PROFESSOR KELLAND ON THE THEORY OF WAVES. 



501 



ANALYTICAL INVESTIGATION. 



Section i. — uniform wave-motion. 



I . We commence with the determination of Wave-motion in a fluid of finite 

 depth, on the hypothesis of parallel sections. 



Let PQ be a portion of the surface of the fluid, PM, QN vertical planes at 

 right angles to the direction of transmission : and let AM = x, MP = z, MD = y, 

 MN = Bx. We shall also retain the notation in common use according to which 



u or 



~ represents the velocity parallel to x, v or -j- that parallel to y. In 



dx 



or- 



der to avoid unnecessary length, we must adopt without demonstration the re- 

 suits which have been arrived at for fluid motion in general. The demonstrations 

 may be found in Poisson's Traite de Mecanique, 2d edition, tome ii. liv. 6, chap. I ; 

 in Moseley's Hydrodynamics, chap. vii. ; in Pratt's Mechanical Philosophy, Hy- 

 drodynamics, chap. i. ; or in Webster's Theory of Fluids, chap. x. ; to aU of 

 which we shall give references, for the sake of saving trouble to the reader. 



To find the motion of the portion PN. 



Let p be the pressure on an unit in PM ; p' that on an unit in QN ; then 

 the pressure on Yl^=fodyp; 



M N 



the pressure on Q.N^r^"^ "" di/ (p + ^§ x\ ; 



=f?^p-f?y (i.+^^^.r) -y;"-'^^^ (^+^^-) 



'dp 



dx ^ 

 ; + — dx 



= -8xf '^d^-fl'^^ .pd^. 



Let us suppose that all the parts in a given vertical move forwards equally 

 at a given time ; then u =-^ is independent of y. 



VOL. XIV. PART II. 4 



