OF AIR OBSERVED IN KUNDT’S TUBES. 
5 
k . 
u=u 0 e int cos kx\ — sink ky-\- cosh hy—e 
-lc'y 
• • (15). 
v=u 0 d nt sin kx \—^ cosh ky-\- sinh ky-\-^-e [■ .... (16). 
These are the symbolical values. If we throw away the imaginary parts, we have 
as the solution in real quantities by (12), 
7 , cosh ley , i \ , 
*]j=u 0 cos kx\ — ^^2 cos \ nt ~i 7r ) + 
-Pv 
sinh ky , e , . 
----- - cos nt-\- cos 
• (17), 
u=« n cos feci — - cos (?2t—^77-) + cosh ky cos —e ^ cos ( nt—/ 3 y ) . 
(IS), 
v=u Q sin fee j — cos (nt—\ 7r)+ sinh ky cos cos i. nt ~i 7r ~~l^y) | (19)- 
This is the solution to a first approximation. At a very small distance from the 
bottom the terms in e~^ become insehsible. 
Although the values of u and v in (18) and (19) are strictly periodic, it is proper 
to notice that the same property does not attach to the motions thereby defined of 
the pai’ticles of the fluid. In our notation u is not the velocity of any particular 
particle of the fluid, but of the particle, whichever it may be, that at the moment 
under consideration occupies the point x, y. If y+q be the actual position at 
time t of the particle whose mean position during several vibrations is x, y, then the 
real velocities of the particle at time t are not u, v, but 
du j. du t dv j. . dv 
v +lJ + l^ ”+& f+ V' ; 
and thus the mean velocity parallel to x is not necessarily zero, but is equal to the 
mean value of 
in which again 
du t du 
dx dy 
From the general form of u, viz., 
cos kx F (y, t), it follows readily that 
= 0 . 
For the second term we must calculate from the actual values as given in (18), (19). 
Thus 
