SEAWAY 



15 



Table 4 Estimated Energy Balance in Observed Waves 



Legend: , ■ r 



Case 1. Johnson and Rice (1952) — Kuii la. d estimated as mean of first four lines of Table 3, assuming C, = 0.0028 on basis of 

 Table 1, Model 4. 



Case 2. Francis (1951) — Severe Condition: Generator and Fan — 12 mjjs. Data estimated Ironi curves; Cd + C, = 0.0242 from 

 Table 2, Cr estimated as 0.0028. 



Case 3. Francis (1951)— Gentle Condition: Fan only, 5 mps; Cd + CV listed in Table 2 would leave miprobably small value for 

 Cd. Cd = 0.0052 is assumed from Thijsse's (1952) experiments. 



Case 4. Roll (1951 ) — Open Air (Observations, Nos. 24 and 26. Conditions at two fetches and wind are means of several observations 

 and do not represent a simultaneous set of conditions. C,j estimated as for Case 3. Roll's (1948(1, Table 1 ) wind-profile measurements 

 give the unreasonably low value of Cd + (V = 0.0025. 



Case 5. Roll ( 1951 ) — Observations 29 and 32. In shorter fetch and stronger wind than Case 4. In view of mean steepness of waves, 

 Cd is assumed as the mean of Cases 1 and 2. 



Case 6. Neumann (1950, page 41) — A typical sea condition in North-East trade winds ciuoted from Larisch-Wind Beaufort 5 to 6. 

 Cd assumed from Thijsse's (1952) experiments. 



6 of Table 4 yieWs values of K of 0.022, 0.022, 0.010, 

 0.0008, 0.073 and O.OOOOd'.), respectively. RowtJen, on 

 the basis of swell attenuation, gave the order of magni- 

 tude oi K as5 X 10"=, which checks well with the figure 

 computed for Neumann's case 6 in trade winds. The 

 variability of the coefficient K indicates that the problem 

 is not yet solved. It appears to the author that Bow- 

 den's dimensional reasoning was faulty in that T and X 

 are not independent, and X and a are not dimensionally 

 distinguishable. It is suggested that the derivation 

 based on \-on Karman be investigated further. 



3.2 Summary of the Energy-Balance Problem. Sum- 

 marizing the foregoing, neither the problem of energy 

 transmission from wind to waves nor the problem of 

 energy dissipation in waves has been solved. The solu- 

 tion of the fir.st hinges primarily on evaluating the celer- 

 ity of the wavelets overlaying larger waves.* The 

 solution of the second requires theoretical and experi- 

 mental determination of the function form of the turbu- 

 lent viscosity n*. Flume and open-air wave observations 

 have not been reported in a form useful for evaluating 

 the wave-energy balance. New observations and flume 

 tests are needed. These must be planned specifically 

 with a suitable analytical procedure in mind. It cannot 

 be too strongly emphasized that wind-flume and small- 

 scale natural waves differ radically from natural ocean 

 waves of significant size in the ratios c/U and a/X. 

 Small values of the former and large values of the latter 

 predominate in small waves. Small-scale tests and ob- 

 servations, therefore, cannot be used as direct representa- 

 tion of open-sea conditions. The objective of the small- 



* A more exact formulation of this statement was given on page 

 9 (last paragraph) and it will be discussed further in Section 

 4.4. 



scale observations should be, therefore, the develtjpment 

 of rational relationships which then can be applied to any 

 wave system. 



The decrease of wave steepness a/X with increase of 

 wave length X is one of the most reliable relationships ob- 

 served in ocean waves. The ciuantitative relationship 

 between these C[uantities was first noted by Sverdrup and 

 Miuik (194G, 1947) and later c<jnfirmed "by Roll (1951) 

 and Neumann (1950a and b, 1953a and b). Bowden ap- 

 pears to be the first to offer a rational explanation of this 

 observed fact. His formula, etiuation (35), indicates a 

 rapid increase of the energy-dissipation rate with wave 

 height and length. The formula for the transfer of the 

 energy from wind, on the other hand, depends on C\ 

 which is a function of the wave form, but is independent 

 of wa\'e height. Thus the balance between the energy 

 received and dissipated can only be achieved in higher 

 waves at a lower a/\ value. While formula (35) appears 

 to have exaggerated the influence of the wave size, the 

 significance of this factor can scarcely be doubted. 



The exposition of the energy balance in the growth of 

 sea waves has been presented simply, in order to call at- 

 tention to the basic elements of the problem. Both the 

 transfer of the energy from the wind and the dissipation 

 of the energy by turbulence and internal friction must be 

 considered in defining the growth of waves. A proce- 

 dure analogous to the foregoing, but much more elabo- 

 rate, was employed by Neumann (1952a, 6, c). It is re- 

 viewed in Appendix B. Although it is listed under 

 "Practical Approach" because of the many empirical 

 and intuitive steps involved, it is the most complete dis- 

 cussion of the fundamental principles involved in the 

 energy balance in waves available to date. It served as 

 a basis for the wave-forecasting method of Pierson, Neu- 



