CALCULATION OF SECOND ORDER TRANSFER FUNCTIONS 



So far the problem of obtaining the second order transfer functions has not been 

 discussed. These may be obtained either from experiment or calculation. 



Measurement of drift forces in regular waves is difficult, and two-wave 

 tests are even more difficult. One problem is that the forces involved are small 

 compared to the first order forces. Dalzell (ISy*) has applied cross-bi -spectral analysis 

 to obtain this data for added resistance. This involves very long test runs and equally 

 lengthy statistical analysis. 



Calculation of the steady drift force is relatively easy, since this may be obtained 

 using the first order velocity potentials for the incident, diffracted, and radiated 

 waves. Maruo (I960) and Newman (I967) have derived formulas for the steady drift force 

 and moment based on conservation of momentum. 



These results can be applied using potentials obtained by a variety of methods. 

 For example, Faltinsen and Loken (1978) use a strip theory potential including a correct 

 "Helm holz" diffraction potential in Newman's result, Molin (1979) applied Maruo's 

 formula using a potential obtained from a 3-D finite element approach, and Faltinsen 

 and Michelson (197^*) used a potential from a 3-D panel method with an intermediate 

 result of Newman (1967)- Because the steady drift forces are to be covered elsewhere, 

 we will not consider this topic further here. 



To calculate the second order force due to the waves of different frequency, it 

 is necessary to integrate the second order pressure over the hull, tal<ing account of 

 the change in hull shape due to the wave elevation and the motion of the hull. This 

 has been done by Faltinsen and Loken (1978) for beam seas by means of a strip theory. 

 Kim and Dalzell (1979) apply a strip theory to find the result in oblique waves, but 

 omit the second order potential. This is not serious for small difference frequencies, 

 as will be discussed shortly. Pinkster (1979) has used a 3-D panel method for the 

 first order potential and an approximate second order potential. This approximation involves 

 including the second order term in the incident wave potential, and finding the 

 resultant diffraction potential, which can be done quite easily. Second order contribu- 

 tions from the interaction of the incident, diffracted, and radiated waves, and from the 

 hull boundary condition are not included. 



Faltinsen and Loken's beam seas calculations, and Pinkster's approximate calculations 

 show that the second order potential contributions go to zero as the difference 

 frequency goes to zero, while contributions obtained from the first order potential 

 become increasingly important. Thus Pinkster's approximation, and Kim and Dalzell 's 

 neglect of this term is reasonable, if only the low frequency drift forces are desired. 



CONCLUSIONS 



Recent developments in predicting slowly-varying forces on a body have been reviewed. 

 An approximation is available for calculating this using only the mean drift forces. 

 Evidence for the usefulness of this approximation is sparse but encouraging. More accurate 

 prediction methods are also available, although the complete second order solution is 

 available only for beam seas acting on a cylinder. It should be noted that all of these 

 developments are for long-crested (un i -d i rect ional ) seas. Information regarding short- 

 crested seas is not available. 



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