3. A DIRECTIONAL WAVE SPECTRUM 



The power spectrum S(£,m,f) is the directional wave spectrum of 

 n(x,y,t). If ri(x,y,t) is to be real, every infinitesimal of the form 

 given in the continuous case above presumes the existence of its com- 

 plex conjugate. Let us consider the one-sided power spectral density, 

 S'(£ m ,f ), of a single real wave where _< f^ < °°. For such a real 

 wave°elemeSt of length X^, from a direction 6^, 1 6^ < 2^, the value 

 S'(£o,%.fo)' = Sa^,m^,fJ + S(-£o,-m -f^), where ^^ = K cos e„, 

 m = K "siSe^, and K^ = l^A^. If the above real wave came from a 

 direct?on (2it - 9) , the power density would be S ' (H^.-m^, fo) - S{1^, m^,!^; 

 + S(-Jio,ii^,-fo). 



Figure 2 illustrates a real wave of length Aq, from a direction 9^, 

 in two-dimensional spacial frequency (wave number) space. 



Wave Direction 



FIGURE 2. REAL WAVE IN WAVE NUMBER SPACE 



If the wave number relation 



K=EitfVa'TaviW(aitKlr.) 



(3.1) 



holds, refer to Kinsman (1965 - p. 157) and Munk et al (1963 - p. 527), 

 where h = water depth, g = acceleration of gravity, and K = wave number 

 = 1/X X being the wave length; then a relationship between f and (&,m) 

 is implied that requires a wave frequency fo to have a unique wave num- 

 ber k^. From this we have the general one-sided spectral form for waves 

 where f = f^ of 



