Chapter VI 



Long Waves in Canals and Standing 



Waves in Entirely or Partly 



Closed Basins 



1. Long Waves in Canals 



(a) Canal of Uniform, Rectangular Cross-section 



In the general wave theory it has already been shown that, when the water 

 depth becomes small in proportion to the wave length, the nature of the 

 wave motion is changed completely, and that the propagation of the waves 

 at the surface of such a water-mass obeys another law as surface waves or 

 short waves, do. The equations applying to "long waves" can easily be 

 derived from the equations of motion, if the characteristic features of these 

 waves are considered at the time the basic equations are formulated. These 

 characteristics are chiefly the following: on account of the small water depth 

 relative to the wave length, the vertical motions become less important than 

 horizontal motion, and the vertical acceleration of the water particles can 

 be neglected. This means that at any point the pressure is in each case equal 

 to the statical pressure exercised by a column of water extending from the 

 free surface to the depth of the point under consideration. From the neglect 

 of the vertical acceleration follows that the horizontal motion is always the 

 same for all water particles in a vertical plane perpendicular to the direction 

 of propagation of the wave; in other words, the horizontal velocity u is 

 a function of the direction x and of the time t only. 



In a straight canal with a horizontal bed and parallel vertical sides and 

 with a constant, regular cross-section we place the x-axis parallel to the 

 length of the canal, the r-axis vertical and upwards. 



Let the ordinate of the free surface corresponding to the abscissa x, at 

 time /, be denoted by h + rj, where h is the ordinate in the undisturbed state. 

 Then the pressure at any chosen point will be 



where p is the (uniform) external pressure. Hence 



