24 



THEORY OF SEAKEEPING 



A crucial step in the analysis leads to definition of the 

 critical elevation s, at which the air velocit_y u(z) is equal 

 to the wave celerity c; i.e. u(z) — c = 0. This step is 

 glossed over quickly with a definition of an appropriate 

 Reynolds number and with reference to Lin.'* The 

 principal result is that the energy transfer (from wind to 

 waves) is proportional to the curvature of the velocity profile 

 at that point in the profile where the mean air speed is equal 

 to the wave speed. It follows that (quoting Allies) : 



"(1) Only those wa\'es having speeds in that range of 

 the wind profile for which —dhi/dz~ is large may be ex- 

 pected to grow; the lower limit for (the celerity) c may 

 be imposed by the existence of a sub-layer of linear profile 

 or by the interaction of the waves with the wind profile, 

 while the upper limit will be rather less than the wind 

 speed outside of the boundary layer ; 



"(2) in the initial phases of wave formation, those 

 waves having speeds well down in the profile (large 

 — d'-u/dz-} may be expected to predominate; 



"(3) experimental measurements of aerodynamic 

 forces on stationary wave models (Stanton, et alia 1932; 

 Motzfeld 1937; Thijsse 19.51) may not yield significant 

 values of such parameters as Jeffreys' sheltering coef- 

 ficient, since the point at which u{z) = c then occurs right 

 at the boundary." 



The reference speed Ui is defined bj' the relationship 



c = U, log, (£./0 (51) 



In application to Roll's observations with the "anemom- 

 eter height" 2 = 2 m (6.5 ft), JMiles indicates that 



l\ = C'/IO 



The pressure coefficient is expressed as an integral 

 which is evaluated by a serial expansion in terms of the 

 ratio zjX. Tiie maximum value of 13 occurs at z, = 

 0.006 X. At z, = 0.05 X, 13 drops to about Vs of its maxi- 

 mum value, and it practically vanishes at z, = 0.16 X. 



The pressure coefficient /3 and the energy transmission 

 from the wind drop to nil at c/Ui between 9 and 11, de- 

 pending on the values of zn and f'l. Thus, in the ab- 

 sence of energy dissipation the waves are expected to 

 cease growing when the wave celerity approaches the 

 wind velocity. 



Miles made two attempts at verification of his theory. 

 In the first, he treated the case of wave initiation with 

 dissipation of energy by molecular viscosity. Taking 

 the probable \'alue of 2o, he obtained a minimum wind 

 velocity of 80 to 100 cm/.sec with a corresponding wave 

 celerity of 40-50 cm/sec. These figures are in agree- 

 ment with Jeffreys' (1925, 1926) observations. 



In another attempt, by superposition of his simple 

 wave results he calculated the drag coefficient of a com- 

 plex sea based on Neumann's spectrum in the form used 

 bj' Munk (1955). The computed coefficients were ap- 

 proximately double those of Munk. Miles strongly em- 

 phasized that mathematical approximations on one hand 

 and uncertainty in estimating the roughness parameter 



18 Lin, C. C, "The Theory of Hydrodynamic Stal)ility," Cam- 

 bridge University Press, 1955, chapter 5. 



zo on the other might make the results off by a factor of 2 

 either way. It is interesting to recollect, however, that 

 Jeffreys (1925, 1926) has shown that the drag coefficient 

 for a short-crested sea is half that of a regular long- 

 crested sea. The drag coefficient of bodies in three-di- 

 mensional flow is generally lower than that in two-di- 

 mensional flow, the simplest illustrative case bemg that 

 of a .sphere and a cylinder. This may well explain the 

 factor of 2 in comparing Miles' and Munk's results. 



Miles' theory is based on the action of pressure; i.e., 

 of the normal force, on the wave surface. It considers, 

 therefore, the transfer of the energy of potential air flow 

 to potential wa\'e energy. The obser\-ed data on the 

 roughness parameters zo, on the other hand, include the 

 total tangential drag; i.e., the pressure drag plus fric- 

 tional drag. In mild wa\-es these two drag components 

 may be of equal magnitude (see Motzfeld's Table 1). 

 The computed coefficient j3 and the rate of energy transfer 

 in such a ca.se should he approximately double the true 

 ones. 



Another important aspect to mention is that Miles 

 sought to a\'oid the limitation to small wa^'e heights by 

 measuring elevations z not from a level surface but from 

 a wavj' streamline. This procedvu'e is equivalent to 

 assuming that the entire velocity profile u{z) shifts up 

 and down with the wave elevation without change of 

 form. This may be an admissible a.ssumption in the 

 ca.se of long and low waves (small H/\ ratio), to which 

 Miles' results should, therefore, be limited. 



In the case of steeper waves, the u{z)-c\n-Ye appears to 

 follow the logarithmic law at sufficient elevation above 

 the wave but at lower z it \'aries with position along the 

 wave profile. In particular, it is drastically changed at 

 wave crests. This should have a large effect on the pres- 

 sure distribution (and therefore on the drag and energy 

 tran.sfer), since Miles demonstrated how small the z^ 

 is in comparison with the wa\'e length when the pressure 

 coefficient is significant. An extension of Miles' work 

 to take into account this effect would make an interesting 

 and important project. 



5 Growth of Waves in Wind — Practical Approach 



The words "rational" and "practical" are u.sed in this 

 work conditionally for convenience of reference. The 

 difference is in the degree to which empirical information 

 and hydrodynamic theory are used. In the "rational" 

 approach empirical data are limited to form-drag co- 

 efficients in simple wa\'es and to the obser\'ed statistical 

 distribution of .some quantity, say wave directions, in the 

 actual sea. The effect or the mutual interrelation of all 

 other A'ariables is then obtained by obser\-ing the laws of 

 mechanics and hydromechanics. It is necessary to pur- 

 sue this approach in order to gain complete understandmg 

 of the nature of the seaway in all its details. However, 

 this development progresses slowly, and for immediate 

 practical use many short cuts become necessary. In 

 these, theoretical reasoning and empirically obtained 

 data are intei'mixed in \'arious forms and proportions 



