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



pression that this circumstance increased the velocity-oscillations, but a detailed discussion 

 seems to give a different result; in any case it has not altered the oscillations essentially. 



By simple considerations, the conditions under which the velocity-oscillations will be 

 small or great may easily be found. For this purpose, we may regard a wave which 

 follows immediately after the vessel, as made up of two waves, one of them being the 

 wave which should at the actual moment be there only by the effect of the transmission 

 of waves from behind: the other wave is then the vessel's direct contribution to the 

 wave-motion. The sum of the heights of these two waves is equal to the height of 

 the actual wave. Let us now make the very reasonable assumption, that the wave- 

 height of the wave-contributions directly generated by the vessel, depends only on the 

 instantaneous velocity of the vessel (and on her shape, etc.) but not upon the waves al- 

 ready created. [This is true, as long as the motions are small, so as to be conformable to 

 linear differential equations. In general, the assumption may therefore be expected to be 

 approximately true]. The wave-making resistance depends only upon the contribution 

 which the boat must pay to the wave-energy, i. e., to the square of the wave-height. Now 

 suppose the vessel to move at a steady speed and the instantaneous height of the trans- 

 verse wave, nearest behind the stern of the vessel, to be li. After having been propagated 

 one wave's length, this wave will have a height = H \r , r being the ratio of transmis- 

 sion of wave-energy (see pp. 30 and 43) ' . During the same time, the boat has generated a 

 transverse wave of a certain height h, depending upon her instantaneous velocity v; and 

 as the waves have invariably the same position relative to the boat, the wave nearest be- 

 hind the stern will have a height equal to the sum of the above elementary waves, i. e. 

 h + H^r. The resistance due to transverse waves consequently varies as 

 (h + H \'r~y - {H ir~f = 2 Hh Vr + h\ . . . (a) 



As h and r depend only on the vessel's velocity, the resistance then increases with 

 the actual wave-height H, when the boat is moving at a steady speed. When the mo- 

 tion has become stationary, the wave-height should also be invariable, and consequently 

 h + H V»" = H- If the value of h or of H, found from this equation, be put in the expres- 

 sion (a), the resistance will be as 



H*(\ - r) (b) 



or as 



*»!±€ (0 



1 -ir 



Equation (6) shows that waves of a given height, cause less resistance, the greater be the 

 ratio of transmission of wave-energy; equation (c) shows that when the wave-generating 

 effect of the vessel is given, the wave-making resistance increases with the ratio of trans- 

 mission of wave-energy. 



The effect of the diverging waves upon the resistance, is quite analogous. Their 

 velocity-component in the direction of the vessel is, however, only v sin 2 a, a being the 

 angle between the boat's keel-line and the crest-lines of the waves (see p. 37). That propor- 

 tion of the whole wave-energy moving along with the vessel, is in consequense, r sinV, and 

 this must be put instead of r, in the expressions (a), (6), and (c). The expression (a) then 

 shows, that the resistance due to diverging waves, increases in only a comparatively small 

 degree with the height of the actual waves, and more exclusively than the resistance due 

 to transverse waves, depends upon h, i. e. upon the velocity of the vessel. The diverging 

 waves therefore contribute less than the transverse waves, to the velocity-oscillations. The 

 transverse waves, again, will give rise to the greatest oscillations, in the case when r is 

 nearly 1, *'. e. when the boat is moving at nearly the critical velocity. 



1 This and the following conclusions are, obviously, but approximate. For the laws of 

 propagation of simple harmonic waves, hold exactly true, only in the case of an end- 

 less series of waves, or, practically, for the middle waves of a large series. 



