HYDRAULIC INVESTIGATIONS, J ]Q 



Tpagnitude, at the moment of the greatest elevation, will be 

 twice as great as that of a single wave in similar circumstan- 

 ces, and the height, as well as the quantity of matter, will 

 be doubled, so that either the actual or the potential height 

 of the centre of gravity of the fluid seems to be essentially 

 altered, whenever such an interference of waves takes place. 

 This difficulty deserves the attentive consideration of those, 

 who shall attempt to investigate either the most refined 

 parts of hydraulics, or the metaphysical principles of the 

 laws of motion. 



V. Of the Effect of a Contraction, advancing through a 

 Canal. 



If we suppose the end of a rectangular horizontal canal, End of a rec- 



partly filled with water, to advance with a given velocity, Gonial velocit" 



less than that with which a wave naturally moves on partly filled 



the surface of the water, it may be shown, that a certain Wlthwate f ™~ 



f . vancing with a 



portion of the water will be carried forwards, with a surface given velocity. 



nearly horizontal, and that the extent of this portion will be 

 determined, very nearly, by the difference of the spaces de- 

 scribed, in any given time, by a wave, moving on the sur- 

 face thus elevated, and by the movable end of the canal. 

 The form of the anterior termination of this elevated por- 

 tion, or wave, may vary, according to the degrees by which 

 the motion may be supposed to have commenced; but 

 whatever this form may be, it will cause an accelerative 

 force, which is sufficient to impart successively to the portions 

 of the fluid, along which it passes, a velocity equal to that 

 of the movable end, so that the elevated surface of the 

 parts in motion may remain nearly horizontal : and this pro- 

 position will be the more accurately true, the smaller the 

 velocity of the movable end may be. For, calling this ve- 

 locity f, the original depth a, the increased depth .r, and the 

 velocity of the anterior part of the wave y, we have, on the 

 supposition that the extent of the wave is already become 



ay 

 considerable, x ~ — v , taking the negative or positive sign 



y i 



according t$ the direction of the motion of the end; since 

 the quantity of fluid, which before occupied a length ex- 

 pressed 



