522 



Internal Waves 



According to the theoretical and experimental studies by Ekman (in 

 F. Nansen, North Polar Exped. 1893-1896, vol. 5; 1904, p. 562) this 

 "dead water" is due to the fact that a slow moving vessel may create 

 internal waves at the lower boundary of a thin freshwater layer, the thick- 

 ness of which is not much less nor greater than the draft of the vessel. The 

 motion of the ship creates in the layered system two kinds of waves, external 

 and internal waves. Their velocities of propagation are given by the equa- 

 tions (XVI. 12). The external waves are very short, and the wave resistance 

 for the ship will be a normal one. With a given speed of the ship, however, 

 there is also the possibility of generation of internal waves. With the 

 second equation of (XVI. 12) it is easy to prove that the quantity x = 2tt/A 

 is only real if 



c l < 



gh' 



(XVI. 16) 



This means that there is "dead water" when the speed of the ship remains 

 below this critical value. If the speed exceeds this value, then only the ex- 

 ternal waves are generated, and the wave resistance is practically the same 

 as if there were only one single water-mass. This is confirmed by the ob- 

 servation. Table 88 gives the critical values of c, at which there can be "dead 

 water", for different thicknesses of the top layer, whose density is determined 

 by its salinity (at 0°C). The lower layer is assumed to have a normal salinity 

 content of 35°/oo (see V. Bjerknes, 1933, p. 390). 



Table 88. Critical velocities (ml sec) for the generation 

 of "dead water'' 1 waves 



(Lower layer: temp. 0°C, salinity 35°/ o) 



It is obvious that the maximum velocity c of a vessel which will generate 

 "dead water" is very small. For smaller vessels (sailing and rowing boats) 

 the order of magnitude is of a few decimeters per sec (thickness of top layer 

 up to 1 m); for larger vessels (thickness of the top layer 10 m) the order of 

 magnitude is 1 m per sec and more (=1-9 knots). The amplitudes of these 

 internal waves can be very large, and it is these waves which increase con- 

 siderably the wave resistance. 



