158 



The amplitudes, M2, S2, etc., and the phases aj, as, etc., of the 

 several components may vary wdth the distance x, of the station from 

 the origin. This origin is conveniently taken at either entrance to a 

 connecting canal, or at the single entrance to a closed canal. The end 

 so chosen will be termed the initial end. 



Expanding the cosines in equation (190): 



y=M2 cos oil cos mo^— M2 sin ai sin m2^+S2 cos a-z cos S2^— 



S2 sin ao sin S2^+" • • (191) 

 Equation (191) may be written: 



?/=Xi cos m2^+Fi sin m2^+X2 cos S2^+F2 sin S2#+* ' " (192) 



In which Xi, Fi, X2, Y2, etc., are functions of x. 



318. When the flow is frictionless, the current is likewise the result- 

 ant of component currents having the speeds of the harmonic com- 

 ponents of the tide (par. 289). 



The velocity at the time t is then giv^en by an equation in the form: 



y=Fi cos m2^+Zi sin m2^+F2 cos S2i+Z2 sin S2^+- • • (193) 



in which T'l, Zi, V2, Zo, etc., are similarly functions of x. 



The form of the functions. A', Y, Z, and T" necessary to satisfy the 

 equation of fluid motion for frictionless flow, and the equation of con- 

 tinuity, is then to be determined. 



319. When the frictional and velocity head terms are omitted the 

 equation of motion is (equation 176): 



dy/dx+{l/g)c)v/bt=0 



And in a channel of uniform width and depth, with a tidal fluctua- 

 tion small in comparison with the depth, the equation of continuity is 

 (equation 184): 



dy/c)t+Ddv/dx=0 



Substituting the differential coefficients derived from equations 

 (192) and (193), the equation of motion becomes: 



(dAi/dx)cos m2^+(c)ri/dx)sin m2^+(c)A2/dx)cos S2^ 



+ (dF2/c)J:')sin S2^+ • ' ' — (m2Fi/5') sin m2#+(m2Zi/^)cos m2i 



— (s2F2/S')sin S2^+(s2Z2/fir)cos S2if— ' • ■ =0 



or: 



(dA''i/dj^+ni2Zi/g')cos m2if+(dFi/dx— m2Fi/^)sin mgf 

 -f (dAV^^+S2Z2/^)cos s2^+(dF2/dx-S2F2/^)sin S2#+ • • • =0 (194) 



