shallow Water Problems in Ship Hydrodynamias 



justified either by careful asymptotic analysis or by physical reason- 

 ing. Suppose the canal has cross-section area A(x,Z(x)), when the 

 water surface at station x is defined by z = Z(x), z being a co- 

 ordinate measured vertically upward from the equilibrium water sur- 

 face. If, similarly, the ship has section area S(x,Z(x)) at station x, 

 and the water has only an x-component of velocity u(x) , then continuity 

 requires u(A - S) = constant, while Bernoulli's equation applied at the 

 free surface gives ju + gZ = constant. 



If we take Z = far upstream, where u = U, S = and 



u(A - S) = UAq (2.1) 



A = Ag , we have 



and 



iu^+gZ = iu^ (2.2) 



In this formulation the ship is fixed in position and the fluid streams 

 past it in the x-direction. Elimination of u gives 



/U^ - 2gZ (A(x,Z) - S(x,Z)) = UA^, (2.3) 



a transcendental equation from which the water surface elevation Z(x) 

 at station x nnay be determined in principle, for a canal of arbitrary 

 section (not necessarily uniform or vertical sided), and a ship which 

 occupies any proportion of the available canal area at any station. 



In this most general case we should then return to the Bernoulli 

 equation to obtain the pressure on the hull (which turns out to be hydro- 

 static) and integrate to obtain the force and moment on the ship. In 

 principle the net vertical force would exactly balance the ship's weight 

 and the trim moment would be zero, if we had started with the ship in 

 its correct squatting position. In practice we should have to devise 

 some kind of iteration procedure to move from an initial guess to the 

 correct position. Such a general and exact study seems not to have 

 been carried out, although it would be of some considerable interest. 



Most investigators avoid this problem by treating an idealized 

 ship which is a straight- sided cylinder, and ignoring end effects. In 

 that case Z is constant over the length of the ship, and it follows 

 that the ship simply rides up (Z > 0) with the water, maintaining 

 constant displacement. If at the same time we restrict attention to 

 the case when the canal is constant in section area, and in the region 

 of interest at the free surface has a width W Independent of Z 

 (locally vertical sides), then we have from (2.3) that 



/ 



U'^ - 2gZ (Aq + WZ - S) = UAq (2.4) 



631 



