past a stationary vessel) differ from those hitherto asponentsof sumed; and resistance is experiresistance. enced through various causes. Frictional resistance results from the rubbing of the water past the surface of the hull; eddy resistances are caused by local discontinuities, such as shaft brackets; and resistance due to wind is experienced on the hull and upper works. Moreover, the stream-line motion, as will be shown later, causes a diminution in the relative velocity of the water at the ends of the ship; from the energy equation above, it is evident that the pressure is increased, resulting in an elevation of the surface of the water at those places. A wave is thus formed at the bow and stern, requiring an expenditure of energy for its maintenance and entailing additional resistance. Of these components of resistance, that due' to eddy making is usually small; eddying is caused by blunt beginnings or endings, particularly the latter, in the water-lines and underwater fittings. Air resistance also is generally of small importance; in the " Grey-hound " (unrigged) it constituted 1.4 % of the total resistance at to knots to calm weather, and in a large Atlantic liner at 25 knots it absorbs about 4% of the total power. In the case of average ships, unrigged or with moderate top-hamper, the proportion of air resistance is probably less than the latter value. The effect of wind and rough weather on the speed of ships is also largely due to the action of the waves and other motion of the sea, the additional effect of which is indeterminate. The difference between the total resistance and that due to skin friction is termed the residuary resistance ; from the foregoing remarks it appears that it consists principally of the resistance due to wave-making. Since the action of the waves is such as to distort the stream lines near the hull, and the form of the waves is in turn affected by the frictional wake, the frictional and wave-making resistances of a ship are to some extent mutually dependent. It is convenient, how-ever, to neglect the interaction of these constituents, and to assume that the whole resistance is obtained by simple summation of its component parts as calculated independently. Considerable justification for this assumption is furnished by the close agreement between the results of experiments on models and on ships, where the proportion of frictional to total resistance is greatly different. Since the action and the reaction of the water pressure on the hull of a ship are equal and opposite, forward momentum is generated in the water at such a rate that the increase of momentum wake. per second is equal to the total resistance. The water participating in the forward movement is termed the wake; the portion of the wake in the vicinity of the propellers is found to have consider-able effect upon the propulsion of the ship. Experiments *ere made by Mr Calvert (Trans. Inst. N.A. 1893) to determine the wake velocity with a model of length 281 ft. and displacement 2.9 tons. The extent of the wake was measured at various positions in the length, and its maximum velocity was observed to be 0.67 times the speed of the ship. Abreast the screw the mean velocity ratio over an area of the same breadth (3.66 ft.) as the ship and of depth equal to the draught (1.55 ft.) was 0.19, of which about 0.05 was ascribed to frictional resistance. In Rep. Brit. Assoc. 1874 is contained an investigation by Fronde of the extent of the frictional wake and its velocity distribution based on the equality of the resistance to the momentum added per second. It may be here observed that for any ship propelled in the ordinary manner at uniform speed the momentum generated in the sternward race from the propeller is equal and opposite to that of the forward wake due to the hull. The motion of the water as a whole thus consists of a circulatory disturbance advancing with the ship, and having no linear momentum. The whole of the resistance at low speeds, and a considerable proportion of it at higher speeds, is due to surface friction, i.e. to the eddying belt surrounding the hull which is caused by the Frictional tangential frictional action between the water and the out- resistance. side skin. It is nearly independent of the form of the vessel ; and is conveniently estimated from the results of experiments made by towing in a tank planks coated with various surfaces. The most important of such experiments were those made by Froude in the experimental tank at Chelston Cross, Torquay. The object was to obtain the laws of variation of resistance with the speed, the length, and the quality of the surface. A dynamometric apparatus by which the planks were towed was used to register the resistance; the planks were given a fine edge at each end to avoid