174 



The diurnal components of the current in a closed canal have there- 

 for a much smaller ratio to the semidiurnal current components than 

 the diurnal components of the tide have to the semidiurnal tidal com.- 

 ponents. It is easy to see, indeed, that the filling and emptying of the 

 tidal prism of the canal by the diurnal components is at but half the 

 rate of the filling and emptying by the semidiurnal components. In 

 a connecting canal, on the contrary, the ratio of the diurnal to the 

 semidiurnal components of the current may be increased because of 

 the proportionally large acceleration heads set up by the semidiurnal 

 components. 



PROGRESSIVE, RETROGRESSIVE, AND STATIONARY WAVES 



338. The progressive wave. — A special condition of frictionless tidal 

 flow arises when the amplitude of a component of the tide is the same 

 at both entrances of a connecting canal, and the phases of the com,po- 

 nent at the two entrances differ by the angle 7. Taking first the case 

 in which high water at the further entrance is later than at the initial 

 entrance, and placing in equation (226), ^1=^0, and ai = aQ—y, this 

 equation becomes: 



y=Ao cos {at-{-ao) sin (7 — ax/c) /sin 7 

 + A cos {at + ao — y) sin {ax /c) /sin 7 



Expanding by the formula: 



cos A sin B=% sin {A+B) - % sin (A-B) (259) 



y=%Ao[sm (a^ + a:o + 7 — «x/c)— sin (at + ao—y-^ax/c) 

 + sin (af + ao — 7 + ax-/c)— sin {at + ao—y — ax/c)]/sm 7 

 = KA[sin {at-\-ao—ax/c-\-y)—sm (at + (Xo—ax/c — y)]/sin 7 

 =Ao cos (at-\-ao—ax/c) sin 7/sin 7 (260) 



=ylo cos {at—ax/c-\-ao) 



The expression for the current is most readily derived by applying 

 equation (177): 



v=-gj iciy/^x)c)t 



From equation (260): 



()yldx={Aoa/c) sin {at—ax/c-\-ao) 

 Whence: 



v=—g I (Aoa/c) sin (at—ax/c+ao) bt =ig/c)Ao cos (at—axlc+ao) (261) 



