40 EKMAN. ON DEAD-WATER. [noew. pol. EXP. 



cannot move spontaneously at more than a certain speed igd, d being the 

 depth of the canal and g the acceleration due to gravity 1 . This maximum 

 velocity of waves is just the critical velocity at which the resistance of a vessel 

 began to diminish; when moving at a greater velocity, the vessel cannot be 

 followed by any transverse waves (compare p. 35), and as the work required 

 for creating these waves is then saved, tbere must be a corresponding decrease 

 of resistance 2 . If resistance were not caused by friction and by the diverging 

 waves, the vessel would move without resistance, at velocities above the cri- 

 tical one. It will be made clear, in the next chapter, that it is just the trans- 

 verse waves, which are of preponderant influence on the resistance in shallow 

 water, as well as in "dead-water". 



The influence of shallowness is practically insensible, as long as the depth 

 of the water is more than half the length of any wave actually created by 

 the vessel. The velocities of the waves are, in this case, in practically the 

 same ratio as the square roots of their wave-lengths, and the ratio of trans- 

 mission of wave-energy is Va — just as in deep water. But when the 

 velocity of the vessel is increased, the wave-velocity and the wave-lengths 

 correspondingly increase; when the wave-length becomes more than twice 

 the depth of the water, the laws of motion of the waves are sensibly 

 altered: the wave-velocity increases more and more slowly, approaching as- 

 symptotically its maximum value \gd, when the wave-length increases. At the 

 same time the ratio of transmission of wave-energy gradually increases from 

 05 to 1; i.e.: the longer be a wave, the smaller part of its energy will be 

 left behind, and the more slowly will it diminish in height while travelling. 

 Very long waves consequently may travel long distances spontaneously without 

 diminishing very much in height and without leaving waves behind them; in 

 this respect Scott Russell's "solitary waves" comport themselves as very 

 long waves. If the velocity of the vessel be a little below the critical velo- 



1 Scott Russell particularly studied a peculiar sort of wave which he called "solitary 

 wave". It consists of a single elevation travelling spontaneously without altering its 

 shape. If the depth of the canal be d and the height of the elevation be h, the velo- 

 city of such a wave is \g {d + h), that is, somewhat more than the critical velocity 



igd. 



- This is more rigorously established by a mathematical investigation of Lord Kelvin 

 "On Stationary Waves in Flowing Water", Phil. Mag., Oct., Nov., and Dec. 1880. 



