392 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 61.3 



Speed Voo of Ship in Deep Water of Unlimited Extent 



Celerity c„ of Deep-Woter Wave of Speed V^ and Lenqth Lwq 

 Celerity c^, or Speed Vj of Wove of Lenqth L'^qq 



J Resistance 



Rwco Due to 



Wovemokinq in 



Deep Wotsr ot 



Speed Voo 



Resistance 



Rpoo -Due to 



I Friction in 



I Deep Water 



I at SpeedVjj, 



Fig. 61. B Definition Diagram for the Shallow- Water Speed-Resistance Determination of 0. Schlichting 



waves alongside it, of the Velox system, travel 

 any faster than it does. The wave of translation 

 or solitary wave which may go on ahead in a 

 restricted channel does not enter into this dis- 

 cussion. Experience reveals that the crests and 

 troughs of the Velox waves remain in essentially 

 the same positions along the length of the ship 

 as in deep water. However, the ship slows down 

 by the ratio of the wave speeds in shallow and in 

 deep water, given by the ratio c,JV„ of Eq. (61.i). 

 It may be assumed that, despite a change in 

 profile due to increased wave height hw in the 

 shallow water, and other second-order changes, 

 the pressure resistance jB,^„ at the slower speed 

 C/, is the same as it was at the speed V„ in deep 

 water. Here, and in what follows, the subscript 

 / applies to values at an intermediate speed 

 V I — Ch . Again it is assumed that wavemaking 

 is responsible for all the pressure resistance. 



The situation regarding resistance and ship 

 speed is now represented graphically in Fig. 61.B 

 by the point Bi . The pressure resistance Rw^ 

 remains the same as at A, , but the friction 

 resistance is diminished from Rf^ to Rfi , that 

 is, from its value at the point Ei to its value at 

 the point Fi . The total resistance at the speed 

 Vi is diminished from Rt^ to Rn, , solely by the 

 amount of this reduction. The ship speed is 

 diminished from F„ to V i . 



A second effect now enters the picture. Because 

 of the greater augment -\-A.U of stern ward 

 velocity due to potential flow in the shallow water. 



especially in the limited space between the ship 

 bottom and the water bed, explained in Sec. 18.2, 

 the ship in shallow water has to move faster 

 relative to the water which closely surrounds it. 

 In other words, it must overcome a total resist- 

 ance greater than Rn to maintain the speed c* 

 or Vi . The resistance represented by the point 

 Bi in the figure is increased to that represented 

 by the point Dj . Unfortunately, it is not possible 

 to derive a simple expression for predicting or 

 calculating this increase. Schlichting therefore 

 assumes that the resistance Rtu remains the same 

 as at the speed V , but that the ship slows down 

 until its total resistance again drops to Rjh . 

 This involves a ship speed over the ground slower 

 than Vi . The new reduced speed is V^ , repre- 

 sented by the point C, on the shallow-water 

 resistance curve of the diagram of Fig. 61. B. 



The amount of the first speed reduction Ac or, 

 better, the ratio between the speed c^ and the 

 unlimited deep-water speed V„ , is determined 

 solely from theoretical considerations, indicated 

 by Eq. (61. i). The speed C/. or V i is for convenience 

 called here the Schlichting intermediate speed or 

 the shallow-water wave speed. The ratio between 

 this intermediate speed and the shallow-water 

 ship speed V^ being sought, or the further speed 

 reduction AVp due to potential flow, is most 

 difficult to determine theoretically. It is therefore 

 derived from experiment data on models tested 

 in shallow water. The sum of the constant 

 wavemaking resistance R^y^ plus the friction 



