52 Observations and Measurements of Ocean Waves 



6. The Energy Spectrum and the Spectral Analysis of Recordings of Ocean 

 Waves 



The actual complex wave pattern characterized by the distribution of the 



apparent periods T and the apparent heights H is closely related to the 

 spectrum of the wind-generated waves. It is not difficult to derive theoretically 

 this spectrum from very simple basic assumptions (Neumann, 1953/?, 1954). 

 It will be pointed out that theoretically all periods T of the frequencies 

 a = InjT are present in this spectrum. Analogous to the distribution of 

 thermal radiation in the individual frequency intervals of continuous spectra, 

 it is postulated that to a given range AT there belongs an energy interval AU. 

 Then, the spectral energy density of the sea motion can be defined by the ratio 



W T = ^ [erg cm- 2 sec" 1 ] . (III. 1) 



The spectral wave energy AU T for the average period T in the interval AT 

 is proportional to the square of the spectral wave height H T in the same 

 interval. Therefore 



Ht = ^ [cm 2 sec- 1 ], (III. 2) 



where h% is the square of the heights of individual waves in the spectrum. 

 With sufficient accuracy for our purpose and with (11.14) 



1 dhl 

 dU T = W T dT = igQjfdT [erg cm- 2 ] (III. 3) 



can be defined as the mean energy per unit area of the sea surface of waves 

 whose periods lie between T— \AT and T+\AT. The total energy in the 

 wind generated wave pattern will be given by 



U 



:QJ~dT. (III. 4) 



The spectral wave height will be not only a function of the wave period (or 

 wave length), but it also will depend on the wind velocity. 



Again, analogous to the classical theory of thermal radiation, it must be 

 assumed that the spectral wave energy bU T or the energy density W T , re- 

 presented as a function of the period T, will attain a maximum with a certain 

 wave of a period T nmx and, with a further increasing period, it will drop until 

 it practically disappears. This form of the spectrum is also suggested by the 

 frequency distribution of the composite sea motion. 



In order to arrive at a mathematical form of the energy-distribution curve, 

 one may use the empirical relation derived from the statistics of basic values 

 of the apparent waves (Neumann, 1953a, b): 



