328 HYDRAULICS AND ITS APPLICATIONS 



This becomes infinite if v 2 = y h, 

 i.e., if 





if fc = ,-V- . . (4) 



4 7T 2 ft r 2 



At the radius at which this relation holds, a standing wave will be 

 formed. 



The height of this wave may be estimated, as in the case of the wave 

 produced in a rectangular channel. 



If vi and v 2 are the velocities and hi and h% the depths immediately 

 before and after the rise, so that h% hi = x, we have, as on p. 313, 



HL = to +JiVfc 

 g " 2/n 



from which 



ART. 93. CHANGE OF LEVEL PRODUCED BY THE PASSAGE OF A BOAT 

 THROUGH A NARROW CANAL WITH HORIZONTAL BED. 



Let A = cross sectional area of canal. 



,, a = sectional area of vessel amidships, beneath water line, by a 

 plane perpendicular to its axis. 



Let v = velocity of vessel. 



Here the state of affairs may be simplified if we first consider the water 

 to be a perfect fluid. As the vessel moves along through this fluid, the 

 volume displaced by the forepart passes along backwards between the 

 vessel and the sides and bottom of the canal to fill the space vacated in 

 the rear. In this case we get a backward current extending from the 

 prow to the stern of the boat, its velocity increasing as the effective area 



of the channel diminishes, and having a maximum value v I . - ) 



\ A. Ct / 



at the amidships section. 



To produce this current a surface gradient is necessary, the surface 

 falling from its normal level at the prow to a minimum at the amidships 

 section, and from this point rising to its normal level at the stern. 



Due to the adhesion and viscosity of the fluid, however, a mass of water 

 is dragged along with the boat, forming a current confined mainly to the 

 centre and surface of the canal. 



Since, for permanence of the regime, the backward flow across any 

 section of the canal must equal the corresponding forward flow, the back- 



