Murthy 



VII. 1 The Unsteady Potential 



The potential can be expressed in the form 

 *(x,y,z;t;5;/5;«) = 1^ + fi*^ + 



+ e 1(Tt pa* 1()ir ! 



^ 0110 



The displacements will consist of the steady terms constitut- 

 ing the trim of the ACV in calm water together with the oscillatory 

 terms : 



x= 5x 100 +/3x 010 + ae x^ + 5ae * 1Q1 + P«* x QU + 



with similar expressions for z and 6. 



In the above expansions o is the frequency of forced oscilla- 

 tion. The discussion in this section will apply equally to free oscilla- 

 tion due to wave excitation in which case a will be the frequency of 

 encounter of the waves. This is discussed in the next section. 



The steady potentials 3> nn anf i $.,.. have been discussed 

 in the preceding section. The oscillatory potentials $ and ♦ ' 



are derived in Appendix V with an explicit integral representation 

 for the former and an integral equation for the latter which could, 

 however, be simplified and an explicit solution obtained under certain 

 assumptions similar to those outlined in the last section. 



VII. 2 Lowest Order Restoring Forces and Moments 



The lowest order restoring forces and moments are obtained 

 from (5-6), (5-7) and (5-8) and after simplification reduce to : 



el ° t(5aX ioio^ aX oiio ) = - ama2 ^ooi "' 



e ifft (^Z 1010+ ^Z ono ) =ae Ut [pgA (z^ - ^ ^ - 

 - a 2 m z Q01 --(x 0()1 + h G # 001) fL^ dxdyj 



158 



