Coupled Ship Motions 



After some manipulation this reduces to 



Hij(a) = U..(a) - iapHcp. n-jl-dS 



+ p[ fv^ -V^i n-Mj dS + Pj J gi n-Mj dS 



+ ^Wl^i-'^^'^o)] "-^jdS i = 1, 2,3 (2.24) 



s„ j = 1, 2, 3 . 



Although details are omitted, we have assumed in deriving Eq. (2.24) that the 



lim i/zj-Cx, y, z, t) 



t ^ CO 



exists and is equal to the incremental steady flow associated with the body in its 

 displaced position. 



In Eq. (2.24), the potential g-(x, y, z) is defined as 



lim 0j.(x, y, z,t) 

 t ^ 



and is therefore the infinite frequency potential satisfying the boundary condi- 

 tion in Eq. (2.13) on S^. 



The real and imaginary parts of H^. satisfy the Kramers-Kronig relations: 



H^Cc) = ^ f ^^ dc' (2.25) 



Hij(c.) 



i f hj^ a.' (2.26) 



^ J a' - a 



where the bar on the integral indicates the Cauchy principal value. 



Thus from a knowledge of either H- ^ or H. j, the other can be inferred, 

 while as already mentioned, the remaining terms in Eq. (2.24) may be regarded 

 as comparatively easy to calculate. 



For completeness, we include the expressions for H. • , analogous to Eq. 

 (2.24), for the remaining mode pairs. We have 



413 



