Long Waves in Canals and Standing Waves in Closed Basins 147 



b t = b ll 



level remains entirely free; on the contrary, the width b, of the canal above 

 this level must satisfy the equation 



2z\~ 3/2 



1/ 



The origin of the co-ordinates lies in this level, the x-axis in the direction 

 of the canal, the r-axis upwards, the v-axis in a direction perpendicular to 

 the jc-axis and h is the mean depth of the lower part of the canal (Fig. 67). 



Fig. 67. Section of canal to compute the transverse current with longitudinal waves in 



a canal (MacCowan). 



It is not necessary that, like in the figure, the cross-section of the lower part 

 of the canal be symmetrical to the centre line, only the upper part must satisfy 

 this condition. The sharp bend of the slope appearing in the low water line 

 reminds, as Thorade remarks, of the channels of the Watten, but it is doubt- 

 ful whether, cross currents can be completely disregarded in such cross- 

 sections. Thorade (1931, p. 76) has estimated the cross-currents which are 

 possible in such cases, provided there is no transversal slope of the water 

 surface. 



In the shaded part of the cross-section F in Fig. 67 reaching from y = to y, the flow section 

 is F+yrj and the the volume of the flow per second M = u(F+ yrj). Through a second cross- 

 section at a distance 8x flows a volume M+ (dMldx)dx. The amount of water staying per second 

 in the considered volume element is — (dMjdx)dx. Let v be the velocity of the cross-current, which 

 will be equal to zero in the middle of the canal if the cross-section is symmetrical; in the lateral 

 distance y the velocity will be v and through the profile (h+rj)dx flows the volume v(h+ij)8x. In 

 this volume element there remains per second the water volume — v{h+rj)8x. The increase in 

 volume must result in a rise of the water surface drj/dt per second above a surface ydx. This gives 

 following equation: 



(F+ yrj) 



8u 

 8x 



yu 



drj 

 dx 



drj 

 v(h + rj)+ —y 

 ct 







If we add the equation of continuity (VI. 7) in which I is replaced by u (h is the mean depth) and 

 neglect the members of second order, this equation is simplified to 



10< 



