163 



325. Computation of tides and currents produced hy frictionless flow 

 in a connecting canal. — The tides and currents in a connecting canal 

 are determined by the known amplitudes ^o and -^^li, and initial phases 

 cco and ai, of the several components at the two entrances to the canal. 

 As shown in paragraph 239, equation (228) may be reduced to the 

 form: 



y—A cos {at-\-a) 

 by placing: 



A cos a=AQ cos QJo sin (1— x/X)7/sin 7+ A cos a^ sin (x/Z)7/sin 7 (231) 



A sin a=AQ sin a^ sin (1 —x/L) 7/sin 7 + ^1 sin ai sin {xlL)ylshi 7 (232) 



The value of 7 in degrees, for any component of the tide, is given 

 by equation (223): 



7 = aL/c = aL/^JgD 



in which L is the length of the canal, D its mean depth, and a the 

 speed of the component, in degrees per second. Thus the value of a 

 for the M2 component is 28°. 9841/3600 = 0°. 00805. 



The initial phase, a, and the amplitude A of each component of the 

 tide at a point distant x from the origin of distances may be deter- 

 mined from the values of A cos a and A sin a, equations (231) and 

 (232). 



The equation of the a component of the current in the canal at a 

 point distant x from the origin, equation (230), may be reduced to 

 the form: 



v^B sin (a#+/3) 



by a similar procedure. 



326. Instantaneous profiles and wave lengths. — The longitudinal sec- 

 tion of the water surface in a long tidal channel at any instant is a 

 curve designated as the instantaneous profile. The instantaneous 

 profile of a component of the tide at any time, to, in a long connecting 

 canal of uniform cross section with frictionless flow, is derived at once 

 by placing t=tom equation (226). 



This equation then takes the form: 



y=C sin iax/c—y)^C' sin {ax/c) (233) 



in which 



C=—Ao cos (rtio+ao)/sin 7, and C'=Ai cos (aio+«i)/sin 7 



