Ochi 



gives the probability that an impact pressure exceeds a certain magnitude per 

 cycle of wave encounter. That is 



;:;^(p-p*) 



r 



(21) 



f(p) dp 



It is known in general that the waiting time to observe the mth occurrence of 

 an event when a sequence of events is occurring in time following the Poisson 

 process obeys the gamma probability law given by the following equation [8] 



N^ m-i -Ns* (22) 



r(m) 



For the present problem, however, the probability density function must be 

 truncated at mt* (where, t* is the natural pitching period). Then, by using the 

 condition that the probability between mt* and co for the truncated probability 

 function must be equal to one, the following truncated gamma probability density 

 function is derived: 



g(t) = -^ ^—4 , t > mt^ . (23) 



--00 j^ m -N t - * 



r(m) 



dt 



The constant m in the above equation was given in Eq. (21), and m is not 

 always an integer. Hence, the denominator in Eq. (23) cannot be expressed by a 

 practically usable formula. However, the integration can be evaluated as fol- 

 lows: Let Ngt = z/2, and obtain the probability density function of a random 

 variable z. Then, the denominator of Eq. (23) is equivalent to 



r 



— —^ Z'^-^e ^dZ Z > mZ (24) 



2"" r(m) = * 



where mZ* = 2mNgt*. 



The above integral is the probability integral of the incomplete gamma 

 function and a table is available for this integration [9]. The integral values for 

 various m, n^, and t* appropriate for full scale ships were taken from Ref. 9, 

 and are shown in Fig. 16(a). 



The probability that a time T, or more, elapses before the next severe 

 slam occurs can readily be obtained from Eq. (23). That is. 



570 



