NO. 16.] EFFECT OF SHALLOW WATER. 41 



city, the transverse waves, in order to follow the vessel, must be very long, 

 and a comparatively small part of the wave-energy, is therefore left behind. 

 In consequence, the transverse waves, at this velocity, cause a comparatively 

 smaller resistance than equally high but shorter waves at a smaller velocity, 

 would do. On the other hand, as they follow the vessel spontaneously to a 

 greater extent, they will, by accumulation, reach a greater height, than they 

 would at the smaller velocity. The result of these two circumstances, acting 

 in opposition, is, as mentioned on p. 39, that the resistance increases to a 

 maximum at the critical velocity \gd, and then suddenly falls off to a lower 

 value. Lord Kelvin's mathematical investigation (I. c. p. 40) gives substantially 

 the same result, as will be mentioned below. 



C. WAVES IN THE BOUNDARY BETWEEN SALT AND FRESH WATER. 



When a layer of light fluid rests on the top of a heavier one, waves 

 can be created in the boundary between the two fluids, as well as in the free 

 surface. The mathematical theory of such waves has been developed by Sir 

 G. G. Stokes 1 . The chief properties of them, as well as a few simple approx- 

 imate formulae for which we shall have use later, are collected here below; 

 the analysis not inserted here, may be found in Stokes' paper or in Lamb's 

 "Hydrodynamics" Art. 223. All the formulae are founded on the supposition 

 that the waves are of small or moderate heights, so as to be approximately 

 conformable to linear dynamical equations. 



Such "boundary waves" may easily be demonstrated as follows: half fill 

 a tumbler (or better, a square glass-vessel) with a salt solution, and lay on the 

 surface a thin disc of cork. Then, by help of a glass rod or a lead-pencil, 

 pour fresh water in a fine stream on to the disc, and, with a little care, 

 there may be obtained a quite sharp boundary between the two water layers. 

 If now the glass be given a suitably slow oscillatory movement to the right 

 and to the left, the boundary between the two water-layers is put into great 

 uninodal oscillations while the upper surface remains almost still. The less the 

 difference of specific gravity between the two water layers, the slower are the 



1 On the Theory of oscillatory waves, Math, and Phys. papers, vol. I, pp. 212—219. 



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