Kaplan 



where "A^ is the ratio of the amplitude of the heave -gene rated two- 

 dimensional wave to the amplitude of heaving motion of the ship 

 section, and | z - r) | is the absolute value of the relative heave mo- 

 tion. It is seen that this expression is thus proportional to the square 

 of the incident wave amplitude, with the force only occurring due to 

 the relative motion between the heave and the incident wave. It can 

 be shown that the mean value of this force in an irregular sea 

 characterized by a wave spectrum such as the Neumann spectrum is 

 given by 



F = \ -^ A^co) dco (124) 



2 ... 



where A (co) is the Neumann spectrum representation given in 

 Eq. (43). 



The results obtained in the basic derivations for determina- 

 tion of the drift force have been carried out for the case of a single 

 sine wave at a fixed frequency. In the real case, the waves are 

 composed of many frequencies in a band, and for the purposes of 

 simplification of the following analysis it will be assumed that this 

 bandwidth is relatively narrow. If a combination of two different 

 frequencies is present in a wave, and hence in the relative heave 

 motion represented by 



z - r| = b sin co.t + bg sin (cogt + <t>) (^25) 



the square of this termi is given by 



(z - T|)^= b, sin^cojt + bg sin (wgt ^ ^) + 2b,b2 sin o),t sin (cogt +^) 



= i|b^ +h\ + 2b,b2 cos[(co2- co,)t + <^]| 



- 4{b^ cos 2co,t + b|cos 2((o^+4>) - 2b, bg cos [(co2 + io,)t +4']/. 



(126) 



It can be seen that this expression is made up of a group of terms 

 that are essentially constants and slowly varying terms (due to the 

 narrow band assumption), and another group of terms representing 

 higher frequency oscillations, i.e. at higher frequencies than the 

 wave terms. If a time average of this quantity is made, it will be 

 seen that the combination of the constants and the slowly varying 

 term remains and the higher frequency terms drop out, and that this 

 first grouping of terms can be represented as the square of the 

 envelope of the combined signal given in Eq. (125) (see [ 19] ). 



1070 



