14 



THEORY OF 5EAKEEPING 



where Ei and E^ are the energies per square foot per 

 second carried over through the planes at Fi and F«. 

 From ecjuation ((30) of Appendix A 



E, = (l/4)p(/ciai2 

 in any consistent set of units, or 



£'1 = 16 Ciar 



(27) 



(28) 



in foot-pound units for sea water sit pg = 64 pcf. Here 

 fi is the wa\'e celerity and en the wave amplitude at Fi. 

 An identical expression with subscripts 2 will hold at Fn, 

 The foregoing rate of change is ecjual to the difference 

 between energy E^, received from wind, and £,,.. dis- 

 sipated in internal friction; i.e.. 



dE/dx = Fp - E,, 



(29) 



Ep is evaluated on the basis of normal pressures acting on 

 water as 



E, = C,^ {U - cr c 



(30) 



where for "standard air" p' = 0.00'2o7 pounds per cu ft. 

 The energy Ej, dissipated in internal friction is given 

 for classical gra\'ity waves by equation (66) of Appendix 

 A in terms of the molecular coefficient of viscosity p. 

 G. I. Taylor (1915), in his study of atmospheric turbu- 

 lence, introduced a coefficient of turbulent viscosity 

 which is larger than /j. This coefficient will be designated 

 by n*. Neumann (19496) also shows that the effective 

 turbulent coefficient of viscosity n* in wave motion is 

 many times larger than fi. The turbulence responsible 

 for this increase results partly from the energy- transmit- 

 ted from wind by skin friction and partly from the kinetic 

 energy of wave motion dissipated in the process of the 

 breaking of wave crests. Assuming for the present that 

 the classical expression is valid with a new coefficient m*. 



E, 



2 fi* k'c-a- 



(31) 



where k is the wave number, 'Iir/X. Expression (31) 

 is valid in any consistent set of units. In the foot- 

 pound system, and taking n = 2.557 X 10~^ for sea 

 water of 59 F (15 C), it becomes 



E,, = 0.065 >i(a/\)- 



(32) 



where n denotes the ratio m*/m- 



Since the wave height, H = 2a, and wave length X 

 are reported in all wave observations, the energies E\ 

 and Ei can be computed readily. The energy received 

 from wind, E,,, also can be computed, pro\ided the drag 

 coefficient Cj is known. (',; can be reasonably estimated 

 from the data of the foregoing section. The only 

 quantity completely ^mkno^^^l is the ratio n. It can be 

 computed on the basis of equations (26) and (29) as 



n = 



1 



0.065(a/X)2 



X 



C, ^{U - cr-c- (E,- E,)/{F, - F,) 



(33) 



Data and computations for six cases found in the 

 literature are shown in Table 4. This is but a small 

 sample of data of varying reliability (for the present 

 purpose) but nevertheless a few conclusions can be 

 drawn : 



a) There is little purpose in analyzing in this man- 

 ner more of the data found in the current literature. 

 Data must be obtained specifically with this type of 

 analysis in mind for the results to be reliable. 



h) The ratio n is not a constant but varies with 

 (a/X) ratio and with wave height. Expression (31) 

 must therefore be modified by including a proper func- 

 tional relationship for ^i*. 



c) In the minute and mild wa^'es (case 3) n is about 6. 

 For essentiall.v the same wave height but greater steep- 

 ness and hence larger Cj in case 5, n increases to 41. 



d) The ;i-\'alues of 207 and 439 in cases 1 and 6 are 

 apparenth' exaggerated by the excessi\-e influence of the 

 wave celerity c in the factor {V — c)-c entering into 

 Ep. In Table 4, the c-values are based on the reported 

 mean or significant waves. It appears more probable 

 that the energy-transfer calculations should be based on 

 the smaller waves or ripples by which the significant 

 waves are overlaid. This would call for smaller X and c 

 in the foregoing calculations. 



c) If the same rea.?oning were applied to wave dissipa- 

 tion, the calculations also would have indicated greater 

 energy dissipation, since smaller waves of lower c/U-ratio 

 usually have higher a/X-ratio. 



/) Case 2 should be omitted. A comparison with 

 case 1 shows too low a X for a similar c/U-ratio. Ap- 

 parently, the waves preformed by the wave generator 

 were too long for the amliient wind conditions. 



3.1 Energy Dissipation (by Bowden, 1950). The de- 

 velopment indicated by paragraph b) of the foregoing 

 section was attempted b.y Groen and Dorrestein (1950) 

 who, on the basis of the previous work of Piichardson 

 (1926) and Weizsaker (1948), assumed p* to be propor- 

 tional to X'*''^. Bowden (1950) showed that Weizsaker's 

 reasoning is not applicable to waves and, on the basis of 

 dimensional reasoning, derived a new relationship. If 

 p* depends on wa\'e proportions, it should be a function 

 of wave length, amplitude and period, so that 



EX'tt^ry 



(34) 



It follows that a + i3 = 2 and 7 = — 1. Bowden took 

 the simplest assumption that a = fS = \ and wrote 



,* 



M 



K 



(34a) 



where K is a nondimensional coefficient. The rate of 

 energy dissipation is then 



£",, = 2 p K k^c'a^ (35) 



Bowden confirmed the hjregoing results by a deri\-ation 

 based on von Karman's (1930a and b) similarity hy- 

 pothesis for shearing flows. 



The application oi the foregoing equation to cases 1 to 



