46 EKMAN. ON DEAD-WATER. [norw. pol. EXP. 



First consider a simple case, which can be treated by mere statics. The 

 vessel may be supposed to be dragged slowly in a narrow square-sectioned 

 channel, and her bottom to reach into the salt water. She may be quite flat- 

 bottomed and with vertical sides, fitting to the walls of the channel, so as 

 to allow no water to pass except than under her bottom. When the vessel 

 begins to move, the fresh-water must then be heaped up ahead of the vessel 

 and become thinner aft of her (see Fig. 6, PI. VI), and as there must be sta- 

 tical equilibrium in the salt water below the vessel, the upper surface rises 

 ahead and sinks aft. The vessel is in consequence, resisted by a sternward 

 pressure easily calculated. The depth of the fresh water, fore and aft, may 

 be fej and h 2 respectively, and the densities of fresh and salt water q and 

 q-\- Jq respectively. In consequence of the statical equilibrium, the differ- 

 ence of level of the salt water, fore and aft, is then found to be 



] h = (ft, _ } h ) X q/(q + Jq) , 

 and the difference of level of the upper surface 



h l —h i — h 3 = {h x - h 2 ) X ^2% + ^2) • 



The breadth of the channel being 6 and the acceleration due to gravity g, 

 the resultant of pressure against the vessel is when Jqlq is small, then 

 found to be approximately 



gbJq ' h % *' (7) 



As the vessel moves along, the difference of levels — and the resist- 

 ance — grows larger, until the fresh water ahead of the vessel reaches a 

 little below the bottom of her. It then runs aft below the vessel, also resist- 

 ing her by friction. If the speed of the vessel be not too small, the fresh 

 water will not come to rest when buoying up aft of the stern, but, on account 

 of its inertia, continue to swing above and below its level of equilibrium 

 thus giving rise to a series of waves in the boundary (Fig. 7, PI. VI). As 

 energy is needed for the creation of these waves, they contribute to the resi- 

 stance (by increasing the sternward pressure-resultant). 



Now suppose that the vessel does not reach, although nearly, the salt 

 water, and that the vessel begins to move slowly. Then, the fresh water 

 passes freely by the vessel, and the latter cannot be resisted by any statical 

 pressure as in the former case. But the passage for the fresh water below 



