36 EKMAN. ON DEAD.WATER. [norw. pol. exp. 



the vessel be increased. If the waves were shorter than indicated by the 

 above rules, they would move more slowly than the vessel, and would gra- 

 dually move away from her; whereby the wave-length and the velocity of the 

 head-most wave would increase until the wave be long enough to follow the 

 vessel. Similarly, the next waves, one after the other, would have their wave- 

 length and velocity increased. 



In calculating the work done by the vessel in maintaining the above de- 

 scribed wave-system, it must be observed, that part of the energy wanted, 

 is constantly transmitted from behind, by the waves already created; but 

 another part of the energy of the latter is always left behind, and a cor- 

 responding quantity of energy must constantly be procured by the vessel. 

 This may be illustrated by a very common phenomenon. If a stone be 

 thrown into smooth water, there is initially formed one circular wave which 

 quickly expands; but one will very soon observe that there are two waves, 

 one in the place of the original wave and one outside it. Both these waves 

 are lower than the original one. This partition of the waves proceeds inces- 

 santly; the wave-rings become larger and larger in number and smaller and 

 smaller in height, the middlemost wave being always the highest. But if the 

 motion of the original wave-ring be carefully followed, one will find that it 

 always remains the outermost one. Thus, the average velocity of transmis- 

 sion of the wave-energy, is smaller than the velocity of the waves themselves 



— really, the ratio is exactly one half in water of infinite depth 1 . This ratio 



— i. e. the ratio of the energy really transmitted across an imagined vertical 

 plane in the water, to the energy which would be transmitted if the waves 

 propagated themselves undivided with their actual velocity — may be called 

 the ratio of transmission of wave-energy 2 . The ratio of transmission of wave- 

 energy is consequently 0'5 in water of infinite depth. 



From the waves following a vessel in uniform motion, the wave-making 

 resistance may then be estimated for the case of deep water, in the following 

 simple way. Imagine two straight lines AA' and BB' (Fig. 3 PL V) stretching 

 in the water-surface at a distance I from each other across the wake of the 



1 See Lamb, Hydrodynamics, Art. 221; see also a highly interesting paper on the subject 

 by Prof. 0. Reynolds, "Nature", XVI p. 343 (1877). 



- Not to be confounded with "the rate of transmission of wave-energy", which is an- 

 other notion. 



