8 
LORD RAYLEIGH ON THE CIRCULATION 
7/ ^ mil 
«=- ' v [e _% { - sin ft]—i cos #/+i/3y cos $y—\$y sin fiy—\e~ py ] 
+ fe-^{l-2%}J . (25), 
and 
~ 2 ^ [c-^{| cos /ly+f sin /3y + i7fy sin %+iV*} 
4- e -2%I _A|_j_3^j] . (26). 
To obtain the mean velocity parallel to a; of a particle, we must add to (25), the 
terms previously investigated and expressed by (20). If we call the total u , we have 
sin j3y—\e ~^ y }-f fe -2 ^{1 — 2% } ] . . . (27). 
At a short distance from the bottom e~ Py becomes insensible, and we have simply 
U ' = f«s™ 2kx e-™y (1-2 ky) .(28), 
V= -Lfv- e-H-Ti+iPy) .(29). 
The steady motion expressed by (28) and (29) is of a very simple character. It 
consists of a series of vortices periodic with respect to x in a distance For a given 
x the horizontal motion is of one sign near the bottom, and of the opposite sign at a 
distance from it, the place of transition being at y=(2£) -1 = \/47r. The horizontal 
motion of the first order near the bottom being by (18) u= u 0 cos kx cos nt, we see 
that it is a maximum when kx=0, n, 2 tt, ... If we call these places loops, and the 
places of minimum velocity nodes, (29) shows that v is negative and a maximum at 
the loops, positive and a maximum at the nodes. The fluid therefore rises from the 
bottom over the nodes and falls back again over the loops, the horizontal motion near 
the bottom being thus directed towards the nodes and from the loops. The maximum 
horizontal motion is simply %up/V, and is independent of the value of v. We cannot, 
therefore, avoid considering this motion by supposing the coefficient of viscosity to be 
very small, the maintenance of the vortices becoming easier in the same proportion as 
the forces tending to produce the vortical motion diminish. 
To ascertain the character of the motion quite close to the bottom, we must include 
the terms in e~ Py . When y is extremely small 
u'=u 0 2 V -1 sin 2 kx{—j/3y-\- . . .}■.(30), 
so that the motion is here in the opposite direction to that which prevails when 
can be neglected. 