eddy making, and were fully immersed in order that no waves should be formed. The results are given in the Reports of the British Association, 1872 and 1$74. In the following extract n is the index of the speed at which the resistance varies, A the mean resistance per square foot of surface over the length stated, and B the resistance per square foot at the after end of the plank; both A and B refer to a velocity of 10 ft. per second in fresh water. Length of Surface in Feet. These results are in-.accordance with the formula-R=fwS• - ; R being the frictional resistance, S the area of surface, V the speed, w the density of the water, f a coefficient depending on the nature and length of the surface, and n the index of the speed; the values of f and n can be readily obtained from the above table. It is seen that the resistance varies as the density of the water, but is independent of its pressure; it diminishes as the length of the surface increases, on account of the frictional wake, which reduces the velocity of rubbing between the water and the surface towards the after end. The index n is 1.83 for a varnished surface equivalent to the freshly painted hull of a ship. The results of Froude's experiments are closely corroborated by similar experiments under-taken by the late Dr Tideman. When applying the data.to ships of length greater than 50 ft., the coefficient B, denoting the resistance 50 ft. from the bow, is assumed to remain unaltered at all greater distances astern. The velocity of rubbing is assumed equal to the speed of the ship, any slight variation due to stream-line action being neglected. The wetted surface S, when not directly calculated, can be estimated with sufficient accuracy by the formula- S =1.7LD+D where V is the volume of displacement, L the length, and D the mean draught. The resistance due to wave making, although inconsiderable at low speeds, is of importance at moderate and at high wave speeds; it constitutes the greater portion of the total resistance. resistance in fast ships. 2 ft. 8 ft. 20 ft. 50 ft. n. A. B. n. A. B. n. A. B. n. A. B. Tinfoil . . . 2.16 .30 .295 1'99 •278 .263 1.90 •262 .244 1.83 .246 .232 Paraffin . . 1.95 .38 .370 1.94 .314 '260 1.93 '271 .237 •• •• Varnish . . 2.00 •41 .390 1.85 .325 .264 1.85 •278 .240 1.83 .250 .226 Fine sand . . 2.00 •81 .690 2.00 .583 .450 2'00 '480 .384 2.06 .405 .337 Calico . . . 1.93 .87 '725 1.92 •626 .504 1.89 .531 .447 1.87 .474 .423 Medium sand . 2.00 .90 .730 2.00 •625 .488 2'00 .534 .465 2.00 .488 .456 Coarse sand 2.00 1.10 •88o 2.00 .714 .520 2'00 '588 '490 .. .. By impressing, as above, a suitable velocity on the whole system of ship and water, the problem is reduced to one of steady motion in a stream flowing past a stationary ship. The stream tubes, originally of uniform width, become broader on approaching the bow of the ship, and attain their greatest breadth close to the stem. Proceeding aft, the tubes contract, and near amidships they become smaller than they were originally; an enlargement in the tubes again takes place near the stern. The changes in size and velocity in the stream tubes lead to corresponding alterations of pressure in accordance with the energy equation, which alterations appear as elevations and depressions of the surface forming what is termed the statical wave system. If this were a permanent system, no resistance to the motion of the ship would be caused thereby. The surface disturbance, however, is subject to the dynamical laws underlying the propagation of waves; in consequence the wave formation differs from the " statical wave," the crest lagging astern of the " statical " wave crest, and the ship being followed by a train of waves whose lengths are appropriate to the speed attained. The energy within the wave system travels backward relative to the ship at one-half its speed; the resistance experienced by the ship is due to the sternward drain of the wave energy which requires work to be done on the ship to replace that absorbed by the waves. The form of the wave system is not susceptible of complete mathematical investigation; but the circumstances are approximately realized and the conditions considerably simplified when the actions of the bow and stern of the vessel are each replaced by the mathematical conception of a " pressure point." This consists of an infinitely large pressure applied over an indefinitely small region of the water surface; it is assumed to move forward in place of the ship through still water, or, equally, to be stationary in a uniform stream. The resulting wave system has been investigated by Lord Kelvin and others. It is found to consist of a local disturbance surrounding the pressure point and depending on the pressure distribution combined with a series of waves which are