SEAWAY 



39 



Table 7 



Amplitude Height 



Average wave O.SSUVE i'Tl^'E 



Mean of 'A highest waves lAl&^E ■2.S32VE 



Mean of Vio highest waves 1 . 800 V^ 3 . 600 V^^ 



Mean of V™ higliest waves 2.359^^ 4.718v'j^ 



equal to the sum of scjuares of the amplitudes a of the in- 

 dividual component wave trains which go to make up the 

 actual wave motion as it is observed. It is related to the 

 total wave energy by the expression 



/? = 2a= = U{2/pg) (82) 



The reader's attention is called to the fact that so far in 

 the discussion wave heights were used, but now E is de- 

 fined in terms of amplitude. Darbyshire (1952) has de- 

 fined an "ecjuivalent wave" height H as that of a simple 

 harmonic wave of the same energy content as the complex 

 seaway. If the amplitude of this wave were designated 

 by .4,' then .1 = VE. It follows that E = 'A of the 

 area of the spectrum drawn in terms of the wave heights, 

 as is shown bj' Fig. o4. Fig. 30, on the other hand, 

 shows the spectrum drawn in terms of wave amplitudes, 

 and in this case E = the area of the spectrum. 



The value of the foregoing definition lies in that the 

 area of the spectrum, or the c|uaiitity E, is connected 

 with \'arious statistical properties of the observed sea- 

 way; i.e., with the distribution of the apparent wave 

 heights. Longuet-Higgins (1952) has shown theoreti- 

 cally that the statistical relationships given in Table 7 

 exist 



It follows from Table 7 that theoretically 



H,y/Hm = 0.()25 

 Hyw/Hu, = 1.27 



According to a sunuuary by Alunk (1952), the fore- 

 going ratios computed from wa\^e records are 0.65 and 

 1.27. The first ratio is also confirmed by the observa- 

 tions of Darlington (1954). Thus, very close agreement 

 is found between the theoretical relationships of Longuet- 

 Higgins and the wa\-e relationships obser\'ed at sea. 



The foregoing relationships serve as the connecting 

 link between the energy spectrum of wa\'es (in terms of 

 amplitudes) defined by its area E, and the observable 

 properties of the sea defined most frecjuently by the 

 height of the "significant wave"; i.e., the mean of the Vs 

 highest waves. It is now possible to evaluate E, and 

 from it the constant C on the basis of a^-ailable sea-wave 

 obser\-ations. For this purpose, Neumann uses the re- 

 sults of his own visual observations made during a \'o.yage 

 on the j\IS Heidberg. Fig. 40 shows the plot of wave 

 heights versus wind speed. Vertical lines show the range 

 of variation of the observed waves. In the earlier 

 work — Xeinnann (1952a) — these were described as 

 "characteristic" waves, and it was mentioned that at 

 15-16 nips, for instance, the heights of characteristic 

 waves fluctuate between 4 and 9 m. The upper limit of 

 these fluctuations is now interpreted as the mean of '/m 

 highest waves, and the empirical .solid line 



Fig. 

 wind 



1 Z 345678 10 

 U(^/sec) ^ 



40 Wave-height observations at different 

 speeds (MS Heidberg) (from Neumann, 

 1953) 



//i/io = 0.000009 U'- 



(83) 



is fitted. The broken line, representing the significant 

 wave, is drawn parallel to it with the coefficient reduced 

 in accordance with the Longuet-Higgins relationships. 

 By the use of these relationships also the average wave 

 height is expressed as 



//■ave = 0.492 //i/io = (0.443 X 10-^) IP'' (in cgs units) 

 and from this 



E 



H„ 



1.772 



0.443 



1 



.n2 J 



or 



U = - pgE = 3.125 X lO-'' t'= (erg cm -5) (84) 



and by comparison with equation (81). the constant C is 

 evaluated as 8.27 X 10"^ 



