SEAWAY 



43 



b) Abandonment of the data on which the spectrum's 

 form was based, Fig. 'M\, in fa\'or of \'isual observation 

 data, Fig. 40, in evahuiting the constant which governs 

 the height of the predicted wa\'es. 



The explanation given by Neumann for the first of 

 these is not clear. In two papers, Walden and Piest 

 (1957, 1958) attempted to clarify this step as well as to 

 impro\'e the precision of various details connected with 

 the spectrum. They came to the conclusion that the 

 validity of Neumann's deri\ation can be neither proxed 

 nor disproved liy these considerations and rests on the 

 gathering of sufficient em])irical data. 



The experience of Roll and I'ischer-'-' indicates that, had 

 Neumann consistently used Fig. 36 for the evaluation of 

 the constant as well as the spectral form, he would have 

 arn\ed at about doultle the actual wa\'e height. Instead, 

 he substituted the wave heights found by his pre\'ioiis 

 visual ob.servations. Fig. 40, and on this basis arri\'ed at 

 an acceptable constant. 



The foregoing remarks, taken with the fact that Neu- 

 mann's constant (' has the dimensions of \/t, lead to a 

 plausible hypothesis. Were Neumann's spectral form 

 accepted as invariable for all significant wave heights, 

 the "constant" C could be taken as variable and a func- 

 tion of the mean period T. For the light sea conditions 

 on which Fig. '.U\ was based, the con.stant may possibly 

 have double its \'alue for the mean sea conditions shown 

 in Fig. 40. This hypothesis appears to have its con- 

 firmation in the trend of the observed points in Fig. 40 

 toward a ciu-\'ed rather than a straight line. The cor- 

 responding trend can al.so be seen in Fig. 36. A more 

 exact curve-fitting would have given a concave curve, 

 rather than a straight line. 



Roll and Fischer (1956) called attention to the ap- 

 parent discrepancy in the mathematical form of the 

 spectrum given by e(iu;itions (76) ami (77). Both e(|ua- 

 tions are meant to describe the same wave form, and on 

 first thought one would expect that the same wave length 

 or period 7',„:,x woukl be indicated in both as correspond- 

 ing to the wave component of maximum energy. T„,^^ 

 can be found by differentiating eciuations (76) and (77) 

 with respect to period T and frequency w. respectively, 

 and by ecjuating the derivative to zero. When this is 

 done, two different values of T^max are found: 0.641 U 

 and 0.785 [' (with T in sec, U in mps). Neiuiiann used 

 the second value only, without discussing the subject. 



Such an apparent discrepancy and a method for dealing 

 with it had been known pre\i()us]y in the theory of ther- 

 mal radiation. It will be recollected that the energy can 

 be defined only for an interval A7' or Aco, at a certain 

 value of T or oj. The apparent discrepancy arises be- 

 cause of the difference in the corresponding intervals 

 dw and dT: 



f'^ = - "rl 'i'J' 



(86) 



w'hile 



Wr (IT = IF„ da 



This apparent confusion can be avoided Iw defining 

 the periods and frecjuencies in logarithmic form, rf(ln T) 

 = (/ T/T and (/(In co) = dw/w. Equation (76) is then 

 written as 



AUr 



W: 



dT 



Cp 



:V27 



7'< exp 



2TrU 



dT 

 T 



(87) 



and from 



(IT 



T 



it follows that 



doj , ,,r dio ,,. dT 

 — and n„ — = Wr-p^ 



03 W 1 



■^0' 



do. 



(88) 



ML = Ho, — = -Cp V- "~ exp 

 to 2 



The powers of T and l/co are now the same in Itoth 

 etjuations, and both yield 



r„., = 0.641 U (89) 



The total energy is obtained by integration as 



U 



/: 



dlL 



—(- p -^- 



l?-'-^{^) 



do3 (90) 



This expression is integrable, and by substitution of 



Vb 



2g\ h 



Tj., , •'■'; rfo) = - ^dx 

 U" CO- .r- 



U is evaluated as 



U = C, 



P^U^ 



1-1 + 



2£- 



exp 



V'oi'- 



and for the fully developed sea 

 Cpir-' 



U = 



16ff 



U'' (erg cm "-) 



(91) 



(92) 



*' To be discussed later. 



It is observed that in this case the energy is shown to 

 be proportional to U*; i.e., the wave height to U-, as 

 against ["'•'' originally derived by Neumann in connec- 

 tion with the spectrum. The dependence of the wave 

 height on the stjuare of the speed has been shown pre- 

 viously b}' Sverdrup and Munk (1947) by the horizontal- 

 ity of the H versus <-curve in Fig. 21. It is likewise in- 

 dicated by the horizontality of the curves at the RH side 

 of Figs. 22 to 25 taken from Neumann (19526). Darliy- 

 shire (1952, 1955) also shows wave height as proportional 

 to wind velocity scjuared. 



It will be recollected that Neumann u.secl Fig. 36 in 

 evaluating the form of the spectrum, but abandoned it 

 without explanation in evaluating the constant (' in 

 fa\'or of a plot of his ow-n visual observations in Fig. 

 40. Roll and Fischer (1956) now return for e\-aluation of 

 the constant to the original empirical relationships (67) 

 and (68) established by Neumann on the liasis of Fig. 36. 

 They then follow Neumann in assuming that the rela- 