confined within two straight lines drawn backwards through the pressure point and making angles of about 20° (tan`12 , 2) with the line of motion. The waves within this region extend indefinitely astern with crests crossing the line of motion perpendicularly. The crest lines are slightly curved, convex to the pressure point, and at the bounding lines form cusps whose tangents are inclined to the line of flow at an angle of about 36° (tan 1 2) . The crest lines afterwards curve forward towards the pressure point. The distance apart of the transverse wave crests is equal to the length 1 of wave appropriate to the speed v, as expressed in the formula v2=gl/22r. These results are of interest since they are in agreement in many respects with those of actual observation for ships and models. In fig. 32, reproduced from a paper in the I.N.A. 1877, read by Froude, is shown the bow-wave system obtained from a model, which is also illustrative of that produced by ships of all types. It appears therefore that two types of waves accompany a ship—(1) diverging waves having sharply defined crests placed in echelon, the foremost wave alone extending to the ship; (2) transverse waves limited in breadth by the diverging crests and reaching the sides of the vessel through-out its length. These compare with the crest lines obtained in the above hydrodynamical investigation ; the transverse and diverging waves correspond to the different portions of the crest lines which are separated by the cusps. Since the bow diverging waves are not in contact with the ship except at the bow, the energy spent in their maintenance travels away from the ship and is lost. A diverging wave system of similar form but of smaller dimensions attends the passage of the stern; and the resistance due to the diverging systems of waves is therefore the sum of its components at the bow and stern, following a regularalthough unknown law, increasing with the speed, and depending considerably on the shape of the bow and stern. On the other hand the interference between the transverse bow and stern wave systems produces a stern wave in contact with the ship; the resistance due to the resultant transverse wave system depends therefore on the phase relation between the waves of the component systems. The effect of interference on the wave resistance was investigated by Fronde (Trans. I.N.A. 1877) by means of experiments on a series of models having the same entrance and run, but in which the length of parallel middle body was varied. At constant speed curves of residuary resistance on a length base consisted of humps and hollows, whose spacing was constant and approximately equal to the wave length appropriate to the speed ; the amplitude of the fluctuation diminished as the length increased. For a given length the residuary resistance in general increased at a high power of the speed; but it was also subject to a series of fluctuations whose magnitude and spacing increased with the speed. The results of these experiments were fully analysed in 1881 by Mr R. E. Fronde, who showed that a reduction in the resistance occurred when the trough of the bow wave coincided with the crest of the component stern wave, the resultant wave system being of relatively small dimensions. Conversely, the resistance was abnormally, increased when the crests of the bow and stern systems coincided. he fluctuation in the resistance thereby obtained was smaller when the length of middle body became greater, owing to the greater degradation of the bow wave system at the stern through viscosity and lateral spreading. For very considerable lengths of middle body, the height of the bow wave system at the stern was insufficient to produce interference or affect the resistance. The speed in knots (V) of a wave is related to the length in feet (1) by the formula V2=1.81. If L' be the distance apart of the component bow and stern waves (which is generally rather greater than the length of the ship), relatively small resistance would be anticipated when V2 is approximately equal to 3.6 L' or any odd sub-multiple of 3.6 L'; on the other hand when V2 was not greatly different from 1.8 L', or any submultiple of 1.8 L', abnormal wave resistance would be developed. This result is to a great extent con-firmed by experience with ships of all classes; for economical propulsion at a speed V, the length L of a ship should be generally equal to or slightly less than V2, corresponding to the " favourable generally 2 value of about 1.2 of the ratio torpedo-boat destroyers and similar vessels of extremely high speed constitute an exception, the 2 value of the ratio being then frequently as great as 4, which ap- 2 proximately coincides with the highest "favourable" value of L,. The foregoing description of the resistance experienced