OF AIR OBSERVED IN KUNDT’S TUBES. 
13 
u = {B — 2k( A + By)) e 2ky sin 2kx, 
v— — 2&(A+By)e -3 ^ cos 2 kx. 
Determining the constants as indicated above we get 
2>v 2 
?/■ = —■ (1— 2fcy)e~ 2 ^ sin 2kx .(51), 
lev * 
v— — (11 + 6/5y)e~ 2ky cos 2 kx .(52). 
The velocities given by (51), (52) are the only part of the motion of the second 
order which is sensible beyond a very small distance from the vibrating plate. The 
nodes of the plate (where sand would collect) are at the points given by kx— 0, n, 2v ..., 
and the loops at the points kx=^n, § tt . . . At the former points v is negative, and at 
the latter positive. For kx=\ir, u is positive, and for Jcx=%tt,'u is negative. 
! — ! —-—i—— J 
0 t > 7 t 7r -|7r 
node loop node loop 
The magnitude of the vortical motion is independent of the coefficient of friction. 
The complete value of u to the second order of approximation (except the terms in 
2 nt) is obtained by adding together (37), (46), and (51), and it will contain the term 
divided by k in (46), whose appearance, however, is misleading. The objectionable 
term will be got rid of, if we express the mean velocity of a particle, instead of as in 
(46), the mean velocity at a point. For this purpose we are to add to (46), (51), the 
mean value of 
du 
dy 
as calculated from the first approximation where 
du 
As in the former problem the mean value of is zero. 
el'll; C 
Multiplying together- , and I vdt as found from (37), (38), and rejecting the terms 
dy’ 
in 2 nt, we get with omission of k 2 , 
