52 EKMAN. ON DEAD-WATER. [norw. POL. EXP. 



plane; the pressure in the surface will, of course, no longer be uniform, but 

 the differences of pressure in a horizontal plane will be almost exactly un- 

 altered. We may therefore, with very small error, assume the water-surface 

 to be held rigidly plane by suitable extraneous pressures. The gravity would 

 then have no influence upon the motion, provided the density of the water 

 were uniform ; and in the case of a fresh-water layer of density q resting on 

 the top of salt-water of density q + Jq, the motion will take place just as 

 if the gravity did not act at all in the fresh-water, and in the salt-water only 

 upon its excess of density Jq. The small difference between the inertia of 

 fresh- and salt-water is of very little influence on the motion. We may there- 

 fore assume the density to be the same (= q) in the whole fluid, but, as- 

 sume instead, that the gravity varies, being nil in the fresh water and 

 dQ./q X 9 i }l th e sa ^ water 1 . In this way we are virtually able to change 

 the gravity to a considerable degree, by just varying the specific gravity of 

 the lower water-layer. We then obtain, by putting 1=1 in equations (b) 

 and (d) p. 50, the following rule, which — as far as the pressure exerted by 

 simple harmonic boundary-waves is concerned — is a consequence of Stokes' 

 theory of these waves, namely: 



If the difference of density Jq of the two water-layers (Jq always 

 supposed to be small) be increased in the proportion y, and if the velocity 

 of the vessel be increased in the proportion V y , the geometrical similarity 

 of the motion will be kept, and the resultant of pressure experienced by 

 the vessel (the wave-making resistance) will increase in the proportion y. 



The two rules above were deduced on the assumption that the viscosity 

 of the water could be neglected. We may therefore — in the same way as W. 

 Froude 2 — determine the frictional resistance separately, and add it on to 

 the pressure-resultant deduced according to the above rules. Rigorously the 

 wave-making resistance, the eddy-making resistance and the frictional resi- 

 stance cannot be quite independent of each other, and the method should 

 consequently not be allowed ; but the experiments seem to show — as might 

 be expected a priori — that it is practically correct. 



1 This artifice may often be of use in solving problems concerning the motion of the 

 water-layers in the sea. 



2 See White's 'Manual', p. 479 seq. 



