Shallow Water Problems in Ship Hydrodynamics 



have been noticed by pilots and ship's captains, who are generally- 

 aware of the problem of squat as a danger to the vessels under their 

 control, but are prepared to adopt a linear riile-of -thumb such as 

 "a foot for every 5 knots" ! 



In shallow water, squat is obviously a most important prob- 

 lem, if only because it may cause a ship to actually scrape the bottom. 

 But not only is this danger present, but the magnitude of the sinkage 

 is actually increased by the proximity of the water bottom. This is 

 clear physically, since the effective channel of flow is constricted by 

 the additional boundaiy, leading to even greater acceleration of fluid 

 particles across the middle sections of the ship. 



Another interesting additional phenomenon in shallow water is 

 a result of a close analogy between long-wave theory and linear aero- 

 dynanmics, in which the Froude number F = U/Vgh based on water 

 depth h plays the role of the Mach number. Thus we expect extra- 

 ordinary effects in the neighborhood of F = 1, c.f, the "sound 

 barrier," and indeed both theory and experiment confirm that this 

 critical speed is of crucial importance. Theory, at least in its 

 linearized form, predicts infinite sinkage at F = 1 , while in both 

 experiments and in observations at sea we obtain a dramatic increase 

 in draft associated with the generation of a type of permanent wave 

 or bore accompanying the ship near F = 1 . 



The squat problem is discussed in Sections 2-4, each section 

 presenting a different aspect of the problem. In Section 2 there is a 

 lateral as well as horizontal constriction to the field of flow, and a 

 hydraulic -type theory applies, while with the removal of the lateral 

 boundaries in Section 3 the true shallow-water theory can be used. 

 In Section 4, for the only time in this paper we use a finite-depth 

 approach and present calculations of sinkage and trim which are 

 close to the shallow- water results when the water depth/ship length 

 ratio is reasonably small. 



Problems of ship motions in shallow water, i.e. of flows of 

 an unsteady nature, are of special interest again because of the 

 dangers inherent in large motions when there is very little water 

 beneath the keel. However, even if this danger of grounding were 

 not present, one might be wary of using present theories of ship 

 motions for cases when the water depth is known to be significant, 

 since theories such as strip theory deal (successfully) with infinite 

 water depth only. 



For definiteness we concentrate here on a particular mode of 

 fluid flow, that involved in pure sideways (sway) motion of the ship 

 or of force on it. This mode is of interest for a number of reasons, 

 some practical, some theoretical. From the practical point of view 

 we expect this mode of motion or force to be of great significance 

 when a ship is berthed or positioned in such a way that the dominant 

 seas are from abeam, or when it is being manoeuvred sideways by 



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