OF AIR OBSERVED IN KUNDT’S TUBES. 
11 
that these prescribed velocities apply in strictness not to y— 0, but to y=^sin kx sin nt. 
Substituting the latter value of y in the expressions (37), and (38), we find 
u 
= v /2./3 H cos kx | —ky cos (^+e+^7r) + \/2./3y cos {nt-\-e-\-^Tr) j 
l ~ sin 2 kx sin j —cos (wt + e-hi 77 ) + cos ( w ^ + e + 2 7r )| 
-in (e+i 77 )— sin (e+i 77 ) | + ten 
/3\H • 9Z J k 
— sm 2kx\ „ sm 
2n 
:ms in 2 nt. 
The first term within the bracket is of the second order in kj{3 relatively to the 
latter term, and may be omitted. Thus 
/3 2 %H . 7 
u— — —— sm 2 kx cos e. 
The terms in 2 nt we need not further examine. From (39), (40), H cos e=— v 0 /2/3, 
very approximately, so that we may write 
ftvd . 7 
u——— sm 2kx 
4:71 
(41). 
To the same degree of approximation, v=v 0 sin kx cos nt, simply. 
We have next, as in the first problem, to consider the complete equation 
4 , _n dy 2 , v 
vV 
v~ dxdt v 2, dydt 
(42) 
m the right hand member of which we use the approximate values given by (36), 
(37), (38). Thus 
— ?iH cos kx e ^sin (nt-\-e—fiy), 
etc 
and (42) becomes 
. nk/3H 2 sin 2kxe Py \ 1 2& . n ^ \ . „ i /jr> . 
V 4 ’/'=-— 2 -j e M ~ sm fiy— sm /3y— cos (3y ) + 2e }• . . (43). 
It will be found presently that the term divided by k disappears from the final 
result, and thus we have to pursue the approximation further than might at first 
appear necessary. We may however neglect terms of order Z; 3 //3y in comparison with 
the principal term. Thus v 4 may be identified with —, and the equation becomes 
