38 



THEORY OF SEAKEEPING 



Fig. 39 Stairway approximation to wave spectrum. Area of a given rectangle is proportional to square of height of sine 



wave with associated frequency (from Neumann, 1953) 



bU^/bo} for a fully arisen sea at a wind velocity of 20, 30 

 and 40 knots is shown in Fig. 38 as a function of cycles per 

 second / = l/T = w/'Itt. These curves are computed 

 from eriuation (77). The ordinate,-' IT'/ = dUf/d/, is 

 proportional to the spectral wave height. It is given 

 with an arliitrary scale, since it depends on the constant 

 C. 



6.21 Total v^ave energy and evaluation of C. The 

 total wave energy in the spectrum lietweeii periods and 

 T or between frequencies <» and co is gi\-en by 



U 



r dU^ = -Cpcfir' roj-i^exp (- 



2g= 



h -(ji 



(78) 



After substitution of h = 2g''/U'' 



b/o 



and 



f/o, = -(6"V.r2)rf.r, 

 equation (78) can be evaluated as 



'^ Subscript / indicates tliat IF and U are functions of fretjuency 



/. 



u 



^^"■'■{i^sr "-"'"■■• -9 



2+1' 



where the integral 



X 



{c-'r-dx = ^$(.r) 



(70) 



(80) 



and ^(.r) is the error integral which can be taken from 

 tables. 



The total wave energy in the case of fully arisen sea 

 follows by integration between limits co = co and oj = 0; 

 i.e., from x = to <» : 



U=(cp^-^±](lAlA (81) 



32^= 



or 



U = (const) U^ 



The constant C can now be evaluated if the relation- 

 ship lietween the total energy U^ and the wa\-e height is 

 estal)lished. First, it is found convenient to omit the 

 constant quantities p, g and the numerical factors in the 

 foregoing, and to replace the energy U by the quantity E 