by ships through wave making makes it evident that the conditions under-lying wave resistance are too complex to enable its amount Lew of to be directly estimated as is possible in the case of frictional resistance. Experiments also show that there is Com no simple law connecting wave resistance with size, form parison. or speed. The effect of size alone, i.e. the scale of the experiment, can, however, be eliminated by means of the " principle of similitude " enunciated by Newton, which is applicable with certain limitations to all dynamical systems. The extension of this principle forms the foundation of all methods employed practically for estimating the residuary resistance and horse power of ships. The principle states that in two geometrically and mechanically similar systems, whose linear dimensions vary as the squares of the velocities of the corresponding particles, and whose forces vary as their masses, the motions of the two systems will be similar. A proof of this theorem follows at once from the equations of motion for any particle. The law of comparison, which is the application (origin-ally made by Fronde) of the principle of similitude to the resistance of ships, is enunciated as follows: " If the linear dimensions of a ship be n times those of its model, and the resistances of the latter be R1, R2, Rs, . at speeds V1, V2, Vs, . . . , then the resistances of the ship at the 'corresponding speeds' Vi Jn, V2 4n, Vs 4n, ... will be R,ns, R2n', Rsns, .. . and therefore the effective horse powers at corresponding speeds are increased in the ratio n'i: 1." It is necessary to ensure that the conditions underlying the principle of similitude are satisfied by all the components of resistance, when the law of comparison is employed for the purpose of obtaining the ratio between the total resistances of two ships at corresponding speeds. Residuary resistance, consisting of that caused by wave making, eddies, and air resistance, is attributable to normal pressures on various surfaces caused by changes of velocity in the water or air. It appears from Bernoulli's energy equation that the pressures per unit area are proportional to the square of the velocity, i.e. at corresponding speeds, to the linear dimensions. The total pressures are therefore proportional to the cube of the linear dimensions, i.e. to the masses, thus complying with the primary condition regarding the force ratios. Frictional resistance, which varies with the length of surface and as the 1.83 power of the speed, does not satisfy this condition. In the application of the law of comparison to ships and models where the linear ratio is considerable, the residuary resistance alone should be compared by that means, the frictional resistance being independently calculated for ship and model from the results of Froude's experiments. The law may, however, be extended with-out appreciable error to total resistance when the corresponding linear dimensions of the ships compared are not greatly different. If it be assumed that the residuary resistance of a ship is capable of being expressed as the sum of a number of terms of the form W'"V", where W is the displacement, it appears from the law of comparison that 6m+n=6 for each term of the expression; and in the construction of approximate formulae of this type for residuary resistance, the indices m and n must satisfy this equation. The values of the indices are found to vary irregularly with the speed and type of ship; at uneconomical speeds n may be equal to or greater than 5, and at " favourable " speeds its value may be as low as 1.5, 4 being an approximate mean value for n at moderate speeds. A fact pointed out by Professor Biles in a paper read before the Institution of Naval Architects in 1881 is interesting in this connexion. When the resistance of a ship varies as the 6th power of the speed, an increase in the displacement by a proportionate enlargement of dimension will not cause an increase in the resistance for the same speed ; and if the resistance varied as a higher power of the speed than the 6th, the resistance would actually be reduced by increasing the displacement. The accuracy of the law of comparison was verified by the " Grey-hound " resistance experiments carried out by Froude on behalf of the Admiralty (Trans. I.N.A., 1874). The " Greyhound " was a twin-screw sloop 170 ft. long and of about t 16o tons displacement; the trials were made over a range of speeds extending from 3 to 121 knots, and with varying draught and trim. She was towed from the end of a spar 48 ft. in length projecting over the side of the towing vessel, H.M.S. " Active "; this ensured that the wave system and wake of the " Active " were prevented from reaching the " Greyhound " and influencing her resistance. A dynamo-metric