June 12, 1879] 



NATURE 



149 



will the body experience in terms of her momentary de- 

 viation from it?" Mr. Froude has demonstrated both a 



will 

 viation 



priori and experimentally that to a stabihsed particle 

 floating at any point on the upper surface of a wave, the 

 position of momentary equilibrium is that which would 

 place the axis of equilibrium normal to the wave surface 

 at the point where it floats, and that to another similar 

 particle, floating or suspended below the surface, the posi- 

 tion of momentary equilibrium is that which would place 

 the axis normal to the sub-surface of equal pressure 

 passing through the point where it is placed. If we take 

 account of the aggregation of particles which a ship dis- 

 places, and for which she herself is substituted, and of 

 which she accepts the aggregate dynamic condition, we 

 know that her position of momentary equilibrium must 

 be the mean of the positions belonging to the various 

 particles displaced ; and we may assume, with a close 

 approximation to the truth, that this is the position which 

 would place her axis of equilibrium, or her masts, at right 

 angles to one of the wave sub-surfaces of equal pressure. 



The sub-surface of equal pressure through the centre 

 of gravity of a ship's displacement may be regarded in 

 theory as a sufficiently close approximation to the efTective 

 wave surface ; and it follows that when a ship deviates 

 from the normal to this surface the effort by which she 

 endeavours to conform to it depends upon the momentary 

 angle of deviation in the same manner as her effort to 

 assume an upright position when inclined in still water 

 depends on the angle of inclination. Hence her stability 

 or effort to become vertical in still water, measures her 

 effort to become normal to the effective wave surface in 

 wave water. The equations of motion for a floating body, 

 oscillating in still water, which had previously been inves- 

 tigated, could therefore be applied to undulating water 

 by introducing the condition that the position of equi- 

 librium changed with the direction of the wave slope. 

 Mr. Froude was not able, at first, to solve the resulting 

 equation by adopting the trochoidal hypothesis. He 

 therefore substituted the curve of sines for the trochoid, 

 which gave him a form of equation he could deal with. 

 Prof Rankine afterwards solved the equation obtained by 

 using the trochoid, but the results agreed with those 

 arrived at by Mr. Froude under the conditions to which 

 the investigations applied. 



The assumptions made in order to adapt the problem 

 to mathematical treatment were (i) that the ship is rolling 

 passively in the trough of the sea ; (2) that she is exposed 

 to a regular series of similar waves ; (3) that the wares 

 are so large as compared with the ship that she may be 

 assumed to accept the motion of the part of the wave she 

 displaces ; (4) that the variations of apparent weight may 

 be neg:lected in comparison with the actual weight; (5) that 

 the ship is of such a form as to make her still water oscilla- 

 tions isochronous — this being approximately the form of 

 the old line of battle ships ; and (6) that the rolling is un- 

 resisted — the effects of resistance in modifying the motion 

 being separately considered. 



The equations thus obtained by Mr. Froude, representing 

 the oscillations of a ship among waves as compared with 

 those performed in still water, are most interesting ; but 

 our space will not admit of giving a full analysis of them. 

 Their general character may, howjver, be appreciated if 

 we call attention to some of their most striking features. 



One critical case is that of a ship rolling among waves, 

 whose periodic time synchronise, with her own time of 

 oscillation. It maybe readily deduced from Mr. Froude's 

 fundamental equations that, if it were not for the resist- 

 ance to rolling caused by surface friction and form, a 

 ship placed broadside on to waves which have her own 

 periodic time, must ultimately roll completely over, how- 

 ever small the wave maybe. It is not uncommon to find 

 the length of a half-wave ten times the height. Such 

 waves would increase the angle of roll by 141" at each 

 inclination, so that six successive impulses, or three com- 



plete waves, passing a ship would produce almost a com- 

 plete overset. Though this conclusion requires to be 

 greatly limited by introducing the element of resistance, 

 it is obvious that such synchronism of wave-period and 

 ship's-period must produce most formidable effects. Mr. 

 Froude produced the result thus indicated by his theory 

 by direct experiment with floating bodies of such form as 

 would give approximate cases of unresisted rolling. He 

 immersed a sphere to two-thirds of its radius ; a prolate 

 spheroid to about the same proportion of its major axis ; 

 and a body like a flattened orange was wholly immersed, 

 having only a very narrow neck projecting from it above 

 the water-level, like the stem of a hydrometer. By an 

 ingenious arrangement for regulating the period of the 

 waves it was found that, when the oscillations of the 

 floats and the wave-period were made to synchronise, all 

 the floats were upset after the transit of a very few waves, 

 while a very small change in the natural period of one of 

 the floats, made by slightly altering the position of its 

 centre of gravity, made its behaviour plainly exceptional 

 as compared with the two others. It now refused to be 

 completely overset by the series of waves which would 

 upset the two others almost at the same moment, though 

 it was itself overset by a series slightly quickened or re- 

 tarded according as its own period was quickened or re- 

 tarded by the altered position of its centre of gravity, the 

 other two being at the same time released from all danger 

 of capsizing. 



Another critical case is when the ship possesses infinite 

 stability ; or an infinitely small radius of gyration or 

 moment of inertia. This is not a practical possibility, 

 but is noticed on account of the indication it gives of a 

 ship's tendencies according to the degree in which it may 

 possibly be approached. In this case the ship will be 

 perfectly quick in her movements and will follow pre- 

 cisely the slope of the wave. The movements of a flat 

 board laid on the water are a practical illustration of this 

 condition. The periodic time of such a board may be 

 practically treated as = o ; and if a ship could be so con- 

 structed as to fulfil this condition, there might be some 

 wisdom in attempting it. It is impossible, however, to 

 construct a ship that will even approximately fulfil this 

 condition, and as an approach to it could only be effected 

 by giving her the greatest possible stability, she would 

 only the oftener meet waves with which she would 

 synchronise and experience the evil consequences of that 

 condition. 



One other critical case, the conditions for which are 

 deducible from the equation of rolling motion, is that in 

 which a vessel exposed to a series of waves performs her 

 oscillations not in her own period but in that of the some- 

 what different wave period. In this case the still water 

 oscillations would not synchronise with the wave period ; 

 but a relation subsists which enables the increasing slope 

 of each wave to just counteract the growing inclination of 

 the ship. At the wave hollow and crest a ship under 

 such conditions would be upright : and she would reach 

 her greatest inclination to the vertical when she was in 

 an intermediate position upon the greatest slope of the 

 wave. She would roll so that her masts would always 

 lean towards the wave. 



A general feature of the theory as deduced from the 

 equation of rolling motion is that when the natural time 

 of oscillation of a ship and the wave period do not 

 synchronise, and when the rolling has not become per- 

 manent, the ship's oscillations will pass through phases 

 analogous to the action of a pendulum when subjected to 

 a series of impulses partially synchronous with its own 

 excursions; and, as we have seen, this deduction is in 

 accordance with the observed phenomena of rolling. 



The resuhs given by Mr. Froude's equation for un- 

 resisted rolling give, so far as character is concerned, a 

 generally correct view of what actually occurs. But, 

 quantitatively, the angles of oscillation indicated arp 



