178 



travels through the canal with changing amphtiide and varying 

 speed. The currents may be correspondingly resolved. 



344. The resolution of the frictionless tides in a connecting canal 

 200,000 feet in length and 30 feet in mean depth, into component 

 waves, when the equation of the tide at the initial and farther entrances 

 are respectively: 



y=4: cos nizt y=2 cos (m2^+60°) 



gives the equation: 



y=1.30 cos (m2^-m22;/c-46°10')+3.24 cos (m2^+m22:/c + 16°470 



It may be seen, therefore, that the amplitudes and phases of the 

 progressive and retrogressive VMves in a long connecting canal generally 

 differ widely from the amplitudes and phases of the tides at the en- 

 trances, and have no simple relation thereto. 



345. The stationary wave. — In a closed canal, and in certain special 

 cases in a connecting canal, as has been seen, the phase of the tide and 

 of the current produced by frictionless flow is the same at all points, 

 so that high and low water and the strength and turn of the current, 

 each occurs simultaneously throughout the canal. When the tide in 

 a closed canal, derived in equation (251), is resolved into progressive 

 and retrogressive waves by the process indicated in paragraph 343, its 

 equation becomes: 



?/=K^o cos (ai—oa^/c + 0:0+7) /cos 7 



+ K^o cos (at-\-axlc-\-ao — y)/cos 7 (265) 



These two component waves have the same amplitude, Ao/2 cos 7. 

 The retrogressive wave may be regarded as the reflection of the pro- 

 gressive wave from the end of the canal. The resultant of the 

 progressive and retrogressive (or reflected) waves of the same ampli- 

 tude is a wave which neither advances or retreats, but remains 

 stationary. It will be noted that when a stationary wave is produced 

 by frictionless flow, the current turns at high and low water; while if 

 a simple progressive wave is produced, the strength of the current at 

 each station along the canal occurs at high and low water, and the 

 current turns at midtide. 



Frictional resistance must modify the conditions of flow in a long 

 closed canal, since the lag of the current increases as the currents 

 decrease toward the head of the canal. From another viewpoint, the 

 absorption of energy by friction reduces the amplitude of the reflected 

 wave. The tides at the head of a closed channel are therefore always 

 later than those at the entrance, and a completely stationary wave is 

 never found. It is nearly realized in such deep channels as the fiords 

 of Alaska. Thus in the Portland Canal, a fiord from 600 to 1,000 feet 



