172 



would occur at mean tide. The tides and currents at the 

 entrance, the middle, and the end of a closed canal 30 feet in mean 

 depth and 200,000 feet in length, which would be produced by a 

 simple harmonic fluctuation of the tide at the entrance with the speed 

 of the M2 component, and a range of 8 feet, were the flow frictionless, 

 are shown in figure 56, with the instantaneous profiles through the 

 canal. 



337. Relation of amplitudes of the components of the current to those 

 of the components of the tide in a closed canal. — From equation (251), 

 the amplitude of a component of the tide at a point distant x from the 

 entrance is: 



A=Ao cos (1— a;/i)7/cos 7 (253) 



and from equation (252) the amplitude of the corresponding compo- 

 nent of the current is: 



B=(g/c)Ao sin (1— a;/X)7/cos 7 C^54) 



so that: 



B=(g/c)Atsin (l-x/L)y (255) 



Designating the amplitude of any other component of the current 

 at the same point as Bi, the amplitude of the corresponding component 

 of the tide as Ai, and the value of 7 for this component as 71, then: 



Bi = (g/c) ^1 tan ( 1 - xjL) 71 



and: 



BJB=(AilA) tan (l-a:/L)7i/tan il-x/L)y (256) 



If the length of the canal is a small part of the wave lengths of the 

 components, the angles {l—x/L)y and (l—x/L)yi are small, and are 

 approximately proportional to their tangents. 



Then, approximately: 



BJB=(A/A)iy,/y) (257) 



Since y=aLjc and 7i=aii/c, equation (257) becomes: 



B,/B=A,aJAa (258) 



It follows, therefore, that unless a closed canal is quite long, the 

 amplitudes of the components of the current produced at a given 

 point by frictionless flow, are nearly proportional to the products of 

 the amplitudes and speeds of the corresponding components of the 

 tide at that point. These speeds, it may be observed, may be ex- 

 pressed in any units. 



