Coupled Ship Motions 



Note that the value of J when the q^ have their minimizing values, serves 

 as a measure of the closeness with which the exact potential exterior to s^ has 

 been approximated. 



It is easily shown that the minimizing q^ satisfy the following set of linear 

 algebraic equations: 



L 



k = 1, 



(4.2) 



where 



E* 



(4.3) 



/-J 



n=l 



*kn 



(4.4) 



where the asterisk denotes complex conjugate. 



Once the q^ are determined by solving Eq. (4.2), several methods are 

 available for calculating the hydrodynamic forces and moments on the body and 

 thereby the hydrodynamic coupling coefficients. 



Lagally's Method 



Cummins [7] has derived an extension of Lagally's theorem to time- 

 dependent flows, which can be used to obtain the oscillatory hydrodynamic 

 forces acting on the body. The calculation is exceedingly simple, requiring (for 

 the linearized force in the case of small oscillations) a knowledge of the singu- 

 larity strengths and locations and nothing else. (A simple summation over the 

 singularities must be performed.) However, this method suffers from two limi- 

 tations. One, it is applicable only to fully submerged bodies since the extension 

 of Lagally's theorem to bodies that pierce the free surface does not yet seem to 

 have been accomplished. Two, even for fully submerged bodies, Cummins' 

 method gives only the forces and not the moments. 



Energy Method 



By considering the rate at which energy is radiated out to infinity, one can 

 express the real parts of the complex cross- coupling coefficients for time- 

 harmonic motions as a sum over the singularities. The terms in the sum in- 

 volve the singularity strengths and certain potentials or potential gradients 

 evaluated at the singularity locations. With this technique, the real parts of the 

 coupling coefficients corresponding to both forces and moments can be obtained. 

 Moreover the body need not be fully submerged. Finally, once the real parts of 



421 



