Tuok and Taylor 



9 



F < 1 (3.12) 



~ — - C dxC deB'(x)s'(e)iog |x-e|. 



^y^ r S'(x)B(x) dx, F>1 (3.13) 



and a bow-up pitching moment 

 Zirh-J 



^5 = 



^ ^ — - r dxC de(xB(x))'s'(e)iog |x-e|, f<i, 



"i^^ -^ -^ (3.14) 



P^^ \ S'(x)xB(x) dx, F>1, (3.15) 



/f^- 1 ^-i 



ZhVF 



where B(x) is the width of the ship at the waterline at station x. In 

 fact the force written down in (3, 12) is invariably negative at sub- 

 critical speeds so that a sinkage is to be expected rather than a lift. 



Tuck [ 1966] also gives formulae for the actual sinkage and 

 trim displacements of the ship in response to these forces, assuming 

 equilibrium with hydrostatic restoring forces, and provides some 

 computed results which are in reasonable quantitative and excellent 

 qualitative agreement with experiments of Graff etal. [ 1964] . There 

 is a need for more experiments, especially in the very low water 

 depth range, but it would appear from the comparisons so far made 

 that the theory is quantitatively accurate so long as the depth is less 

 than about one eighth of a ship length, and the Froude number based 

 on depth is less than about 0.7. 



It may be worth observing here that the integrals in (3. 1 2) - 

 (3. 15) are fairly insensitive to the shape of the section curves 

 B(x) , S(x). For instance, the ratio 



i^ f{ dx f{di B'(x)S'(e) log |x-e| 

 X=- -^-^ /-^ (3.16) 



J_^ B(x) dx • f_\ S(x) dx 



is nearly an absolute dimenslonless constant, taking values between 

 2.0 and 2.4 over a very wide range of B(x) , S(x) curve shapes, 

 including actual ships and mathematically defined curves. Thus a 

 nearly universal approximation to the subcritical vertical force is 



638 



