270 



NA TURE 



[July 19, 1906 



LETTERS TO THE EDITOR. 



[The Editor aoes not liold himself responsible for opinions 

 expressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications.] 



The Stability of Submarines. 



Sir William White, in his paper in tlie Roy. Soc. Pro- 

 oeedings (vol. Ixxvii. A., p. 528), discusses tlie liydrostatic 

 forces tending to stability or instability of a submarine at 

 the surface of the water. When the vessel is in motion, 

 hydrodynamical forces come into play from the stream-line 

 action of the water, and these also will affect the stability 

 of the vessel. Sir W. White insists that these forces can 

 only be examined experimentally, and has no data to 

 give as to their magnitude. Although it is obviously im- 

 possible to obtain an e.xact calculation of the magnitude of 

 these hydrodynamical forces, yet it may be worth notic- 

 ing that a very simple calculation will give an approxi- 

 mation to their value, which at least is of importance in 

 that it suggests that the question is one of extreme gravity. 



The principle involved is, of course, the well-known 

 principle by which an ellipsoid moving through still water 

 tends to turn so that its smallest axis is in the direction 

 of motion. 



We may obtain a first approximation to the stream-line 

 action by treating the submarine as a cigar-shaped spheroid, 

 and assuming it to be completely immersed in an infinite 

 ocean. Let a, b denote the semi-axes of the spheroid, and 

 let it be moving with velocity V, its major axis making 

 a small angle 8 with the horizonl.al. The couple tending 

 to decrease is known to be 



JV- fin 28 



>■ 



and D is the displacement of the vessel. 



The figures given by Sir William White for an actual 

 submarine are — length 150 feet, breadth 12-2 feet. If we 

 take alb — 12-^ in our spheroid, we obtain 



-095. 



Thus, for a displacement through a small angle 6 from 

 the horizontal, the stream-line couple produced by a velocity 

 V through the wrter is 



o 95 v= r>e. 



tending to turn the vessel further from the horizontal. 



If li is the metacentric height for longitudinal displace- 

 ment, the h\'drostatic righting couple is 



ghDB. 



Thus the effective righting couple is 



so that the metacentric height is diminished by the motion 

 of the submarine by an amount 0-95 V'/g. This factor 

 does not appear to depend much on the shape of the 

 submarine; clearly, if its shape had been that of a thin 

 long stick, we should inerely have to replace 095 by i.oo, 

 whereas if the ratio a/b had had only half of its present 

 value, the numerical factor would still be about 0-9. 



NO. 19 16, VOL. 74] 



Sir William White gives as the metacentric height nf 

 an actual submarine 37 feet when awash on an even 

 keel, and only 45 per cent, of this, say 165 feet, when 

 trimmed to an angle of 4 degrees. The value of \"/^ 

 when V = io knots, is about 95 feet, so that the effective 

 metacentric height would be reduced to about 28 feet on 

 an even keel, and to 75 feet when trimmed 4 degrees by 

 the stern. So far as can be judged from a diagram given 

 by Sir William White (Fig. 7 in his paper), the meta- 

 centric height would vanish altogether for a trim of 

 about 7 degrees. 



Thus a submarine moving ahead at 10 knots, even under 

 perfect conditions, might apparently be expected to founder 

 if its inclination at any time reached as much as 7 degrees. 



Sir William White mentions that in the case of the 

 subma'-ine A8, the hydrostatic metacentric height had been 

 reduced, at the time the accident occurred, to 85 feet. 

 The further diminution in this height produced by a head- 

 way of 8 knots is about 6 feet — by a headway of 9 knots 

 is about 75 feet, leaving only about i foot of effective 

 metacentric height as the margin of safety. 



Obviously these rough calculations ignore a great num- 

 ber of factors which ought to be taken into account before 

 accurate knowledge can be obtained. The most important 

 of these factors is probably the proximity of the surface 

 and the consequent formation of surface-waves. A cal- 

 culation which omits a factor of this kind cannot lay 

 claim to any value as advancing exact knowledge, but 

 may serve the humbler purpose of suggesting possible, and 

 even probable, dangers, and of emphasising the need for 

 experimental knowledge, before this is forced on us by a 

 catastrophe. J. H. Jeans. 



Trinity College, Cambridge. 



The mathematical investigation which Mr. Jeans puts 

 forward is of great interest, but avowedly rests on the 

 assumption of the complete immersion of a submarine in 

 an infinite ocean. The concluding paragraph of his letter 

 indicates that a great number of factors, which ought to 

 be taken into account, are not represented in the mathe- 

 matical investigation, the most important being near 

 proximity to the surface and the consequent formation of 

 surface-waves. It will suffice, therefore, for me to say 

 that my insistence on the necessity for direct experiment, 

 rather than mathematical investigation, had relation to the 

 case where the submarine was moving at the surface with 

 a small reserve of buoyancy. The slides which I exhibited 

 at the Royal Society reproduced photographs taken in these 

 circutnstances, and showed the singular and irregular 

 character of the surface-waves produced by the headway 

 of submarines under these conditions. These slides 

 furnished conclusive evidence of the impossibility of re- 

 presenting the conditions of practice by purely mathematical 

 investigation, and the absolute necessity for experiments on 

 models and full-sized submarines. 



Mr. Jeans's investigation for the completely submerged 

 vessel has, however, a great practical value, because it 

 furnishes fresh and important reasons (in addition to those 

 urged by myself) against the tendency to increase the 

 under-water speeds of submarines. When submerged, the 

 measure of stability of the vessel for all directions of in- 

 clination is found in the height of the centre of buoyancy 

 above the centre of gravity. We are informed authori- 

 tatively that in the diving condition this height is less than 

 I foot in existing types of submarines. It will be seen, 

 therefore, that a very small value of V — less than 6 knots 

 — might render such a vessel unstable ; if the speed were 

 increased to 10 knots no possible use of water-ballast could 

 give such a hydrostatic stability to the vessel when at rest 

 as would secure the maintenance of stability when sh" 

 moved at full speed. The existence of superstructures on 

 the upper portions of submarines, of course, involves a 

 departure from the cigar-shaped spheroidal form, but can- 

 not be accompanied by any such decrease in the moment 

 of the couple resulting from the stream-line forces as would 

 secure, or even add sensibly to, the safety of the submarine 

 moving at high speed under water. W. H. White. 



