10 



THEORY OF SEAKEEPING 



\elocity of translation of the facets of the wave form on 

 which the predominating air pressures act. Both defi- 

 nitions would be identical if the waves were simple har- 

 monic. The true wave .structure consi-sts, however, of 

 smaller wa\-es carried on the surface of the bigger ones, 

 and the velocity of translation of small facets of wave 

 form is therefore a complicated function of celerities of 

 waves of all sizes. ^ 



The \-alue of the drag coefficient, C^ = 0.02156, agrees 

 well with Motzfeld's model Xo. 4, and with the ap- 

 pearance of the wa\'es shown on Fig. 11. It was de- 

 ri\'ed on the basis of the mean wind speed of 23 fps at 

 which a small C' I'-ratio and large percentage of sharp- 

 crested small waves can be expected. 



2.6 Summary of Cj Data from Previous Sections. 

 It appears from the foregoing data that there is generally 

 good agreement among the f'a-\'alues obtained from 

 pressure measurements in wind tunnel and ffmnes, and 

 deri\-ed from the slope of the water siu'face. In a very 

 mild case (Francis at 5 mps) and in very severe cases 

 (Francis 12 mps and Johnson and Rice) the wave form 

 can be shown to be analogous to Motzfeld's trochoidal 

 and sharp-crested wind-tunnel models. Howe\-er, for 

 the cases of intermediate severity, the problem of defin- 

 ing the sea surface in a form significant for the drag has 

 not been solved. Clearly X/H-ratio is not the desired 

 parameter. Its effect is practically discontinuous; a 

 \'er\' slowly rising value of Cj with decreasing \/H, and 

 then a ciuick and drastic jump to a high \-alue as sharp 

 wa\-e crests are developed. It appears that the Cj 

 values are go\-erned by a statistical parameter depending 

 on the freciuency of occurrence of sharp crests or possibly 

 on the frecjuenc}' of occurrence of steep wa\-e slopes. The 

 problem has to be treated therefore by statistical meth- 

 ods. These will be discussed later in Section 8 of this 

 chapter. 



It can be added here that in the wind-flume experi- 

 ments rather strong wind velocities and short fetches 

 were used, so that the waves were short (of the order of 1 

 ft) and the ratio c V of wave celerity to wind speed 

 was very small (less than Vio)- The predominating 

 waves in the actual sea have a c/V ratio of the order of 

 0.86 (Neumann 1953, p. 21). The wind-flume data, 

 therefore, although \aluable material for the study of 

 various relationships, do not represent ocean conditions 

 directly. The method of analyzing such data thus be- 

 comes particularly important. Analyses in the past 

 were generally inadec[uate because of limitation to o\'er- 

 all drag of the water surface, neglect of wave irregular- 

 ity, and the indeterminateness of the wind velocity to 

 which the data are referred. An important observation 

 in connection with wind-flume tests is that the wave 

 system is very irregular from the outset. The irregular- 

 ity was commented u]3on by Francis (1951), and is dem- 

 onstrated (luantitativoly by Johnson and Rice (1952), 

 who present a number of graphs of the statistical distri- 

 bution of wa\'e heights and periods. These are repro- 

 duced here in Fig. 12. 



* It was derived statisticallj- l)j- Longuct-Higgins (1955, 1957). 



2.7 Estimation of Tangential Drag from Wind- 

 Velocity Gradient.^ The method of estimating the drag 

 coefficient of a sea surface by measuring the velocity 

 gradient in wind is based on the theory of turbulent 

 boundary laj-er at a rough surface. The tangential re- 

 sistance of such a surface causes momentum loss in the 

 immediately adjacent layers of air. The turbulent move- 

 ments of air particles cause a momentum transfer from 

 one layer of air to another, and. as a result of this, the air 

 \-elocity diminishes gradually from the velocity of the 

 undisturbed flow V at a large distance from the rough 

 surface to a velocity u < V as the surface is approached. 

 The air velocity (( is therefore a function of the distance z 

 from the plate; i.e., ;; = u(z). The plot of velocity u 

 \-ersus height z has the general shape shown in F'ig. 13. 

 The tangential drag of a surface is equal to the shear 

 stress in the air layers in close proximity to the surface 

 and is expressed in the general form as 



T = Aidu.'dz), 



(23) 



where r is the shearing (or frictional) force per unit area 

 and the coefficient A is yet to be defined. In the most 

 common usage a coefficient of hydrodynamic force is de- 

 fined in terms of pF-/2, as for example in equation (14). 

 In aerodj'iiamic usage in Great Britain, however, it be- 

 came customary to express it in terms of pV'-. This 

 usage has been generally adopted in the field of oceanog- 

 raphy, and the tangential force coefficient is written as 



or preferably 



7 = r p T 



r pu- 



(2-t) 



In the foregoing expression u denotes the air velocity as 

 measured at a certain specific height z'. It follows then 

 that coefficients Cj* and y are related by 



C/ = 27 



Attention should be called to the fact that Cj* represents 

 the total drag coefficient; i.e., C,, -f C, in the notation 

 used in preceding paragraphs. 



In the fields of aerodynamics and of hydrodynamics 

 (as applied to ships) the distance z over which u{z) is 

 variable is generally small and it is easy to measure the 

 fluid velocity at a distance from the body where u{z) = 

 V. In meteorology and oceanography it is necessary to 

 consider the wind which has blown over a vast distance, 

 and the height z over which u{z) is appreciably variable 

 is so large that it is impossible to measure velocities in 

 the region where uiz) = const, except by means of pilot 

 lialloons. It becomes necessary, therefore, to establish 

 the form of function uiz), as well as certain conventions 

 as to the height at which u should be measured. 



These con^'entions have been only very loosely de- 

 fined. G. I. Taylor (1915, 1916) used the data of pilot- 

 balloon observations over Salisbury Plain in England, 



5 For a more complete treatment of this subject the reader i.s 

 referred to Ursell (M). 



