Shallow Water Problems in Ship Hydrodynamics 



possibility of fluid passing underneath the ship means that we must 

 replace the ''hard" boundary condition (5.6) by a more general con- 

 dition, expr essi ng in effect a relationship between the velocity 

 (94>/9y) + AVg/h of fluid passing "through" (i.e. under) the strip 

 y = 0^, |x| < i and the pressure difference (proportional to potential 

 difference) across the strip, which causes this underflow. Thus we 

 write 



1^ +a/|'= TPc|) on y = Oi, |x|<i, (5.7) 



where P = P(x) is the "porosity" of the ship section at station x. 

 If the ship is actually touching the bottom, then P = and (5.7) 

 reduces to (5.6); at the other extreme, if there is substantial 

 clearance, P — * oo and the jump in potential <^ across the strip 

 tends to zero, leading as expected to zero force on the ship. 



In the following section we indicate how to obtain the porosity 

 P(x) for any given ship and sea bottom geometrical configuration. 

 The problem of solving (5.5) subject to (5.7) is then identical to that 

 for acoustic scattering by a "semi-soft" or porous ribbon with finite 

 acoustic impedance, see e.g. Morse [ 1948] . However, no general 

 procedure seems to be available in the acoustic literature for solving 

 this type of problem, and we present in Section 7 a numerical 

 approach based on an integral equation formulation. 



It should be remarked that as k — ^ the present problem 

 reduces to uniform steady streaming flow across the ship, the free 

 surface being replaced by a rigid wall. This problem was discussed 

 by Newman [ 1969] , who presented solutions for the added mass of 

 the ship in such a flow. The present theory can be considered a 

 generalization of Newman's theory to allow for waves, and gives 

 results which agree with Newman's in the limit as ki — ^ , i.e. as 

 the waves become long conmpared with the length of the ship. 



VI. THE DETERMINATION OF THE EFFECTIVE POROSITY 



The problem formulated in the previous section is to be inter- 

 preted as an outer problem, which provides a solution for the scattered 

 field (j) everywhere except within a beam or two of the center plane 

 y = of the ship. In this latter region, the outer solution must match 

 an inner approxinnation which describes the detailed flow field beneath 

 the hull. This flow can easily be shown (Tuck [ 1970] , Newman [ 1969]) 

 to be locally two-dimensional in the (x,z) plane, and to satisfy the 

 two-dinnensional Laplace equation 



|!j. 1^1=0 (6.1) 



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