524 



PROFESSOR KELLAND ON THE THEORY OF WAVES. 



2. In the case of common oscillatory waves, there ought one to be positive 

 and the other negative, equal in magnitude, 



from which it follows, that bo is a quantity depending entirely on the progressive 

 motion. 



SECTION II. — VARIABLE WAVE MOTION. 



2o. Hitherto we have confined our attention to motion in two dimensions, 

 limiting the channel to one of uniform breadth and depth. 



We proceed next to the more general case, where the breadth is variable but 

 the depth constant. 



To take the same order as before, we will commence with the hypothesis 

 that all vertical sections have the same horizontal velocity at any tune, the sec- 

 tion being perpendicular to the direction of transmission. 



By reference to arts. 2, 6, and 7, the process which follows will be perfectly 

 intelligible, without any lengthened explanation. 



We commence with that case wherein the section perpendicular to the direc- 

 tion of transmission is a triangle, one side of which we may suppose vertical, but 

 it will not affect the calculation. 



Let the axes be called those of oc, y, and z ; the introduction of a third being- 

 requisite in this case. In the former calculations, we 

 adopted z as the representation of the depth : in the 

 present case, we shall use the letter s for the same 

 purpose. 



The annexed figure is supposed to be a section of 

 the fluid perpendicular to the direction of motion, and 

 it is supposed that all sections so made are similar. 

 Our object is to determine the motion of a portion of 

 the fluid enclosed between two planes perpendicular to 

 the direction of translation. 



Let a be the distance between these planes at the time U 

 a + Ba. . . . . at the time ^+^^; 



s, the depth of the fluid at ^, . 



s + 8s Sit t + 8t', 



r the breadth at the top at the time t, 

 r+8rsitt+8t; 

 then, from the property of the triangle, r and s bear a constant ratio : let r=ms. 



Also, since the quantity of fluid between the planes is supposed to remain 

 unchanged, we obtain 



rscx,= (r+8r) (s + 8 s) (a, + 8 a) 



