shallow Water Problems in Ship Hydrodynamics 



(iv) (|) — — S'(x) log r + f(x) + 0(r log r) as r — 0. (4.6) 



The physical interpretation of <|) is as follows. The contribution 

 from the first term "e*'^" inside the square brackets is just the 

 potential of a line distribution of sources, of strength proportional 

 to S'(x), in a fluid which extends to infinity in all directions. The 

 contribution from the second term inside the square brackets cor- 

 rects for the presence of a bottom wall at z - - h, while the last 

 term in the square brackets corrects for the presence of a free 

 surface at z = 0. 



The last property (4.6) indicates that the given solution (4.2) 

 can serve as an outer approximation (see Tuck [ 1964] ) and will 

 match an inner approximation which satisfies the correct boundary 

 condition on a slender hull surface. Thus (4.2) gives the disturbance 

 potential for flow around a slender ship in finite depth of water, no 

 shallowness assumptions having been made. 



The function f(x) in (4.6) is of crucial importance, and may 

 clearly be considered in three parts, arising from the three terms 

 in the square brackets in (4.2), Let us write 



f(x) = foo(x) +g(x) (4.7) 



where 



. , > US'(x) , .,„2 2 JJ C^ , S'(x) -s'ii) ,. ^. 



(Tuck and von Kerczek [ 1968] ) is the corresponding function for the 

 double-body flow in an infinite fluid (no bottom or free surface), 

 while g(x) is the contribution from the second two terms in the 

 square bracket of (4. 2) , and takes the value 



dk kSlk)e'''* A*(k) (4.9) 



where 



.00 



A*(k) = - 2 r -4^— ri+_2 f^ 1. (4.10) 



-"k 7q2 _ k^ L ^ - Kq tanh qh J 



In the integral (4. 10) , if Kh < i there is a pole on the real q-axis 

 at q = qQ(k) , where 



641 



