146 



In steady flow, (3 is by definition the same constant at all cross 

 sections, and equation (181) affords a complete expression of the con- 

 dition of continuity of flow. 



297. Equation of continuity for tidal flow. — In tidal flow, water is 

 stored and released throughout a channel as the tide rises and falls, 

 and Q therefore varies from section to section as well as varying 

 at each section with the time. 



Let So, figure 47, be a cross section of a tidal channel at a distance 

 X from the point chosen as the origin of distance, and *S'i, an adjacent 



cross section at the elementary distance dx from Sq. At section So, 

 and at the time t, let: 



z be the surface width of the channel, 



X the area of the cross section, 



D=X/z its mean depth, 



y the elevation of the water surface above any assumed hori- 

 zontal plane of reference, 



c)y/bt the rate at which y is increasing with the time, 



V the mean velocity in the cross section, 



Q the discharge. 

 The volume of water passing So during the elementary time interval 

 dt is then Qdt. During the same interval the water surface between 

 So and Si rises the distance {(>y/dt)dt. The volume of water passing 

 Si during the interval is then decreased by the contents of the prism 

 whose width is z, whose length dx and whose height is {dy/bt)dt. 

 Designating the rate of decrease in discharge with the distance as 

 — bQjdx, the decrease in the discharge in the distance, dx, between 

 the sections, is —{c>Q/bx)dx, and the decrease in volume of water 

 passing section Si in the time dt is —{c)Q/c)x)dxdt. Obviously, there- 

 fore: 



- {dQ/dx)dxdt=zdxidy/dt)dt 

 whence: 



bQ/dx+zc)y/dt=0 (182) 



