Long Waves in Canals and Standing Waves in Closed Basins 145 



where it is no longer possible to neglect the vertical acceleration. The wave 

 crests overturn and break. 



Table 18a gives the Lagrangian velocity of propagation c for various 

 depths. The appendix contains an extensive table. The bottom line of the 



Table 18 a. Wave velocity according to Lagrange for several depths 



table gives the time needed by a wave to travel over a distance equal to the 

 circumference of the earth 2nR. In order that a long wave complete this 

 distance within one day (24 h), the water depth should be 22 km. This result 

 is important in relation to the theory of the tides. 

 (b) Canal with uniform arbitrary cross-section 



Kelland (1839, p. 497) has extended the results obtained for a canal of 

 constant rectangular cross-section to a uniform canal of any form of section. 

 The equation of horizontal motion (VI.5) remains unchanged, but the form 

 of the cross-section of the canal is introduced into the equation of continuity. 

 If the area of the section is S when the surface is undisturbed, its width at 

 the free surface b, we obtain, in case there are no cross-currents or pressure 

 gradients perpendicular to the canal, the equation of continuity 



a(6S) 



8x 



+ i)b =0. 



(VI. 14) 



If S and b are constant and h = SJb denotes the mean depth of the canal, 

 we can also use in this case the Lagrangian equation 



// 



dx' 



if the mean depth h is substituted for h. If, however, the width b is not 

 constant, when the wave passes by, which is the case for a canal with sides 

 inclined, conditions are changed and the Lagrangian equation cannot be 

 applied. MacCowan (1892) has dealt with a canal with sides inclined and 

 he studied the question if the change of the wave profile caused by the 

 propagation of the wave with a velocity (VI. 1 3) could be cancelled through 

 an appropriate form of the slope, so that the wave form would be again 

 permanent. This can be done, whereby the cross-section below the water 



10 



