Lewis 



Finally, for the present purpose the spectrum must be transformed from 

 amplitude (b) to slope (c) form. In general, maximum wave slope is l-ni^/k. 

 Since k - i-ng/co^, maximum slope can be expressed in terms of w, 



a;2 i 



where ^g ^^ wave amplitude. The square of the amplitude of a wave component 

 is given by:* 



[r(logg QJ)] S loggO) 



where [T(iog^co)]^ represents the wave spectral ordinates on the log^oj base. 

 Therefore, the square of the slope is given by: 



4 



-^ [r(loggaj)]2 S loggO). 



g 



I 



Hence, if the spectral ordinate plotted in (c) represents 



-^ [r(log o))] , 



2 

 g 



an incremental area will represent the square of a component wave slope. Fur- 

 thermore, the area under the spectrum (with finite limits) can be interpreted as 

 a mean wave slope. 



The most obvious difference between the log-slope form and the conventional 

 form of spectrum is the suppression of the spectrum peak which is so prominent 

 in the conventional form of presentation. This calls attention to the fact that the 

 wave components at the peak of a conventional spectrum are usually less steep 

 than at the higher frequencies. It has been found that for many ship motions 

 wave amplitude in relation to length, i.e., wave slope, is more important than 

 wave amplitude directly, or energy. For such motions, the log-slope form is 

 preferable for the study of ship behavior. 



For example, pitch angle is directly related to maximum slope. In fact, as 

 wave lengths become very long and the frequency of encounter is far from reso- 

 nance with the ship's natural frequency, pitch amplitude will approach wave 

 slope asymptotically. 



The manner in which the new form of log-slope sea spectrum may be used 

 in predicting ship responses is shown in Fig. 2 for the case of pitching motion. 

 The figure shows the simple case of a ship heading directly into a long- crested 



'''The original concept of [2] is used here, in which the spectrum represents am- 

 plitude squared. In sonne systenns a factor of 1/2 is introduced in order to 

 represent wave energy. 



190 



