March 25, 1 875 J 



NA TURE 



4C9 



the subject of waves, which is at present occupying so 

 much the attention of all those who, both in this country 

 and abroad, arc endeavouring, by researches into their 

 forms and habits, to improve the theory of Naval Archi- 

 tecture. 



The first paper was on a proposed method of obtaining 

 the outHnes of deep-sea waves, by Mr. W. W. Rundell, 

 the secretary of the Liverpool Underwriters' Association. 

 The important part which photography has recently- 

 played in the observations on the Transit of Venus, and 

 the assistance which it has thus rendered to astronomy, 

 led Mr. Rundell to consider whether it might not also be 

 employed to determine the forms of waves and so supply 

 data for obtaining their chief components. The applica- 

 tion which Mr. Rundell proposes consists of a system of 

 poles about 36 feet in length, painted with alternate bands 

 of red and blue, each band being a foot wide. These 

 poles are spaced 15 feet apart and loosely coupled at one 

 end to yards or spare spars extending to a length of 

 about 600 feet. A similar system of poles intersects the 

 first system at intervals of 90 feet, the different parts 

 being connected together, in moderate weather, while 

 floating on the surface of the water. Weights being at- 

 tached to the spars would cause the poles to sink until only 

 about 12 feet of their length was visible above the water. 

 Mr. Rundell proposes, by the aid of photography, to take 

 pictures of the outlines of waves seen against this system, 

 the photographs being taken either from the crosstrees of a 

 man-of-war or from some elevated position such as the 

 Fastnet, or Skellig Lighthouses. Mr. Rundell thinks 

 that thus the complete history of a gale might be photo- 

 graphically recorded. Mr. Froude, however, seemed to 

 think that there would be greater difficulties to encounter 

 than Mr. Rundell imagined. 



The ne.xt paper, by Mr. Froude, was a description of 

 the graphic integration on the equation of a ship's rolling, 

 including the effect of resistance. Mr. Froude first pointed 

 out that the commonly employed methods of graphic in- 

 tegration, i.e. the semi-geometrical processes by which 

 the solution of intractable mathematical problems is 

 effected, do not readily lend themselves to the treatment 

 of a problem in which the forces which govern the move- 

 ments of the body arise afresh at each instant, as the 

 direct and indirect effects of the verj- movements they are 

 creating, but that his method is perfectly capable of deal- 

 ing with this circumstance. 



The two principal forces taken account of in this 

 method are the ship's " righting force " or " moment " as 

 dependent on her inclination relatively to the wave slope 

 at each instant, taking into consideration any speciality 

 in her curve of stability ; and the resistance she experi- 

 ences while in motion, as dependent on her angular 

 velocity. Taking the equation of rolling motion to be 

 integrated is, in its most complete form, as follows : — 



Here 6 is the ship's absolute inclination, 6' the inclina- 

 tion of the wave, and .'. {6 ~ &) is her inclination rela- 

 tively to the wave slope, or the ship's "relative inclination ;" 

 the tenn / {6 — 6') signifies that function of the relative 

 inclination which in the curve of stability is assigned 

 to the particular inclination, and expresses the righting 

 moment of the ship when so inclined. 



T is the time, in seconds, occupied by the ship in per- 

 forming a single swing when rolling to moderate angles 

 in still water, being half of what is commonly called the 

 " metacentric period." 



R is the effective " moment of resistance " which the 

 ship is at the instant experiencing when rolling with her 

 existing angular velocity, its elementary signification 

 being homogeneous with that in the ship's curve of sta- 

 bility, in which/(e - ff) stands for the righting moment. 

 In both terras, alike, these elements consist in eff^ect of 



" so many foot-tons X rrp— 2'" "'^^""^ W is the ship's weight 



in tons, p her radius of gyration expressed in feet, as » 

 also usually is. The abstract value of R is 



dt - dt- 



where l\ and k„ will have values appropriate to the parti- 

 cular ship in question ; and observe that the ± sign must 

 be understood to mean that the sign of the second term, 

 which, being a square, would in itself be always positive, 

 must change signs in company with the first term. 



A base line being taken to represent time, and divided 

 into equal spaces representing small unit-intervals of 

 time, Aj/, A.y, &c., the inclination at each instant, 

 whether of the ship or of the wave, are to be expressed 

 as ordinates to a scale of degrees ; those above the base 

 line being positive, and those below it negative. A 

 " curve of wave slopes " being drawn, the ship's absolute 

 inclinations, which grow out of the circumstances, as time 

 (and the varying wave slopes which time brings) proceeds, 

 by Mr. Froude's method of graphic integration, are repre- 

 sented by a curve analogous to the '■ curve of wave slopes " 

 in general character. This curve which gradually grows out 

 of the integration Mr. Froude calls the " curve of rolling " 

 or the " curve of inclinations." The difference between 

 the ordinates of these two curves, at any instant, gives 

 the ship's relative inclination at that instant on which the 

 righting force depends. The angular velocity of the 

 ship's change of inchnation is obviously expressed by 

 the tangential direction of the curve, and this circum- 

 stance is of essential importance in the process by which 

 the curve is deduced. 



To carry out the process two auxiliary curves have to 

 be introduced : — 



1. The " ship's cur\'e of stability," which supplies, as has 

 been explained, her righting moment, as due to her relative 

 inclination at any instant. In this, the base is foimed of 

 a scale of angles, this scale being the same as in the 

 " curve of wave slopes " and the " curve of inclinations." 

 The ordinates corresponding with given inclinations ex- 

 press the righting moments at those inclinations to the 

 scale which is employed in the graphic process. 



2. The " curve of resistance," which supplies the 

 moment of resistance experienced by the ship when 

 moving with any given angular velocity. 



As has been already stated, the conditions are— 



R = ^,'^ + ^Vli= 



^dt - at 



The first of these terms is expressed by a straight line, 

 and the second by a parabola which takes that straight 

 line as its base. 



Turning to the employment of these data in the geo- 

 metrical solution of the dynamical equation, by grouping 

 the force terms under the single symbol <^, we may write 

 the equation thus : — 



dl^dS) :di = (f>: 



Tr'dt 



Substituting for the differential terms, small quantities 

 virtually infinitesimal — 



•^2 



A (A ^) : A ^ = (|> : 



r A/ 



where A/ is the unit'of space taken in the curve of wave 

 slopes. 



By a simple geometrical contrivance this ratio is 

 utilised by drawing a base line each way from the foot of 

 the vertical axis in the "curve of resistance," which 



measured by the time scale = -^j ending at -}- P and 

 _ p ^ It at 



Through the end of this line the inclination 6^ is set off 

 with a parallel ruler from the " curve of wave slopes. 

 The height j> at which this line cuts the directrix is pro- 

 portional to the angular velocity. 



