and at mooring depths of 50, 100, 150 and 200 feet below the surface. The 

 results are summarized in Table 3 which presents the root -mean -square 

 values of pitching motion for the various conditions. Values greater than 

 25 degrees are not reported since this implies pitching motion in excess 

 of 90 degrees which is definitely not within the limits of the linear 

 theory used in these calculations. 



The data in Table 3 show that the pitch motion of the MARK 56 buoy 

 is greater than the MARK 57 in the lower sea conditions. The resonant 

 pitch period of the MARK 56 buoy is much shorter than the MARK 57 and, 

 consequently, it responds more to the higher frequency-wave content in the 

 seaway. As the sea state increases, the energy at the lower wave fre- 

 quencies increases very rapidly, and the MARK 57 buoy with its longer 

 resonant period responds with greater pitching motion than the MARK 56. 

 Increasing the mooring depth attenuates the motion for both buoys since 

 the wave energy is attenuated with depth. 



PROBABILITY OF EXCEEDING A CERTAIN PITCH AMPLITUDE (PEAK VALUE) 



The pitch spectra have sufficiently narrow bands that we may 

 assume that the envelope, hence the peak values of the pitching motion, 

 follow a "Rayleigh distribution." 

 The Rayleigh distribution is given by 



P (e) = i^ e - ^'/2°'e 

 ^ e 



2 

 where 9 is the envelope or a peak value of the pitch motion and a is the 



mean-square value of pitch and may be obtained by squaring the values, in 



Table 3. 



The probability that the envelope of the pitching motion exceeds 



a value 6 is obtained by integrating the expression for the Rayleigh 



distribution from 6 to °°, which yields 



P (6 > 9 ) 



2 2 



13 



