177 



shape, in which the cross section diminishes at such a rate as to coimter- 

 balance the friction losses, also takes the form of a progressive wave, 

 but with the current out of phase with the tide. The progressive 

 wave is commonl}^ regarded, therefore, as a normal form of tidal 

 motion in a long channel; but tidal flow does not take the form of a 

 progressive wave in all cases. 



342. The retrogy-essive wm^e. — If the amplitude of a tidal component 

 is the same at both ends of the canal, and its phase at the far end 

 exceeds by the angle y the phase at the initial end, so that high 

 water is later at the initial end, equations (260) and (261) become: 



y=Ao cos {at-\-ax/c-\-ao) (262) 



v= — (g I c)Ao cos {at -\- ax I c-{-ao) (263) 



A wave then retrogresses through the canal at the rate of -^J gD 

 feet per second. 



343. Resolution oj Jrictionless tides in a connecting canal into pro- 

 gressive and retrogressive waves. — The application of equation (259) 

 to the equation of any component of the tide in the form derived in 

 equation (226): 



y=Ao cos (at-\-ao) sin {y — ax/c)/sm 7+^1 cos (ai+«i) sin (ax/c) /sin 7 



gives: 



y=y2Ao sin {at-\- ao-\- y — ax/ c) /sin y — JiAo sin {at-\-ao—y-\-ax/c)/sin y 

 -\-y2A1 sin {at -^ ai-\- ax /c)/ sin y — JiAi sin {at-\- ai — ax/c) /sin 7 



The first and fourth terms of this equation may be combined into a 

 term in the form: 



Wi cos {at—ax/c-\-Wi) 



and the second and third into one in the form: 



Wo cos {at-{-ax/c-\-'W2) 

 giving: 



y=Wi cos {at—ax/c-\-Wi)-{-W2 cos {at-\-at/c-\-W2). (264) 



From the form of equation (264) it is seen that if the flow were 

 frictionless, the water surface in a connecting canal produced by a 

 component of the tide at the entrances would be the resultant of two 

 waves, one progressive and the other retrogressive. Since the speed, 

 ■yJgD, of the waves produced by each component of the tide depends 

 only on the depth in the channel, the resultant of all of the compo- 

 nents of the tide is similarly resolvable into two compound waves 

 traveling in opposite directions through the canal. The combination 

 of these two component waves produces, in general, a wave which 



