NoTvbin 



In Ref . [ 89] Tuck has extended the theory to canals of finite 

 width, in which the ratio of sinkage into the water (or trim) in the 

 canal to the sinkage (or trim) in shallow water is given by a unique 

 curve on basis of a simple width-and-speed parameter. Replotting 

 this curve as in the lower diagram of Fig. 22 Tuck's results are 

 shown to yield a square dependence on the bank- spacing parameter 

 Ti when F^, « 1. 



In canals presenting higher blockage the total sinkage or 

 "squat" is dominated by the contribution from water level lowering 

 as a consequence of flow continuity. From the Bernoullie and conti- 

 nuity equations an approxinnate relation for the hydrostatic ship 

 sinkage in terms of ship lengths is given by 



A 2 



4-- F 



I 



/ 5ik 



V (X 



^ WL 





^ Ac Ac ""nl- 



Here (p^ and ci^\_ are the prismatic and waterline-area coefficients 

 of the ship. Other methods of the practical calculation of squat are 

 discussed in Ref. [90] . 



At low speeds wave making is concentrated to bow and stern 

 of the ship, where changes of the local velocities do not influence the 

 blockage conditions, and it shall be possible to calculate the forces 

 on the ship without regard to wave making. The absolute speed still 

 is a parameter, as it is seen to affect the hydraulic as well as the 

 dynamic squat in a canal, 



Kan and Hanaoka first presented low-aspect-ratio wing 

 results for the calculation of transverse forces and moments on a 

 ship in oblique or turning motions in shallow water , [ 91]. As the 

 theory predicts the same correction factor to be applied to all deep- 

 water values it seems to be essentially a two-dimensional theory as 

 it is in deep water. Newman studied the same problem by use of 

 the method of matched asymptotic expansions and by the assumption 

 of a three-dimensional flow, differently orientated close to the body 

 and close to the bottom (and upper image wall), [ 92], His results 

 bear out the effects of finite length, most obvious in case of mLoments 

 due to yaw acceleration. 



Newman considers the inner flow to be a two-dimensional 

 cross-flow of reduced velocity, at each section depending on a 

 blockage parameter in the velocity potential. The outer solution 

 assumes flow to take place in planes parallel to the bottom wall at 

 nominal transverse velocity as the body is reduced to a cut normal 

 to the flow, this being physically similar to the flow past a porous 

 plate. The results as applied to forces on a wing of low aspect 

 ratio (or to a ship) are given in a simple diagram in [ 92] , and here 



872 



