per unit of frequency and direction and makes it possible to compare results 

 from various experiments. This entity is known as the frequency-direction 

 spectral density of wave energy or, more simply, the frequency-direction (FD) 

 spectrum denoted by S(f„,5j„) , and defined by 



^(^n'^™) = dFd^ (7> 



52. The left side of Equation 7 is often abbreviated as S(f,^) where 

 the subscripts m and n have been dropped from the arguments, suggesting an 

 approximation of a continuous function. 



53. The result expressed by Equation 7 is still proportional to wave 

 energy per unit crestlength for a region of the frequency- direction domain of 

 dimensions df and d9 from a given experiment since df and d^ are the 

 same everywhere in the domain. The conversion from spectral density to energy 

 is multiplication of the elements of Equation 7 by pg-df-d9 . 



54. Generally, energy is found from larger or smaller areas in the 

 frequency- direction domain by adding (integrating) elements (or partial 

 elements) over the region of interest. In particular, one can obtain the 

 conventional frequency spectrum (resulting from measurements at a single point 

 in space and from which the JONSWAP, TMA, etc., models have been derived) from 

 the frequency-direction spectrum by summing Equation 7 over all direction 

 contributions times the incremental arc d^ for each frequency. Denoting the 

 frequency spectrum by S(fn) , the result becomes 



s(f„) = I s(f,,ej dd (8) 



111=1 



55. This equation gives a measure of the total wave energy in each 

 frequency increment where contributions from all wave directions are included. 

 The frequency spectrum can be considered as a frequency-direction spectrum 

 with the lowest possible directional resolution (none) where the "incremental" 

 arc length is the full 360 deg of possible wave directions. The left side is 

 often abbreviated as S(f) . Since this definition is derived from an 

 integration of S(f,^) , it can be called the integrated frequency spectrum 

 (IPS). 



23 



