NO. 16.] QUALITATIVE RESULTS. 73 



The same result can be obtained in another way. If the boat, previously at rest, be 

 moved uniformly at a speed v, the transverse wave first produced will have the height h; 

 the next wave — that which is nearest to the vessel after she has proceeded 2 waves' length 

 — will have got a contribution It ]/r to its height from behind, and will therefore have a 

 total height h (l + VV). Similarly, the third wave will have a height 



h (l + yj7) VF+ h = h (1 + Vr + tfr*), 

 and so on; and gradually, the height of the wave at the stern, approximates to its final value 



n = o ' " 



For diverging waves, it is only necessary to put r sin 2 « for r, just as above. The greater the 



fraction r [or r sin 2 a), the more slowly does the series 2' V r" converge, that is: the more 



slowly will the wave-motion and the resistance reach their final magnitudes. Again, the 

 more slowly the resistance reaches its final magnitude, the more will the changes of resi- 

 stance follow after the changes of velocity, and the greater will be the oscillations, in the 

 case of the boat being put in motion by a constant force. 



This agrees well with the experimental result, that the velocity-oscillations, as well 

 as their time-periods, increased rapidly with the towing-force, i. e. with the velocity 1 . For 

 as the vessel's velocity was increased, the waves seemed to become to a greater and 

 greater extent transverse waves and comparatively less, diverging waves; and at the same 

 time the wave-length and consequently the quantity r, increased. In addition, the velocity- 

 oscillations were more stifled by friction the slower the velocity; for, as will be shown in 

 section F of this chapter, a relatively larger part of the resistance is due to friction, at 

 slower velocities, than is the case in the neighbourhood of the critical velocity. 



The velocity-oscillations of the smaller Fraw-model in the large tank, 

 were somewhat smaller than those of the large model (see diagrams 284—287, 

 PL VIII). This partly depends upon the friction being in the former case 

 comparatively greater; but it also depends upon the different wave-motion. 

 For the smaller boat-model moved in a relatively wider channel than the large 

 model, and the waves created by it, were therefore to a greater extent diverg- 

 ing waves than in the case of the large model. The same will be the case 

 with vessels on the sea, as the water is there practically unlimited in hori- 

 zontal direction; and this possibly explains why the velocity-oscillations, although 

 very strong and evident in the experiments, are not observed by the sailors. 



1 In contradiction to this result, the diagrams 147 and 156 show slower oscillations than 

 148 and 160 respectively. I believe, however, that the variations ol velocity in ex- 

 periments 147 and 156 depended on wave-motions which remained in the water from 

 the preceeding experiment and influenced the motion of the boat. I often required 

 much patience in waiting for the wave-motions in the boundary to subside, otherwise 

 they would disturb the motion. It has happened that by the effect of such residuary 

 waves the boat-model has got free from the dead-water, although, with the same moving 

 force it would have remained in dead-water if there had been no waves when it started. 



10 



