SEAWAY 



23 



Jeffreys' sheltering coefficient s is related to the form- 

 drag coefficient C'j by the relationship 



C, = 2m-k- (48) 



The drag coefficient C^ has been defined by (f4) in 

 Section 2.f , following aeronautical practice, merely as an 

 empirical coefficient to be obtained experimentally for 

 each body (or wave) form. The coefficient s is also 

 found at the end to be purely empirical, but is defined in 

 terms of an erroneously assumed relationship between air 

 pressure ;; and wave slope drj/d.c. The important result 

 of Motzfeld's experiments (Section 2.3 and Table 1) is 

 that Ci is very low and can be assumed nearly constant 

 for smooth waves with rounded crests, but suddenly in- 

 creases several times in magnitude with the f)ccurrence of 

 sharp wa\"e crests. It appears, therefoi'e, that the drag 

 coefficient C^ should be defined in terms of the frequency 

 of occurrence of sharp crests. It is possible that with 

 this fre([uency of occurrence increasing with wind veloc- 

 ity, C'j will be found a smooth function of (/, but not 

 necessarily proportional to cr- or to V. Neumann 

 pointed out that small waves are damped out by turbu- 

 lence after breaking of the crests of larger ones. The 

 percentage of the sea siu'face covered by small sharjj- 

 crcsted waves may therefore be reduced in a strong wind, 

 bringing about a reduction of C^. 



While Munk (1955) has introduced the important con- 

 cept of "facet velocity" he uses only one symbol, C, in 

 his work and does not distinguish clearly between the 

 celerity c of a harmf)nic wave, C of the facet velocity and 

 c of the celerity of significant waves.''' In deriving ex- 

 pression (45), for instance, the celerity c of a small rip- 

 ple, as.sumed to be harmonic, may be negligibly small. 

 The facet velocity C, howe\-er, may be much larger than 

 the small ripple celerity c; it appears to be identifialjle 

 with the "formula velocity" 1', of Keuligan and Van 

 Dorn. The analysis of Van Dorn's data in Section 2.5 

 has indicated that in his case V, was of the order of 0.6f ^. 



The factor {V — c)- in the exjiression iVjr air pressure 

 in the case of irregular waves should clearly depend on 

 the facet velocity C. In expression (43) Munk used the 

 one symbol C not distinguishing between c and C. In 

 reproducing the expression here it was interpreted as 

 f(A), since the entire formula represents a summation of 

 the effects of simple harmonic components; C itself is a 

 function of the superposition of many harmonic \va\'e 

 profiles and therefore the expression C{k) would have no 

 meaning. On the other hand, the factor [V cos 9 

 — c{k}\ also cannot be accepted as valid without further 

 discussion. ([/ — G) has the well-defined physical 

 meaning of relative A^elocity of the wind with respect to 

 the moving water-surface facet. The expression [U — 

 r(/v) ] has, however, no such meaning, since cik) is merely 

 the result of a mathematical analysis, and not a physi- 

 cally visible entity. With c(A) negligibly small in com- 

 parison to U for small ripples and C = O.Gf/^, use of the 

 expression U[U — c(k)] in place of ((/ — C)- appears to 



have introduced something like a six-f<ikl exaggeration of 

 T. It would appear that statistical work should be 

 directed primarily to the evaluation of the corresponding 

 pairs of values of the facet velocity C and the wave 

 slope b-q/bx, for suliseciuent use with the relationship 



{V - cr-. 



Munk's (1955a) work has been directed to evaluating 



wind stress or form drag of waves. In application to 



uergy transfer from wind to waves, the wind stress has 



to be multiplied by the facet velocity; i.e., in a simple 



form, 



E = <K(V - C)-C> 



(49) 



This indicates the extreme importance of developing 

 methods of evaluating the facet velocity C for the realistic 

 irregular sea. '* The coefficient K in (49) also depends on 

 the wave form, and K and C are therefore interrelated. 



4.5 Miles' Theorv. I'ollowing the earlier work of 

 Jeffreys (1925, 192(i) and Munk (1955n), Miles (1957) 

 considered the transfer of energy from wind to water 

 waves as caused by the air pressures resulting from the 

 \\ave form. In defining these pressures, however, he 

 considered the properties of the thick tiu'bulent boundary 

 layer in proximity to the sea surface. A slightly \'iscous 

 air was assumed so that the velocity gradient u(z) had 

 formed,'^ while the analysis based on the velocity po- 

 tential and stream function could, nevertheless, be ap- 

 plied. 



Under the assumption of a two-dimensional sinusoidal 

 wave, the pressures acting on the water surface were de- 

 fined in terms f)f the wave slope /,'?; as 



V 



(a + ili)p'U{'k-n 



(50) 



iki,x-cl) 



where kr) is the local sloj)!.' of the waves, r) = ac 



the real parts of complex (juantities are implied in the 



final interpretation . 



The \'ectorial form of the nonilimensional pressure co- 

 efficient (a -|- t/3) represents a sum of pressure components 

 in and out of phase with the wave. As was shown in 

 Section 2.1, only the pressure component out of phase 

 with 7) contributes to the energy transfer. Nothing more 

 need be said, therefore, about the coefficient a and the 

 primary problem is a rational evaluation of the coef- 

 ficient )3. 



In the elementary discussions in previous sections of 

 this monograph, the pressure was related to {U — c)^'^ 

 Miles, however, like Phillips, treats the "reference 

 velocity" U\ as unknown at fii'st. His paper, therefore, 

 has two objectives ; e\'aluation of both the velocity U\ and 

 the coefficient /3. The coefficient 0, while nondimen- 

 sional, is not a constant but is a function of the wave 

 celerity c and the wave niunber k = 'Iv/X = g/c"-. 



'■* The use of the symbol c is Hniited to this section only. 



'* Valuable material in this connection can now be found in two 

 papers by Longuet-Higgins (1(1.56, 19.57). 



"See Fig. i;5; also refer to Section 2.7 for the elements of 

 boundary-layer properties and the definition of the roughness 

 parameter Zo. 



" The symbol U refers to the air velocity uiz) at an arbitrary 

 elevation z at which an anemometer is located. 



