THEORY OF SEAKEEPING 



I I Smooth 



I ] Wavy rcflec-Viors 



■-" Weak waves visible 



Measured or measurable 

 waves 



Fig. 1 



300 



Frequency distribution of initial waves as function of 

 wind velocity (from Roll, 1951) 



wave components traveling at angles to the wind direc- 

 tion. The existence of these wave components he 

 attributed to air turbulence which produces local tra\-el- 

 ing pressure areas and causes oblique waves (like ship 

 bow waves) to form according to the well-known theory 

 of the traveling disturbance (Lamb, D, art. 256, pp. 

 433-437: Davidson, 1942). 



Becau.se of the glaring discrepancy between the results 

 of Kelvin's analj'sis and observed data, Jeffreys (1925, 

 1926) undertook a new analysis in which both dissipation 

 of the energy in water due to viscosity and modification 

 of the air pressures due to air viscosity were taken into 

 account. In the potential flow of air, maximum air 

 velocity and maximum drop of air pressure are found 

 over wave crests, and minimum air velocity and an 

 increase of air pres.sure occur in troughs. Thus wave- 

 exciting forces exist (causing the instability indicated by 

 Kelvin), but because of the symmetry of the flow on both 

 sides of each wa^'e crest, there is no mechanism for trans- 

 niitting the energy from air to water. Jeffreys assumed 

 that each element of distorted water surface acts essen- 

 tially as a flat plate in a fluid stream; i.e., that the pres- 

 sure acting on it is proportional to its inclination, which 

 can be expressed as cirj/dx. With the wave profile 

 expressed as- 



7} = a sin k{x — ct) 

 the slope of the water surface is 



brii'(ix = ah cos k{x — cl) 



(0 



(2) 



Jeffreys assumed an element of the pressure exerted by 

 wind on an element of the wave surface to be 



p = fip'{V — cYbr)/bx 



(3) 



^ In this case and in the following section the wave formulas 

 differ from those given in Table 1 of Appendix A in that the origin 

 of co-ordinates is taken at the nodal point instead of the crest. 



here p is the pressure, /3 a nondimensional coefficient, 

 p' the density of air, and T' the air \-elocity. 



AVere the air flow potential, the pressure distribution 

 would be symmetrical about the wa\e crest, and the 

 mean value of the coefficient /3 would be nil. Jeffreys 

 assumed, howe^'er. that in a A'iscous flow the leeward 

 slopes of the waves are shielded, so that asymmetry of 

 pressure distribution exists and the coefficient /J has a 

 value (to be determined empirically) greater than 

 zero. 



2.1 Transmission of Energy from Wind to Harmonic 

 Waves. Without making assumptions as to the nature 

 of the air pressure, Lamb (D, p. 625) wrote: "It is not 

 likely that the action of the wind, even on a simple 

 harmonic wa^■e-profile, could be represented by any 

 simple formula. JBut, neglecting the tangential action, 

 which seems to be of secondary importance, we may 

 imagine the normal pressure to be expressed bj' a Fourier 

 series of sines and cosines of multiples of ki^x — ct), and 

 it is e\-ident that the only constituent which does a net 

 amount of work in a complete period has the form 



Ap = C cos k(x — ct) 



(4)" 



In A'iew of the importance of the foregoing quotation, 

 a more detailed de\'elopment of this idea will now be 

 presented. The air pressure acts normally to the water 

 surface. For the purpose of analysis it is convenient to 

 consider the normal pressure \'ector as decomposed into 

 vertical and horizontal components, p cos 6 and p sin 6, 

 where 9 is the inclination of the water surface. This 

 inclination in waves is generally small and these pres.sure 

 components can be e^-aluated in two ways. 



(a) Attention can be concentrated on an element of 

 water area at a certain fixed location x moving up and 

 down with vertical ^'elocity v. The mean work done by 

 the pressme p per second is 



1 r ack C^ 



E = - j pvdl = Y j V cos k{x - ct) dt 



(5) 



A positive sign is assigned to the work of the downward 

 pressure acting on the surface element with downward 

 velocity. 



(b) The alternate method is to consider an element 

 of the wave form mo^■illg horizontallj- with celerity c. 

 The mean work per second is 



E = ^ 



f 



Jo 



p{dr]/dx) cdl = 



i Jo 



p cos k(x — ct) dt 



(6) 

 Let the unknown pressure distribution o\'er the wavy 

 water surface be represented bj' a summation of sine and 

 cosine harmonics, i.e., 



" n 



p = Y, ^-^nsm nk(x - cO + Z) B„ cos nkix - ct) (7) 



For the wave form represented by equation (1) 

 drj/dx = a k cos k{x — ct) 



(8) 



