228 



shape that a closed tidal channel of constant depth must have, in 

 order that a simple harmonic fluctuation of the tide at the entrance 

 will produce throughout the channel primary tides of constant range 

 and primary currents of uniform strength. In the lack of a better 

 term, a channel of this shape may be called an ideal estuary. 



439. Derivation of the Jorm of an ideal estuary. — Taking the origin 

 of distances at the entrance from the sea, the positive direction up- 

 stream, and the origin of time at a high water at the entrance, let: 

 D, be the constant mean depth of the channel at mean tide. 



z, its width at a point distant x from the origin. 



r, its constant hydraulic radius at mean tide. 



C, the applicable Chezy coefficient; also taken as constant. 



A, the constant amplitude of the primary tide. 



B, the constant amplitude of the primary current. 

 a, the speed of the primary tides and currents. 



S, the amplitude of the surface slope at a point distant x from 

 the origin. 

 H°, the" initial phase of the slope at /the same point. 

 ^, the angular lag of the current. 

 The relations established in paragraph 373 show that if B, r, and 

 C have constant values in a given channel, the values of 4> and 8 also 

 are constant throughout the channel. 



The equation of the tide at the entrance is: 



y—A cos at. 



Since the tide at a station within the entrance occurs at a later 

 time, its equation is in the form : 



y=Acos (at—t) (296) 



in which f (zeta) is a positive angle which varies with x. 



The surface slope at a point distant x from the origin, and at the 

 time t, is then: 



S cos (at+ H°) = by/ bx=A sin {at — f) df / bx 



=A cos {at- ^-7rl2)b^/dx. (297) 



Since equation (297) is identically true: 



S=Ab^/()x (298) 



H°=-^-Tr/2. (299) 



From equation (298) 



b^={SIA)()x. 



