Tuck and Taylor 



dominant effect on the ship hydrodynamics, 



A degree of quantitativeness may be attached to this concept 

 of shallowness in a number of ways. Physically and most significantly 

 we require the depth to be so small that the hyd rodynamic part' of the 

 pressure distribution in the field of flow not too close to a disturbing 

 influence such as a ship is to a good first approximation independent 

 of the vertical coordinate z. Thus the only pressure variation with 

 z is hydrostatic. 



If there are waves present, ambient or nnade by the ship, it 

 is consistent with the above that their wavelength be much greater 

 than (say 5-10 times) the water depth, and hence the resulting 

 shallow-water theory is sometimes called a "long wave theory, " 

 However, this terminology can be misleading, since most of the 

 results obtained also apply when free surface effects are negligible, 

 as for a ship moving at very low Froude number, when the wave- 

 length is vanishingly small . An additional requirement for significant 

 effects of shallowness in the ship hydrodynamics context is that the 

 draft of the ship be comparable with the water depth, so that the 

 water bottom does have a profound effect on the ship. 



In this paper we are concerned principally with two major 

 problems, which are apparently quite distinct from each other, but 

 in which similar phenomena appear. The first problem is the classi- 

 cal squat problem, associated with steady forward movement of the 

 ship in calm water, while the second is an unsteady problem associ- 

 ated with the response of, or forces on, a ship without forward motion 

 in beam seas. 



Squat is the change in draft and trim of the ship as a result of 

 hydrodynamic pressure variations over its hull. Acceleration of 

 fluid particles as they pass the middle sections of the ship tends to 

 produce a diminution of pressure there, and hence a downward force. 

 There is also an upward force near the bow and stern stagnation 

 points, but the effect of these forces is small since there is a much 

 smaller area over which the positive hydrodynamic pressure can act. 

 In any case, the developing boundary layer and ultimate flow separa- 

 tion tends to eliminate the upward force at the stern. 



Thus we expect a net downward force, which leads to a down- 

 ward displacement, an increase in draft, or sinkage. At the same 

 time we might expect a nnuch less significant angular trim effect, in 

 which the presence of the small upward force at the bow and the lack 

 of such a force at the stern may give a bow-up trim. These conclu- 

 sions are confirmed qualitatively at reasonably low speeds. 



In fact, of course, being a simple Bernoulli effect, squat is 

 present in any depth of water. An interesting and immediate conclu- 

 sion we may draw is that, like all Bernoulli effects, squat depends 

 predominantly on the square of the ship's speed. This seems not to 



628 



