334 TIDAL WAVES. [CHAP. VIII 



From (9) we find 



and if we substitute from these in (10), we obtain an equation in 

 nly. 



In the case of uniform depth the result takes the form 



where Vj 2 = d 2 /da? + d 2 /dy 2 , as before. 



When =0, the equations (7) and (8) can be satisfied by constant values of 

 u -> v i provided certain conditions are satisfied. We must have 



2-*t (iii) 



2ndx~ 



and therefore =Q ................................. (iv). 



** v^> y ) 



The latter condition shews that the contour-lines of the free surface must be 

 everywhere parallel to the contour-lines of the bottom, but that the value of 

 is otherwise arbitrary. The flow of the fluid is everywhere parallel to the 

 contour-lines, and it is therefore further necessary for the possibility of such 

 steady motions that the depth should be uniform along the boundary (sup- 

 posed to be a vertical wall). When the depth is everywhere the same, the 

 condition (iv) is satisfied identically, and the only limitation on the value of 

 is that it should be constant along the boundary. 



201. A simple application of these equations is to the case of 

 free waves in an infinitely long uniform straight canal *. 



If we assume f=oe* (<*-*) +y, v = ..................... (1), 



the axis of x being parallel to the length of the canal, the equa- 

 tions (7) of the preceding Art., with the terms in f omitted, give 



cu = g 2nu = -gm.... ................. (2), 



whilst, from the equation of continuity (Art. 200 (8)), 



c{=hu .............................. (3). 



* Sir W. Thomson, I.e. ante p. 331. 