apparatus was placed in the bow of the " Greyhound, " and arranged so as to record the horizontal component of the tension in the tow rope; by this means the ship's resistance was measured under various conditions and her effective horse-power obtained. A " log ship" or small board, ballasted to sink a few feet and remain normal to the direction of the pull, was attached to the end of a log line which was allowed to run freely out over the end of a spar during the trials. The slip of the " log ship " having been obtained during independent trials, the speed of the " Greyhound " was estimated from the log-line readings with fair accuracy. From these results curves of resistance on a base of speed were constructed for various conditions of draught and trim; the frictional resistance was estimated from the experiments on planks, and curves of residuary resistance were obtained. A model of the " Greyhound," on a scale ofs full size, was also towed in'the experimental tankunder conditions corresponding to those of the ship; as with the ship, the total resistance was measured, that due to friction was calculated, and the residuary resistance of the model was obtained. It was found, by assuming a particular value for the unknown frictional coefficient of the " Greyhound," that a close agreement occurred between the residuary resistances of ship and model. This coefficient corresponded to that fora mixture of 1 calico and 1 varnish, which was probably equivalent to the condition of the ship's bottom during the trials. Similar experiments were carried out by Mr Yarrow (Trans. I.N.A., 1883) on a torpedo boat too ft. long; it was found that the residuary resistance of the boat was then about 3% in excess of that deduced by the law of comparison from experiments on a model. As a result of the " Greyhound " trials, the accepted method of estimating the horse-power required for a new ship is by running a Model scale model under corresponding conditions in an ex- experi- perimental tank fitted and equipped for the purpose. meats. The law of comparison is applied to the residuary resistance, or, if used for the total resistance, a " frictional correction' is made (see below). In 1871 Froude constructed a tank and suitable apparatus at Torquay on behalf of the British Admiralty. In 1885, six years after his death, the ground occupied by the Torquay tank was required for building purposes, and a new tank was constructed at Haslar, near Portsmouth, from the designs and under the super-vision of Mr R. E. Froude, such improvements being added as experience at Torquay had shown to be desirable. At both these tanks models of propellers as well as of ships were experimented upon, besides a variety of matters connected with the general subject. Similar establishments have now been instituted by several foreign governments and by two private firms in Great Britain, Messrs Denny at Dumbarton and Messrs John Brown at Clydebank. The experimental tank now under construction at Teddington should prove an important and useful addition to the number of such installations in this country. It is intended to be used for general research and to be available also for undertaking such private work as may be required by shipbuilding firms. Its inception is due to a committee composed largely of members of the Institution of Naval Architects, and the cost of installation is being defrayed by Mr A. F. Yarrow. The tank will form a part of the National Physical Laboratory',and its general control will be in the hands of officers of the laboratory. The Admiralty experimental tank at Haslar is nearly 400 ft. long, 20 ft. wide and 9 ft. deep. The main experimental carriage spans the whole width of the tank, and carries a secondary railway on which the subsidiary carriages, which carry the experimental apparatus of different kinds, are adjusted in position. The main carriage runs on rails on the side walls, and can travel the whole length of the tank; it is driven at various speeds by a wire rope from a stationary engine of ample power. Ordinary speeds range from 100 to 800 ft. per minute, while an extreme speed of 1200 ft. per minute can be obtained; the speeds are regulated by a highly sensitive governor. The models, generally from to to 14 ft. long, are made of hard paraffin wax, somewhat overt in. in thickness; they are cast in a mould, with an allowance of about 4 in. for finishing. The model is shaped accurately by being placed bottom up on the bed of a machine in which a pair of revolving cutters, one on each side of the model, cuts out on its surface a series of level lines, whose contours are precisely similar to those on the drawing of the ship whose model is under treatment. When all the level lines have been cut in, the model presents the appearance of