OF AIR OBSERVED IN KUNDT’S TUBES. 
On account of the factor e Py this quantity is sensible only when y is very small. 
We mav write it with sufficient approximation 
nku 0 2 sin 2kx e 
4z rp 
— /3y sin (3y — cos /3y+e Py 
( 22 ). 
The terms in 2 nt, corresponding to motions of half the original period, are not 
required for our purpose, which is to investigate the non-periodic motion of the second 
order. The equation with which we have to proceed is found by equating (22) to v‘h/>- 
The solution will consist of two parts, one resulting from the direct integration of (22) 
and involving the factor e~ Py , the second a complementary function with arbitrary 
coefficients satisfying vh//=0. In the calculation of the first part we may identify v 1, 
with <P/dy 4 , on account of the smallness of Jc relatively to /3. In this way our equation 
becomes 
cl^yjr _ nJcud sin 2kx e 
d/fiy)* 4:v~f3 5 
— /5y sin 
/3y —cos /3y-\- e Py 
(23), 
of which the solution is 
*=' 
nkud sin 2 kx e Py 
4i/ 2 /3 5 
f cos (3y +1 sin f3y+±/3y sin f3y e 
-to 
(24). 
The complementary function, being proportional to sin 2 kx, may be written 
nkud sin 2 kx 
4zF/3 5 
{(A+By)e- 2 ^+(A' + B , y)c +2 ^}. 
If the fluid be uninterrupted by a free surface, or otherwise, within distances for 
which hy is sensible, we must suppose (A / +B / y) = 0, so that by (13) the complementary 
function may be written 
uJ sin 2 kx 
/3V 
(A + By)c 2 A 
The condition that v (equal to — dxfj/dx ) must vanish when y— 0, gives A= —yf. 
For the velocity parallel to x we have 
u~- 
ud sin 2 kx, 
V 
[e &{ — sin (3y — ± cos f3y+±(3y cos /3y~i/3y sin (3y —|e 
+ /3~ l e~ 2 ^ { B — 2k(A + By )f ]. 
In order that u should vanish when y— 0, we must have 
B = 2M+|/3 = f/3-^ = i/3, 
approximately. Thus 
