164 

 This equation is readily transformed into: 



y— W sin {axlc-\-'w) 



(234) 



The graph of this equation, if sufficiently extended, is the sinusoidal 

 curve shown in figure 51. 



Y 



< — 



X 



X 



Figure 51.— Instantaneous Profile. 



The distance, X (lambda) from crest to crest of the profile is desig- 

 nated its wave length. The crests of the sinusoidal curve representing 

 equation (234) are at the points at which: 



So that: 



axo/c-\-w=Tr/2, axi/c^w=57r/2, etc. 



\=Xi—Xo=5Trc/2a—7rc/2a=2irc/a=2T^jgD/a 



(235) 



In which a is the speed of the component, in radians per second, D 

 the mean depth of the canal, and g the acceleration of gravity. The 

 wave length of the M2 component of the tide, in a channel whose 

 mean depth is 30 feet, is, for example 



27rV305f 



0.00014053 



1,389,000 feet ==263 miles 



Evidently, the length of a canal is usually but a small fraction of 

 the wave length of its principal tidal components. 



The form of equation (229) shows that the graph of the instan- 

 taneous component velocities through a long canal of uniform dimen- 

 sions with frictionless flow is a similar sinusoidal curve with the same 

 wave length as the instantaneous profile. 



327. Relation between y and \. — From equations (223) and (235) 



7=aX/c=27ri/X 



(236) 



It may be noted that if the length, L, of a connecting canal is one- 

 half the wave length, X, of a component of the tide, 7=7r and 

 sin 7=0. As subsequently discussed in paragraph 347, the tides and 

 currents in a canal of this length would become infinite if there were 

 no frictional resistance to flow. 



