Tuek and Taylor 



where S is now constant. On squaring, (2,4) gives a cubic equation 

 which may be solved directly for Z. Alternatively, following 

 Constantine [ 1961] , we may treat the problem in an inverse manner, 

 solving for the speed as a function of Z and obtaining in non-dimen- 

 sional form 



^ - Ll - (l-d-s)2j ^^-^^ 



where 



F = U/Vgh (2.6) 



d = - Z/h (2.7) 



and 



s = S/Aq, (2.8) 



with 



h=Ayw (2.9) 



as the mean depth. 



Constantine [ 196i] discusses the nature of the flow predicted 

 by (2.5) and presents curves of F against d. Equation (2.5) permits 

 only a restricted range of Froude numbers F for any given blockage 

 coefficient s, namely 



0<F<F, (s) and F2(s) < F < oo, (2.10) 



where F| (s) , F (s) are critical Froude numbers shown in Fig, 1, 

 No steady flow is possible in the "trans-critical" region F, < F < Fp , 

 and Constantine [ 1961] discusses how an unsteady bore forms ahead 

 of the ship if it attempts to exceed F , Notice from Fig, 1 that the 

 trans -critical regiine becomes narrow if s is small, and as the 

 blockage tends to zero there remains a single critical Froude number 

 F, = F2= 1. 



The last result is relevant to an alternative linearized ap- 

 proach to solution of (2,3) not utilized by the previously referenced 

 investigators, but described in a somewhat different context by Tuck 

 [ 1967] . Instead of specializing the shape of the ship, one now makes 

 the approximation that its section area S is everywhere small com- 

 pared with the CBJial section area A. If we again take for definite- 

 ness the case of a canal whose undisturbed section area A is inde- 

 pendent of X and equal to A^, the water elevation Z will likewise 



* 3 2/3 1 2 



Roots of the equation s=l--pF ^ -j^F , 



632 



