

ON THE MOTION OF WAVES IN A VARIABLE 

 CANAL OF SMALL DEPTH AND WIDTH. 



THE equations and conditions necessary for determining 

 the motions of fluids in every case in which it is possible to 

 subject them to Analysis, have been long known, and will be 

 found in the First Edition of the Mec. Anal, of Lagrange. Yet 

 the difficulty of integrating them is such, that many of the most 

 important questions relative to this subject seem quite beyond 

 the present powers of Analysis. There is, however, one par- 

 ticular case which admits of a very simple solution. The case in 

 question is that of an indefinitely extended canal of small 

 breadth and depth, both of which may vary very slowly, but 

 in other respects quite arbitrarily. This has been treated of in 

 the following paper, and as the results obtained possess con- 

 siderable simplicity, perhaps they may not be altogether un- 

 worthy the Society's notice. 



The general equations of motion of a non-elastic fluid acted 

 on by gravity (g) in the direction of the axis z, are, 



supposing the disturbance so small that the squares and higher 

 powers of the velocities &c. may be neglected. In the above 

 formulae p = pressure, p = density, and < is such a function of 

 x, y, z and t, that the velocities of the fluid particles parallel to 

 the three axes are 



(d$\ 



M== T^ > 

 \dxj 



dp 

 Tz. 



15 



