166 



THEORY OF SEAKEEPING 



rolling moment. The situation is particnlaiiy dangerous 

 for a small ship on an advancing flank near the crest 

 of a large wave. The subtraction of the vertical ac- 

 celeration of orbital water motion from the force of grav- 

 ity leads to a reduction of the righting moment at the 

 time when the above mentioned rolling moment occurs. 

 The wind force acting on the superstructure will further 

 aggravate the situation. 



The analyses can also l)e expected to lead to improve- 

 ment in the rolling characteristics of ships through 

 better proportioning of appendages. Since rolling is 

 caused by waves directly and also is induced by yawing, 

 the resultant amplitude of nAi depends on the magni- 

 tudes and relative timing of these two effects. Certain 

 proportions of the appendages (skegs and bilge keels) may 

 secure the optimum relationship of these effects. Thus, 

 considering the stability of an airplane in circular flight, 

 Korvin-Kroukovsky (1929) pointed out the fact that the 

 ratio (yawing moment due to yawing \elocity)/(yawing 

 moment due to side slip) strongly predominates over all 

 other factoi's in determining the airplane behavior. Fur- 

 thermore this ratio is much more important in defining 

 the stability of an airplane than the absolute values of the 

 moments. Simple emjiirical tiesign rules were then for- 

 mulated, based on this ratio, for proper proportioning of 

 wing dihedral and vertical tail surfaces. This simple 

 action was possible, howe\-er, only because a complete 

 investigation of the problem by Bairstow and his asso- 

 ciates was available. 



Information on miscellaneous coelhcients of equations 

 (13) and (14) can be obtained by: 



i E.xperiments in towing tanks with suitable re- 

 straints on models and the use of oscillators and/or ro- 

 tating arms. 



ii Theoretical methods, following the procedure for- 

 mulated by Haskind (1946, section 2-6) and extending 

 it to evaluate tlcri\-ati\'es in un.symmetrical motion. 

 This is limited to idealized ship forms. 



iii Theoretical method using strip theory, closely fol- 

 lowing the procedure used by Korvin-Kroukovsky and 

 Jacobs (19.57) for motions in the plane of symmetry and 

 extending it to unsymmetrical motions. The supporting 

 work of Ursell (1949), Grim (19.j()), and Landweber and 

 de Macagno (1957) was described in Chapter 2. This 

 method is applicable to normal ship forms. 



3 Ship Motions in Irregular Seas 



Chapter 1 of this monograph is a digest of all the infor- 

 mation currently available on the nature of ocean waves. 

 After the sum total of the knowledge on generation and 

 propagation of waves is distilled, there emerge several 

 basic tenets upon which ship motions investigations may 

 be carried forth. 



There is, hrst of all, no existing analytical expression 

 that describes the behavior of the surface of the sea for 

 any specified time and/or space. There are however 

 .several semi-empirical expressions which purport to de- 

 fine the energy-frequency characteristics of waves. Of 



these wa\e-spectrum descriptions (Section 1-6) none 

 has gained widespread acceptance, although each has 

 been \'erified to some extent by measin-ements. 



Then too there are fairly reliable methods for observa- 

 tion and spectrum anah^sis of waves at sea and these 

 have been thoroughly discussed in Section 1-8. 



Finally, there is a hyi)othesis which suggests that a 

 good model of the sea surface is achieved by the linear 

 superposition, in random phase, of an infinite number of 

 sinusoidal wave components of all freciuencies and in- 

 finitesimal amplitudes. 



Perhaps it will be worth while to iliscuss the implica- 

 tions of these results as they apply to the three methods 

 of studying shiiJ-motions problems (theory, model tests 

 and full-scale tests). The spectral representation of the 

 sea surface forces statistical treatment of ship motions in 

 all three investigative media. Wave measurement sup- 

 plies a time history of the waves, made in a certain way, 

 which can be reduced to an estimated spectrum of the 

 .seaway which in turn may define some statistical char- 

 acteristics of the waves. The wave profile lends itself to 

 a re.stricted deterministic treatment of ship motions both 

 theoretically and in the model tank. The hypothesis 

 defining the composition of surface waves complements 

 the hypothesis that the response of a ship to a sum of .sine 

 waves equals the sum of the ship responses to the indi- 

 vidual sine waves (if the sy.stem is linear). Thus the 

 discussion in Section 2 of this chapter, on the solution of 

 hydrodynamic e(|uations of motion for sinusoidal in- 

 puts, becomes a basic building block, for ship-motions 

 prediction based on the linear superposition principle. 



The intent of this section is to review the work done in 

 the application of present knowledge of waves to ship 

 motions in irregular seas. 



3.1 Probabilistic Versus Deterministic Methods. 

 These two philo.sophies were touched on briefly in Sec- 

 tion 1-8 where the probabilistic approach to the descrip- 

 tion of the seaway was justified on the grounds that a 

 general deterministic representation was unknown and 

 that statistics of waves deri\ed from the energy spectrum 

 were adequate to define the state of the sea. In the case 

 of ship beha\-ior there are argimients in favor of a deter- 

 ministic solution to the problem of ship-wave interaction. 

 There is little that may be said against the derivation 

 of the time history of a ship motion from the given time 

 history of the waves at some point. This is particularly 

 useful in predicting the occurrence of short-duration 

 phenomena such as slamming and beniling moments. 

 There is also the notion of reproducing an actual time 

 history of waves in a towing tank (Fuchs, 1956) so that 

 moilels may be tested in the most realistic conditions 

 ajjpropriate to !ong-cr(>sted irregular waves. Advocates 

 of the statistical apijroacli argue that the basic pertinent 

 information on ship beha\-ior is a\'ailable through energy 

 spectrum considerations, for considerably less effort, 

 and that any pretHcted time history of a ship motion 

 neglects the short-crestedness of the wa\'es and is still 

 unwieldy and must be reduced to statistics before the 

 essence of its potential is realized. 