a series of steps, the bottom angles of which correctly represent the true form the model should possess. The paraffin ridges between these level lines are trimmed off by the use of suitable tools and the outside surface made quite smooth with flexible steel scrapers. The model is ballasted to its required displacement and saddled with a frame, which carries the guiding attachment and also the towing-rod, and is then placed below the dynamometer. The towing-rod at its for-ward end is then in a position to impart horizontal forces bya hard steel surface to a knife-edge on the dynamometer lever within the model at about the level of the water surface. There are various delicate arrangements with knife-edge adjustments, which result in the horizontal forces being transmitted through a spiral spring, the extensions of which are multiplied by a lever and recorded by a pen on a paper-covered cylinder, distance and time being simultaneously recorded. The speed and resistance corresponding to each experiment are deduced from these elements, a most necessary condition being that the speed shall be uniform throughout each experiment. By somewhat similar arrangements on a subsidiary carriage, the action of model screw propellers is tested either in undisturbed water or behind a model, the speed, rate of rotation, rotary resistance and thrust being measured. An interesting account by Dr Glazebrook of some experimental tanks in various countries, together with particulars of some improvements in their equipment, appears in Trans. I.N.A., 1909. Of the very large number of experimental results that have now been obtained from the trials of ships' models in the tanks referred to above, comparatively few have been made public. Experl-In connexion with the Torquay and Haslar tanks some mental few of the reports by the elder Fronde and Mr R. E. results. Froude have been published by order or permission of the Board of Admiralty, chiefly through the Institution of Naval Architects. Amongst these may be mentioned the "Greyhound" experiments recorded in 1874; the " Merkara " results in 1876; experiments on the effect produced on the wave-making resistance of ships by varying the length of parallel middle body, in 1877; results obtained from models of three merchant liners in 1881; papers in 1888 and 1892 on the " constant " system of notation of results of model experiments, used at the Admiralty Experimental Works; and some results of a systematic series of model experiments by Mr R. E. Froude appeared in 1904. Some records of the experiments made at private and foreign experiment establishments have also appeared. Some of the most important of these experiments are described in these notes; it remains to show how they are applied in practice to obtain an estimate of the indicated horse-power required to drive a ship at any speed. If the resistance has been obtained from a model experiment, or inferred by the law of comparison from data obtained with a vessel of similar type, the effective horse-power is known; and by assuming a suitable value for the propulsive coefficient (vide Propulsion) the indicated horse-power is determined. If model experiments or data for exactly similar ships are unavailable, the method of estimating the power which is probably most commonly used is one involving a relation between I.H.P., displacement, and speed, which is expressed by the formula (Speed)' X (Displacement) c I.H.P. C being called the Admiralty coefficient. The value of C varies considerably at different speeds even for the same ship. For it to be constant, the I.H.P. must vary as the cube of the speed; if resistance varied as the square of the speed and I.H.P. as resistance and speed, the condition of constancy would be fulfilled. Actually, owing to variations in the index of the speed to which the resistance is proportional, in the length and form of the ship and" in the machinery and propellers, this method of estimating I.H.P can . only be used with great caution, care being taken that the values of C selected for comparison are taken from ships of fairly similar type, and of corresponding lengths and speeds. Another means of obtaining approximate estimates of the power required for ships of ordinary types is from curves of resistance drawn on a base of simple functions of the speed, length and displacement, the curves being faired through the spots obtained from a large number of results of model experiments with different classes of ships. Curves of this character have been constructed by Mr D. W. Taylor and Mr A. W. Johns (Trans. I.N.A., 1907) ; the former series expresses the residuary resistance per ton of displacement in z terms of L and Lz; the latter gives the residuary horse-power B V2 divided by W in terms of t and the prismatic coefficient