Shallow Water Problems in Ship Hydrodynamics 



Instead of using sophisticated numerical techniques for F2°, 

 in the present case we estinnated F^ by assuming that the under- 

 water hull could be approximated by an equivalent spheroid, with the 

 sanne length and displacement. If we define the slenderness e by 



= iv?- (^-^O) 



which is equal to the beam/length ratio of the equivalent spheroid, 

 the exact infinite fluid force on the submerged half of the spheroid 

 can be obtained from the formula given by Havelock [ 1939] in the 

 form 





fJ'= - pU^ . \ B(x) dx . Cg{e) (4. 21) 



where 



2 i 4-./TT72 



1+yr^ 



Cs(e) =1 +1(1 - 36^ + Ze^ (l - f log ^+Vl-e_ \ ^ (4^22) 



In fact, since e is generally small , an adequate slender -body approxi- 

 mation to (4.22) is 



Cs(e) = - e^(log^e +|) + e^ + o(e^log e). (4.23) 



The result (4.22) is of course in exact agreement with 

 Havelock's [ 1939] more general formula for ellipsoids whose sections 

 are not circular. On the other hand (4.23) is within 10% of computa- 

 tions based on Havelock's formula for general ellipsoids, providing 

 e < 0.2 and the half-beam/draft ratio of the general ellipsoid lies 

 between 0.65 and 7.0, a range of parameters which includes the 

 usual ship dimensions. Some preliminary numerical computations 

 using the theory of Tuck and Von Kerczek [ 1968] have shown that 

 (4.23) is a good estimate for non-ellipsoidal geometries, while 

 Havelock [ 1939] himself made satisfactory comparisons between 

 his ellipsoid estimates and experiments of Horn [ 1937] on actual 

 ship models, so that there are grounds for believing that (4.21) 

 subject to (4.23) gives a useful prediction of the infinite fluid zero 

 Froude number sinkage force. 



The finite depth computations were carried out for water 

 depths of 100, 60 and 30 feet. The results for the smallest of these 

 depths are in very close agreement with the shallow-water theory 

 of Section 3, shown dashed on Fig, 2, over the complete range of 

 Froude numbers shown. This indicates that a water depth/ship length 



645 



