PROFESSOR KELLAND ON THE THEORY OF WAVES. 1Q9 



73 2D 3 2 2DD 3 



- J * 



whence, omitting small quantities, 3 S 4 6 S 2 =D 3 . 



In order that this may hold, it is necessary that A be much longer than 2 nr h, 

 a circumstance which appears not likely ever to be satisfied. We conclude there- 

 fore that k=Q for all Avaves in which the pressure is a complete differential of 

 an and y, and the motion oscillatory. 



Writing /fc=0 the expression becomes 



- b D cos 6 - 9- a t> 3 cos 3 6 - 4 If c sin 2 6 + {2 S - S 3 cos 6- S cos 3 6} 

 a a 



+ 4a6eS 4 sin20-8a&cS 2 sin40 



+ 4 a b c S 4 sin 2 0-8 a 5 c S 2 sin 4 6 + 6 c 2 S cos + 3 a c 2 S 3 cos 3 6 = 0. 

 We have already obtained this equation in a more general form, in the for- 

 mer Part. 



v 



SECTION V. 



64. We turn our attention next to the case of waves proceeding along a canal 

 of given or of infinite width, but of variable depth, in the direction of motion. We 

 will commence with the case in which the depth gradually diminishes, so that 

 the bottom of the fluid is an inclined plane. It will be more simple in this case 

 to assume the plane which passes through the centre of the wave to be the plane 

 of x z. If, then, we write the values of u and v as originally obtained, they will be 



= b (e* y+ h - x + *- *) sina (x-cf) 

 v=-b(e" 2/ + /t -^_ tf y + *-) cos a (x-cf), 



where h ax is the height of the plane of xz above the bottom of the fluid, or is 

 the statical depth at the point in question. 



Let us conceive, then, that the above equations represent the velocities in the 

 case in question at the surface of the fluid when z is written for y ; z being now 

 very small. As long as the waves retain their /orw, this must be true very nearly 

 at least. 



It must be observed that the same thing is not true to all depths, for the very 

 obvious reason, that at the bottom of the fluid the motion is no longer perfectly 

 horizontal. There will, therefore, be a perpetual impulse tending slowly to alter 

 the form of the wave. But, whatever be the form, it is possible to expand it in a 

 series of the following nature, 



VOL. XV. PART I. G g 