Volume of Displacement the frictional resist-Area of Immersed Midship Section X Length' ance is calculated independently by Froude's or Tideman's tables. To furnish data for estimating the I.H.P. of vessels covering a considerable range of type, a series of experiments on systematically varied forms of-hull were made by-Mr R. E. Froude. The results-were published by him in the Trans. I.N.A., X904; and are given in figs. 40 to 51. The forms of hull dealt with may be primarily divided into two groups, A and B, differing in Beam and Draught ratio; Beam Draught - being equal to 2.59 and 3.48 for A and B respectively. Each group is further divided into 6 types, differing in block coefficients, and the table following gives particulars of the coefficients for the models tried Stern snubbed, Bow snubbed, forward body after body as Type 1. as Type 3. Type. I. 2. 3. 4. 5. 6. Block coefficients .495 .505 .516.522 ' 529 .542 or Volume of Displacement Length X Breadth X Draught Largest section coefficient •951 or' Area of immersed'midshipsection Breadth X Draught J The hull characteristics for A are shown in figs.. 33 and 34; and the. mode of presenting, these indicates the way in which the several types were formed, each being obtained from the type i model by successively cutting back its stern and bow. This cutting back is termed snubbing. A curve of areas of transverse sections is given (fig. 35, Plate I.), as well as the sheer draught. The lines of group B can be derived from A, by altering beam apd draught scams in the ratio of 56 and Zo,4 respectively.' Each of, the i.. forms which embodied these lines was the generator of a 'series, differing only in length. proportion. The curve of areas is an important item in the hull characteristics. Experiment shows that the resistance of a hull of given curve of areas, beam and water-line entrance, is practically unaltered how-ever the lines ate varied (solong as they are -kept ship-shape, and no-unfair features are introduced). It follows, therefore, that although the data correspond to a given type of lines, yet (consistently with the preceding conditions) they are capable of applicatiop.-over a wider field than at first sight seems likely, covering variations of draught, form of profile and transverse sections- Regarding the foregoiifg statement of permissible variations of lines, alteration in Beam ratio has some effect. Comparison of Draught the two groups A and B gives the effect of the variation in. the Beam ratio tried ; and _ it is found that (caeteris Jaribus) in-Draught Beam creasing Draught by 34% (i.e. from 2.59 to 3.48) increases the E.H.P. by about 4%. A brief and approximate statement of the results of some ex eriment with models of va in Beam ratio, P 11 g Draught by Lieut.-Colonel G. Rota, R.I.N. (see Trans. I.N:A., 1905), is that i These lines differ from those tried in the' models which are given in Trans. I.N.A.; 1904 W.v.).. Those now given have the same curve of areas and beam, but age modified in respect of draught, profile and shape of transverse-sections, these latter being filled out so as more closely to represent modern forms. However, a model has been tried recently, embodying the modifications, and the results found to be practically identical with those obtained for the original lines.causes about 1% to 2.5 % increase in resistance (the lower value being appropriate to the higher speeds, and vice versa). This result accords with that deduced from the A and B groups. By the aid of the law of comparison (and a correction for skin friction), the information providedcan-be used to obtain the E.H.P. for any size of ship of form included in the experiments (or covered by the possible extensions, vide supra). The I.H.P. follows by using a suitable propulsive coefficient. An example is given below as an illustration. In practical application it is important to notice that the lengths used in reckoning the proportions must be the total length of immersed form (i.e. of the curve of areas) and not the distance between perpendiculars arbitrarily placed. The data are here given (,figs. 4o-51, Plates III:-VI.) in the form of curves of E.H.F. for ships of 1000 tons displacement, plotted for a given speed on a base of immersed length. The range in abscissae shows the amount of variation in length proportion tried in the. experiments; and as regards speed range the group B is for generally higher speeds than group A. The curves may be termed standard E.H.P. curves. The block coefficients of the forms dealt with are lower than those of the greater proportion of merchant ships, and hence the data are not directly applicable to these. At higher speeds, however, the E.H.P. might be approximately estimated from 'these curves, by assuming a further degree of snubbing appropriate