Shallow Water Problems in Ship Hydrodynamics 



F = - X P^ ■ ^ \ B(x) dx \ S(x) dx (3.17) 



with a fixed value of \. From this follows a similar approximation to 

 the actual sinkage, say a displacement of 6 downwards, where 



C i 

 . ^2 J t S(x) dx 



Finally, introducing the displaced volume 



V = \ S(x) dx (3.19) 



and making a further assumption (justified in most practical situations) 

 that F « 1 , we have 



where L = 2i is the ship length. 



In practical terms, if 6, L and h are in feet, V in cubic 

 feet and U in knots , and if we insert reasonable (conservative) 

 values for \ and g, (3.20) innplies 



6 = 0.13^Vt • (3.21) 



hL 



We put forward this formula (3.21) quite seriously for practical use 

 by anyone interested in a quick estimate of squat in a wide expanse of 

 shallow water. One should note the quadratic dependence on forward 

 speed, the inverse dependence on water depth, the proportionality to 

 displacement (at fixed length) and inverse square dependence on length 

 (at fixed displacement). 



In a subsequent paper. Tuck [ 1967] extended the 1966 work to 

 the case where the ship is moving along the center of a rectangular 

 channel of width w, considering only the sub-critical case. The 

 assumption made was that w is comparable with the ship length L; 

 however the results obtained were uniformly valid, in the sense that 

 the infinite -width results were reproduced as w/L -*• co, while as 

 w/L —- we obtain predictions which may also be obtained by ele- 

 mentary (linear) one-dimensional theory as in Section 2. An inter- 



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