54 Observations and Measurements of Ocean Waves 



frequencies between a = 083 and a = 300, that is, in terms of periods 

 between 12 and 3 sec. The maximum of spectral energy is concentrated in 

 an optimum band around o- max = 01 24 or r max = 81 sec. 



With increasing wind speed the range of significant frequencies extends 

 more and more towards smaller a values, and with a 30 knot wind the 

 significant range is between about a = 048 and a — 0-24, or between 

 T = 21 and T = 4 sec. The optimum band is displaced to lower frequencies 

 at <r max = 00826 or r max = 121 sec. 



The frequency of the optimum band, <r max , is found from equation (III. 6) 

 by differentiating the energy distribution function with respect to a. One obtains 



ffmaxtf = } hg. (III. 7) 



The law that the product of the frequency of the energy maximum and the wind 

 speed is constant (= j/f.g), corresponds to Wien's law in the theory of 

 thermal radiation. It is one of the theoretical results which can easily be 

 checked by observations, if in the future reliable spectral analyses are available 

 from sea surface wave records of pure fully arisen wind generated sea. 



Equation (III. 7) can also be written 



7W = 0-785« (III. 8) 



where T max means the period of the energy maximum in the energy curve 

 for the wind speed v. With increasing frequency the spectral energy rises 

 rapidly to the maximum <r max , whose position is given by (III.8). Beyond this 

 value it drops approximately inversely proportional to the sixth power of a. 

 The total wave energy in the spectrum between the period and T, or 

 between the frequency oo and a is obtained by integrating 3 12 between these 

 limits (Pierson, Neumann and James, 1953): 



u ™= c ^\lV^ { 



(III. 9) 



e~ x% [jc 3 3x) 

 ) b 5 \2 4) 



where b = 2g 2 /v 2 and .y 2 = b 2 /a 2 ; (III. 10); &(x) is the error integral. The 

 function (III. 9) is called the co-cumulative power spectrum of waves and the 

 curve which represents this function in a graphical form is called the CCS-curve. 

 Examples of such CCS-curves are shown in Fig. 33 for wind velocities between 

 20 and 30 knots. The ordinate is scaled in ^-values which are related to the 

 wave energy U by equation: 



U = IgQE . (III. 10) 



E has the dimension of a (length) 2 . This is done in order to facilitate the 

 practical use of the CCS-curves. 



The total wave energy in the case of fully arisen sea follows from equa- 

 tion (III. 6) by integration between the limits a = oo and a = or from 

 equation (III. 9) with x ■ = oo: 



