Long Waves in Canals and Standing Waves in Closed Basins 143 



The equation of horizontal motion, viz.: 



du du 1 dp 



dt ox Q dx 



is further simplified in the case of infinitely small motions by the omission 

 of the term u(du/dx) which is of the second order so that 



- = -S^. (VI.3) 

 dt g 8x K } 



Now let 



t 



I =judt, (VI. 4) 







i.e. we introduce the horizontal displacement £ instead of the horizontal 

 motion u for the water particles in the .v-direction until the time /. Equa- 

 tion (VI.3) may now be written 



— = -*^ (VI 5) 



dt 2 g dx' K } 



which is the equation of motion for long waves. The ordinary form of the 

 equation of continuity 



Thus 



du dv _ 

 dx dz 



»!*--*£■ (V,6) 



»• 



if the origin be for the moment taken in the bottom of the canal. This formula 

 shows as a consequence of our assumption that the vertical velocity of any 

 particle is simply proportional to its height above the bottom. At the free 

 surface we have Z = h + rj and the equation of continuity in a canal with 

 a rectangular cross-section is found by 



,— *|. (VI. 7) 



If we eliminate from equations (VI. 5) and (VI. 7) either r t or £ we obtain 



3-*3-- S-*£- (VI - 8) 



It is easy to prove that its complete solution has the form 



I or tj = F(x - ct) +f(x + ct) (VI. 9) 



in which c = j/(gA), and where F and / are arbitrary functions. In other 

 words, the first term is a progressive wave travelling with velocity c in the 



