NO. 15.1 EMPIRICAL LAWS OF RESISTANCE. 123 



was small, the boat however, generated a so called solitary wave, which on 

 account of its considerable height, may have had higher velocity than the cri- 

 tical velocity (3) p. 43. The velocity of solitary waves in shallow water is 

 given by the same formula as that of long waves of small amplitude, if the depth 

 of the water be only reckoned to the top of the wave (see foot note 1, p. 40). 

 Rough experiments indicated, as might be a priori, expected, that solitary 

 boundary-waves follow the same law. ff the maximum wave-velocity be cal- 

 culated in this way by measuring the depth of the surface-layer at the crest 

 of the solitary wave, it will be found that the velocity corresponding to the 

 maximum dead-water resistance, has a nearly constant ratio to it (about 0"73). 

 That this ratio remains unchanged when the difference of spec, gravity 

 between the water layers is varied, is a consequence of the rule on p. 52 

 and equation (3) p. 43, combined. 



The influence of the sharpness of the boundary. 



The actual conditions are in reality not so simple as they were arranged 

 in the experiments; the transition from heavy salt-water into light fresh-water 

 is in general not quite sharp, but is extended over a considerable region in 

 which the salinity decreases upwards. Experiments 165 — 184 pp. 82—83, 

 are intended to show the influence of this circumstance; and the results are 

 represented in Fig. 2 PI. XI. 



The vertical distribution of spec, gravity in these experiments, is repre- 

 sented by accessory, step-shaped diagrams in the same figure. In Case (2), 

 for instance, the spec, gravity is 1*000 down to 4 cm. below the surface, 1*010 

 from 4 to 5 cm., l - 020 from 5 to 6 cm., and 1*030 at 6 cm. below the sur- 

 face and down to the bottom. In both cases the water-layers may be ima- 

 gined as formed by the partial mixing of an originally fresh, water layer and 

 a bottom-water of spec, gravity 1*030. These imaginary, sharply defined 

 water-layers are represented on the diagrams by dotted curves (the spaces 

 between the dots are unfortunately too small, so that they look almost like 

 full-drawn curves). In Case (1) the mixing is very thorough, in Case (2) it is 

 less complete; in both cases, the mixing proceeded during the experiments, 

 so that the broken lines on the diagrams, passed into smooth curves. The 

 water-layers seemed on the whole to be more apt to mix in these experi- 



