Long Waves in Canals and Standing Waves in Closed Basins 223 



Especially interesting are the conditions in an infinitely long, uniform canal. 

 If we neglect in (VI. 130) the friction, we find first a solution in the form of 

 Kelvin waves (VI. 103); if rj = 0, then 



iy=0, v = 0, ?■= ±- = e*W»F(x±ct), (VI. 132) 



in which c = ygh and F is an arbitrary function of the argument (jc=F ct). 

 Proudman, however, has also given a solution for the case that an atmos- 

 pheric pressure wave with a velocity V travels in the positive direction of 

 the canal. In this case, the solution is 



! = e -(flv)yF(x-Vt) 



and 4 = 



1 



Vg 



c- 



■WV)y 



'F{x-Vt) 



(VI. 133) 



This is a wave disturbance which travels also in the + x- direction; the 

 factor 1/[1 — (V 2 /c 2 )] shows, however, that large amplitudes are to be expected, 

 when the atmospheric pressure disturbance travels with the same velocity 

 as the free water waves in the canal. The form adopted in (VI. 133) for the 

 pressure disturbance in the y- direction is somewhat special; it is difficult 

 to transfer it to arbitrary pressure distributions. But if the canal is not too 

 wide, the e -power in (VI. 133) can be disregarded, which means neglecting 

 the earth's rotation. The application is then limited to narrow canals. 



Let us now assume that the canal be closed at x = and extends infinitely 

 in the x- direction. The wave disturbance generated by a travelling pressure 

 disturbance now must also fulfil the boundary condition u = for x = 0. 

 This can be done by combining equations (VI. 132) and (VI. 133) (/= (o = 0) 

 and we obtain: 



rj = 



rj=F[t 



1 



v = and 



(F 2 /c 2 ) 

 V/c 



'-3-?'H 



(VI. 134) 



g l-(F 2 /c 2 ) 



Ft 



F^K 



A free wave is added to the forced wave. For x = 



1 

 l+V/c 



F(t) 



and 



'/ 



1 



rj 1 + V/c 



= constant 



(VI. 135) 



