SEAWAY 



Substituting equutions (7) and (8) into equation (5) 

 leads to integrals of the following forms. 

 For the first harmonic 



Jo 



sin k{x — ct) cos k{x — ct)dt 



and 



Jo 



cos- k{.r — ct)dt 



and for higher harmonics 



• 7' 



sin n k{x — ct) cos /i(.i' — cl)<U 



x 



and 



i: 



cos n k(x — ci) cos k{x — ct)dt 



(10) 



All of the foregoing integrals \-anish upon evaluation, 

 except the one for the first harmonic containing cosine 

 squared in the integrand. This proves equation (4) 

 given by Lamb. The pressure components in phase 

 with the wave, the sine harmonics, transmit no energy 

 to water, and the entire energy transmission is due to the 

 first harmonic of the cosine, i.e., to the out-of-phase 

 component. Retaining only the first harmonics, equa- 

 tion (7) can be rewritten as 



p = A sin k(x — ct) + B cos k(x — ct) 



= C sin k{x - ct + i) (11) 



This shows that the transmission of energy from wind 

 to water occurs because a sinusoidal pressure distribu- 

 tion differs in phase from the wave form. The dis- 

 tortion of the pressure-distribution cur\-e by higher 

 harmonics apparently has no effect on the energy trans- 

 mission. In a potential flow the pressure is in phase 

 with the wave, e = 0, and there is no energy transmission. 

 In a fluid of very low visco.sity, as is the case with air, 

 the flow is generally potential, but the boundary con- 

 dition at the water surface is modified by the presence 

 of the boundary layer. This latter thickens locally in 

 the adverse pressure gradient on the lee sides of waves, 

 and thus brings about the phase shift e and the resultant 

 energy transmission. This will be discussed further in 

 connection with experimental wind-tunnel data. 



Evaluation of equation (5) after sub.stituting the 

 cos k{x — ct) pressure term from equation (11) gives the 

 total energy transmitted by wind to waves, per second 

 per unit of sea surface. 



E = B ac k/2 



(12) 



As can be seen from equations (7) and (11) the coeffi- 

 cient B is the amplitude of the out-of-phase component 

 of the air pressure. Taking it as proportional to the 

 square of the relative velocity between air and wave 

 propagation (V — c), B can be expressed in terms of a 

 nondimen.sional pressure coefficient Cp as 



where the coefficient Cp depends on the wave form and 

 height. Following Lamb's example of equation (4), 

 the symbol ACp is used to emphasize that only part of the 

 total air pressure, the out-of-phase component, is con- 

 sidered. 



In practice it is more convenient to measure the hori- 

 zontal drag force acting on the water surface. Express- 

 ing it in terms of the drag coefficient C'd, as defined cu.s- 

 tomarily in aerodynamics: 



Drag = (■„(p72)(F - c)"- 



(14) 



The total energy per second transmitted by wind to 

 water is 



E = mrag) 



C. 



{V - c)-c 



(15) 



2.2 Effect of Skin Friction. Before proceeding further, 

 it is important ti) comment on the remark: ". . . neglect- 

 ing the tangential action which seems to be of secondary 

 importance . . ." in the pre\'ious quotation from Lamb. 

 In the case of slightly viscous fluids, such as air and water, 

 the flows are generally well described Ijy the potential 

 theory, and a slight dissipation of energy by viscosity 

 does not .sensibly modify the results of this theory.' 

 The primary effect of \'iscosity (jn the general flow is 

 indirect; it appears as a modification of the boundary 

 conditions l)ecause of formation of the boundary layer. 

 At the Ijoimdary of a fluid and a .solid, or at the air-water 

 interface considered here, there is a strong velocity 

 gradient, and the viscous effect becomes important in 

 the form of "skin friction." The essentially potential 

 flow can be affected directly only l)y normal pressures, 

 and not by the tangential .skin friction. This latter 

 represents the dissipation of kinetic energy in the form 

 of eddy-making first and finally of molecular motion, or 

 heat, so that this part of the energy is no longer available 

 to the potential motion of waves. It is therefore quite 

 proper to neglect the tangential frictional force in the 

 theory of wave formation just outlined, and to consider 

 only the effect of normal pressure.'' 



Confirmation of the foi'egoing statement can be found 

 in the theory of wave resistance of ships, as dc\'eloped 

 primarily by Havelock, Weinblum, and Guilloton, and in 

 experiments conducted in connection with this theory. 

 The particularly rele^'ant references are Wigley (1937), 

 Havelock (1948, 1951), Shearer (1951), Birkhoff, 

 Kor\-in-Kroukovsky and Kotik (1954) and Korvin- 

 Kroukovsky and Jacolis (1954). The entire treatment is 

 based on the potential theory, yet it gives results in good 

 agreement with towing-tank test data (in the case of 

 fine boats formed of parabolic arcs). Despite the fact 

 that possibly 2.^ of the total towing energy is dissipated 

 in skin friction, the effect of this on waves (i.e., on the 



B = ACp'/.,p'{V - c) 



(13) 



' Lamb (D, art. 346, p. 62:5) also Appendix A art. 4.4; the tre- 

 mendous success of aerodynamics has been based on this postulate. 



■• It is interesting to note that frictional drag is also neglected in 

 the recent advanced work on energj' transmission from wind to 

 waves by Miles (1958) and Phillips (1958). The air viscosity 

 is considered only in the .sense of defining the effective wind ve- 

 locity. 



