142 EKMAN. ON DEAD-WATER. [norw. pol. exp. 



From this expression, which is identical with the final result in Lord 

 Kelvin's paper, the wave-making resistance R is easily found in the way indi- 

 cated on p. 37. The wave-energy per unit area of the water-surface is 



q being the density of the water and H the amplitude of the waves. The 

 wave-making resistance is 



R = B(i — r)E, 



where B is the breadth of the ridge, i. e. of the channel, and r is the ratio 

 of transmission of wave-energy (see p. 36), found by means of a well-known 

 formula of Lord Rayleigh (see for instance Lamb, Hydrodynamics art. 221). 

 From this, it follows 



1 4g i 



R 



_ , 2(7. —la. 



gqBA 2 e '— e ^ 



D 2 



/>. 



! -)V 



2 / \ ' (7 2 S 2 ff, 2 S 



The broken curve in Fig. 2 PI. VI represents the resistance calculated in 

 this way, as a function of s, that is of v/v,„. The unit of resistance is chosen 

 arbitrarily. It is seen that the resistance is inappreciable as long as v is less 

 than \v m ; it increases with v and is a maximum when v = v m , at higher 

 velocities it is, as before mentioned, nul. 



In a similar way the waves and the resistance in the case of dead-water, 

 may be calculated. The boat may be imagined to be replaced by a small 

 ridge stretching in the upper surface, right across the channel and moving 

 along with a velocity v. The disturbances of the salt-water fresh-water boun- 

 dary (reckoned downwards from its equilibrium level) are then given by 

 the same formula (1), where however the function f (a) is somewhat different. 

 If the two water-layers are equally deep, y is found to be given by (2) as 

 before, s being the ratio of the velocity of the ridge to the maximum velocity 

 of the boundary-waves (3, p. 43) K If the salt-water is by comparison infi- 



The fact that the laws are the same in homogeneous shallow water and in the bound- 

 ary between two different water-layers of equal depths, gives an interesting and con- 

 venient method of studying wave-motion on a small scale, since the boundary-waves 

 can be made to move at a conveniently slow speed and — even when of small 

 dimensions - without the disturbing effect of capillarity. 



