Linearized Potential Flow Theory for ACVs in a Seaway 



lar cylinder with its axis along the z-axis and extending downwards 



to the bottom of the deep ocean. The radius and depth of this vertical 



cylinder are assumed to be very large. If the intersection of this 



cylinder with the plane z = is the circle L 3 , it is obvious that 



L, (and therefore L., ) will be contained well within it. Also, /. 



4 x H ' 3 



is the ring-shaped domain on z = lying between the circle L_ 



and the closed curve L. 



We may now apply Green's theorem to the closed domain in 

 z > bounded by ]T, i.e. by J^ + X + E + E ' with ( x » Y' z ) 

 lying within V* ° 1 



^(x, y, z) 



— [[ \g h± 



4tt J J L an 



# 



2 ~~ 3 



_a_& 



^n 



dS 



(V. 10) 



E + E+ZL+E, 



O 1 2 3 



The boundary conditions satisfied by ^ are given by (V.4), 

 (V. 6), and (V. 8). 



Considering first the integral over \^,^ the base of the large 

 cylinder, the integrand tends to zero in view of the assumed behaviour 

 of 4> (and therefore of ^ ) and of G as T — ><x>. The integral over 

 the lateral surface^] -is also zero as the radius r — >.oo since the 

 radiation conditions are specially selected (and considered physically 

 reasonable) to ensure that this is so. A full discussion of this matter 

 will be found in Appendix V of Reference 1. 



We are therefore left only to deal with the integrals over 2-i 

 and 2_/ ^ . As regards the later, 



2^3 



a n 



- * 



a& 

 an 



2^3 



and on substituting for ^ .. from (V. 9) the integral becomes 



4 7Tg 



//[ 



*(V G^ + 2i<rVG£ - * G) - G(V $,,- 2iaV*£-cr *) d£dr) 



V (*G^ - GtfvJ + 2icrV (*G£ + G%) d^drj 



= -i g -/j r "ar [ yZ ( * G * - G ^ + 2i ° Y * G ] 



d£dr, 



221 



