NO. 15.] QUALITATIVE RESULTS. 71 



taneously with the changes of the vessel's velocity, but the waves and the 

 resistance increase gradually towards their final values, after the boat has 

 acquired a given speed; and this circumstance was evidently the reason of 

 the oscillations. It will be shown that the diverging waves are less apt to 

 cause velocity-oscillations, but the transverse waves more so especially if the 

 boat is moving at a speed near to the critical speed at which the resistance 

 is a maximum. 



To make all this clear, first suppose that the boat at each moment creates all the 

 wave-energy which we find within the space moved through during that moment. The 

 wave-motion near the boat and the resistance to her, would then depend only on her 

 instantaneous velocity, so that, directly after the boat has her velocity increased, waves 

 of the height corresponding to the new velocity, are generated. In this case no oscillations 

 in the velocity would be possible; the boat would, on the contrary, be accelerated by a 

 force, equal to the difference between the towing-force and the resistance corresponding to 

 her actual velocity, and she would gradually attain the velocity commensurate with the 

 propelling force. She would never exceed this velocity, for as soon as it were attained, 

 the resistance would be exactly equal to the propelling force, and her velocity would, in 

 consequence, not be further accelerated. 



Actually, the waves once created, propagate themselves to a certain extent in the 

 direction of the vessel's motion, and thus the wave-energy is partly supplied from behind, 

 the rest, only, being directly generated by the vessel. It is then clear that the waves will 

 be smaller at the beginning, when there are no waves behind them; and if the boat be 

 moved at a steady speed, some time must elapse, before the waves have obtained their 

 final height. For the same reason, if the boat has her speed incessantly accelerated, the 

 waves never reach the height corresponding to her instantanous velocity, but the changes 

 of wave-motion must always follow on the changes of the vessel's velocity. Similarly, when 

 the velocity is being retarded. With regard to the changes of the resistance, when the 

 boat begins to move at a steady speed, three cases are conceivable: either it depends only 

 on the speed of the vessel and not on the waves already created, or it is greater at the 

 beginning when the vessel must create waves in water, before motionless, or it is smallest 

 at the beginning and increases with the height of the waves. In either of the two first cases 

 there would be no velocity-oscillations: for as soon as the boat ceases to be accelerated, 

 the resistance then does not further increase, and therefore a higher speed than can be 

 constantly sustained, cannot be given by a constant force. It is therefore proved by the 

 velocity-oscillations, that the resistance increases with the height of the waves, when the 

 boat begins to move at a stationary speed, and that in consequence the changes of resi- 

 stance follow on tlie changes of the vessel's velocity. 



The velocity-oscillations are now easy to understand. When the boat, initially at 

 rest, is set in motion by a constant force, the resistance does not balance the moving force 

 and stop the acceleration, before the boat has already a larger velocity than could be con- 

 tinually sustained by the moving force. The waves and resistance continually increase, and 

 the velocity begins to be retarded; the resistance increases until it has reached a value 

 corresponding to the actual velocity of the boat (but larger than the moving force). After 

 that, the velocity and the resistance both diminish. As the changes of the latter are con- 

 tinually behind hand, the resistance continues to be diminished even after it is equal to 

 the moving force, and the velocity has ceased to be retarded; the boat consequently be- 

 gins to be accelerated again. The velocity-oscillations are then a necessary consequence of 

 the law of wave-generation; only that they will be more or less stifled by viscosity. The 

 matter will really be somewhat more complicated, because the waves during the oscillations 

 alter their position relative to the vessel. When making the experiments, I had the im- 



