119 



The surface head in the section is then: 



hs=yi—yo=A.i cos (at-\-ai)—Ao cos {at-]-ao) 

 =Ai cos {at+ai)-\-Ao cos (at+ao^ 180°) (135) 



Since two components of the same speed unite into a single com- 

 ponent of that speed, equation (135) may be written: 



h,=^H cos {at+W) (136) 



In which H is the amphtude, and W the initial phase, of the fluctua- 

 tion of the surface head during the tidal cycle. 



239. Computation of H and H^. — The amplitude and phase of the 

 surface head in the section readily may be determined from the 

 amplitudes and phases of the tides at the ends of the section, through 

 the relation established in equations (135) and (136) : 



H cos (at+H^)=Ai cos (at-{-ai)—Ao cos (at+ao) (137) 



Equation (137) is identically true for all values of t. By placing 

 at^O, the equation of condition is derived: 



H cos H^=Ai cos ai— ^0 cos ao (138) 



and by placing at=—90° 



H sin H^=Ai sin ai~Ao sin ao (139) 



The values of i?° and H may then be determined from the equations: 



tan H'^^H sin HyH cos W (140) 



H=H sin i^Vsin H''=H cos i^Vcos H' (141) 



The amphtude, H, is directionless. The quadrant in which H^ hes 

 is determined by the algebraic signs of H sin H° and H cos H'^. 



240. Example. — The curve showing the average height of the tide 

 at station 180+30 on the Cape Cod Canal, after the time of a lunar 

 transit, prepared from observations during the period September 28 

 to October 6, 1932, is represented by the equation: 



2/=3.74 cos (m2H-58°340 



and at station 225 by: 



y=3.18 cos (m2^+61°10') 



