142 



The relation between the surface head and the velocity in a short sec- 

 tion of a channel, derived in this chapter, affords, however, a basis for 

 estimating the currents in a projected long canal, by a procedure which 

 is developed in detail in chapter VIII. 



CURRENTS IN A SHORT SECTION OF CHANNEL WHEN THE 

 FRICTIONAL RESISTANCE IS NEGLIGIBLE 



287. If a channel is so deep, and the current velocities are so small, 

 that the flow is essentially frictionless (par. 257), the currents pro- 

 duced in a short section of the channel by any fluctuation of the tides 

 at the ends of section have a simple relation to the amplitudes and 

 speeds of the harmonic components of these tides. Designating the 

 amplitudes of the several harmonic components of the tide at the 

 initial end of the section as M2', S2', etc., and at the other end as M2", 

 S2", etc., the equation for the tide at the initial end becomes: 



yo=M2 cos (m.2t-j-ai')-{'82 cos (s2^ + q;2') + ' " " 



and at the other end: 



7/i = M2" COS (m2^ + «/0+S2" cos (s2i+a2") + - • " 



The surface head through the section is then: 



hs^yi-yo-=M2" cos (m2i+a/0-M2' cos (m2i+«i') 



+ S2" cos (s2ti-a2")-S/ COS (Ssi + aaO 



+etc. (175) 



288. Since the respective pairs of components of the same speed 

 unite into components of that speed, equation (175) reduces to one in 

 the form: 



hs=H, cos {m2t+Hi°)+H2 cos (sot + H.2°) A- ■ ■ • 



In which the amplitudes. Hi, H2 and the phases Hi°, H2° of the com- 

 ponent surface heads could be computed by the process indicated in 

 paragraph 239. 



\^Tien both the velocity head term and the friction term in equation 

 (112) are dropped, this equation becomes: 



by/bx+(l/g)c>vldt=0 (176) 



Whence: 



'J 



v=-g\ {^y/^x)c)t (177) 