to the required block coefficient; but at speeds which correspond to those of ordinary merchant ships (which are the lower speeds given in the diagrams) the effect of snubbing is variable, and depends really upon the actual speed-length ratio (i.e.v L/ of the ship we are dealing with. \ ) In this connexion it may be noted that the diagrams not only afford a means of determining the I.H.P. of a given ship, but they may also be used in designing, and so enable the best form to be chosen, to fulfil the given conditions of displacement and speed, &c. For example, suppose a ship of given displacement is required to obtain a given speed, with a given maximum E.H.P. (or I.H.P. assuming an appropriate propulsive coefficient). First bring the given particulars to the proper scale for 1000 tons "displacement (n, the ratio of the linear dimensions, is equal to ( loo° ) a and hence E.H.P. becomes Dispt. f woo 1 rs and speed ( 1000 1.1 times the given values). An E.H.P. Dispt. \Dispt.J curve for the given speed is easily interpolated on the diagrams, and we can at once obtain for the given E.H.P. (i') the length for each type,' (2) the type which gives the most suitable length, (a)` the economy resulting from any additional length, (4) the type for a.given fixed length which gives the speed with least E.H.P., and (5) by inspectionat lower speeds; how alternative forms compare at these speeds. The following points may commend themselves, from consideration of an instructive comparison shown in fig. 4, where for the f3 group, E.H.P. curves for types 1, 3 and 6 are drawn-together. In draw= ing conclusions, it must be clearly remembered that the E.H.P.'s, speeds and lengths are for a standard displacement, viz. 'boo tons; and so in applications for different displacements, these quantities all undergo a numerical change, dependent upon the change in displacement. The first point is the effect of length on E.H.P.; this is most marked at high speeds; and even at low speeds, for the shorter lengths the E.H.P. begins to increase rapidly with decrease in length. At these low speeds if, on the other hand, the length be increased beyond a certain point, no economy at all results, but the reverse. The reason for this is clear. At the low speed-length ratio we are considering, the wave-making resistance is practically nil, the resistance being almost entirely due to skin friction and eddy making, &c. It is obvious that by continually reducing the transverse dimensions of a ship of constant displacement, we increase the wetted skin (in the limit when the transverse scale is zero the surface is infinite) ; hence the resistance due to skin friction increases, and so therefore does the total resistance. This point would be more evident if the diagrams had been continued to a greater length and lower speed. A second point is the effect of alteration in block coefficient. At all speeds above 20 knots, snubbing within the limits shown is beneficial as regards performance. At lower speeds the effect depends on the length. Since it is at these lower speeds the ordinary type of merchant ship works, we may say that the effect of snubbing is doubtful for these, and depends upon the speed-length ratio. A better result might be obtained in such cases if the method of increasing the block coefficient were by the insertion of parallel middle body and not by an extension of snubbing. (For fuller information on this point see Mr R. E. Froude's 1904 I.N.A. paper.) A third point is the effect of change in speed. For a given length, the.rate of increase of Ed-I.P. with speed grows with the speed,. but increases least for the more snubbed type. As an instance consider group, B, types i and 6 at a length of 300 ft. (see fig. 36, Plate I.). The following table gives the increase in E.H.P. for the corresponding changes in speed, and the index of the speed, representing the variation of E,,.H.P. with speed. The figures in columns (4) and (5) are the means obtained from the individual pairs of speeds; at intermediate speeds these may have different-and constantly changing values:— B beyond- a value of "Beam 'Draught increase of 10% in ra ug Draught =2'5 Draught M H w The variation of the rate of growth of I.H.P. (or E.H.P.) with the speed is a result of the interference of the bow and stern wave systems, and is dependent upon the speed-length ratio (vide " Wave Resistance,” above). A good illustration is afforded by taking the case of a vessel such as a torpedo-boat destroyer, which is run over a considerable range of speed. Fig. 37, Plate I I. shows, for such a vessel,three curves plotted to a base of speed, the ordinates being respectively
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ROBERT ROLLOCK (c. 1555-1599)

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