0.875 and L24, the true value will lie between tlj.ese bounds four times in ten. 

 The contours in figure H,28 can be smoothed by tciking these facts into 

 consideration and by assuming that th.e true spectrum is basically a smoothly 

 varying fujiction. The resulting smoothed spectrum, is shown in figure 11.29. 

 An attempt to indicate the very steep forward face has been made. In order to 

 obtain this smoothed version, it was ordy necessary to go outside the 40 percent 

 bounds about 10 times in. the area -wdiere the estimates were greater than 0,0050. 

 Analytic representation of the directional spectrum 

 The curves shown in figure 11„22 ajid the data tabulated in Table 11.6 pro- 

 vide a w^ay to find an analytic representation for the directional spectrum of the 

 sea. The results of the frequency analysis show that the theoretical Neurnann 

 spectrum as a function of frequency fits the data as summed around semicircles 

 quite w^ell. 



The spectrum as a function of frequency ajud direction can therefore be 



written as equation (11^16), 



T 2 * 2 2 



(11.16) [A(tx. e)]2 = f^-^ — g^^. [f(Hi,e)l 



4 

 where c = 3.05 x 10 and all values are in. c, g, s. units. 



The function, f(|j., 0) should have the property that it is zero over half 



the plane, that 



/ir/2 



(11.17) J fCfx, e)de=i 



and that f{\i., 9) > 0. 



225 



