PROFESSOR KELLAND ON THE THEORY OF WAVES. 



501 



ANALYTICAL INVESTIGATION. 



Section i. — iinifoem wave-motion. 



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

 depth, on the hypothesis oi parallel sections. 



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

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

 MN = 8x. We shall also retain the notation in common use according to which 



dip 



d(p 



u or -j^ represents the velocity parallel to w, v or -^ that parallel to y. In or- 

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

 sults 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. 1 ; 

 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 all of 

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



To find the motion of the portion PN. 



Let 2) be the pressure on an unit in PM ; pi' that on an unit in QN ; then 

 the pressure on VM=fodyp; 



the pressure on ^^ ^' ^ '^ "^^ (^ + ^ ^ »^) ; 



. •. the moving force =r dyp ~J l^ ^' " '^' ^ \P + -£.^ A 



=f^dyp-fyy (/. + g^-) -jf^'^dy (^ + g^-) 



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

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



VOL. XIV. PART II. 4 



