175 



339. From equation (260) it is soeii that the component tide has, 

 under these conditions, a constant amplitude throughout the canal. 

 At a point distant x from the origin its high water occurs when 

 at—axlc-\-ao = 0; or when t — xlc — aoja. 



The time of high water therefore increases with the distance x at 

 the uniform rate of c = -ylgD feet per second. Successive instantaneous 

 profiles when sufficiently extended, have the relation shown in figure 

 57. 



in I 31 



Figure 57. — Successive instantaneous profiles of progressive wave. 



Evidently, in this case, a wave jpr ogresses through the canal with a 

 speed, -^gD, which depends only on the depth, D, of the canal and is 

 independent of the speed of the tidal fluctuations and of the wave 

 length of the component, and independent also of the maximum cur- 

 rents in the canal. 



The current velocities (equation 261) similarly have the constant 

 amplitude, Aoglc=Ao'\JglD throughout the canal. At every station 

 alang the canal the strength of the current occurs at liigh w^ater. 



340. Example of a frictionless progressive wave. — In a canal 200,000 

 feet in length, with a m.ean depth of 30 feet, the value of y for the M2 

 component of the tide has been found to be 51°50'. The tidal flow 

 through a canal of these dimensions will then have the form of a pro- 

 gressive wave if the tides at the entrances have a simple harmonic 

 fluctuation with the speed of the M2 component, the same amplitude, 

 and the phase of the tide at the initial entrances is 51°50' larger than 

 that at the other entrance. The tide at the farther entrance is then 

 51°.83/28°.98 = 1.79 solar hours, or 51°. 83/30°= 1.73 lunar hours later 

 than at the initial end. If tidal range at the cmtrances is 8 feet, the 

 equations of the tides and currents in the canal are, when the origin 

 of tim.e is at the time of high water at the initial entrance: 



y = 4: cos (m.^— 0°.000259a:) 



i'=4.14 cos (mo^-0°.000259j) 



The currents and tides at the entrances and at the midpoint of the 

 canal, and the instantaneous profiles at successive lunar hours, are 

 sho^v^l in figure 58, page 176. 



