92 
THE HON. J. W. STEUTT ON THE THEOEY OF EESONANCE. 
the reduced length of the tube, including the corrections for both ends, being denoted 
by L. Thus rarifaction in S 
=& 
A'Q cos kL, sin 2-nnt 1 tty 2vnA! sin A'L 
S 
lim 
d l dt 
sin 2 Tilt. 
This is the condition to be satisfied at the inner end. It gives 
o 2 kQ Q 
tan Jih— 
4 wot 2 S kiS’ 
(18) 
When 7tL is small, 
tan ^L=/(*L-f ^(/i’L) 3 = ; 
5 
n= 
V&HVHV 
Q 
E(>S + £LQ) 
(19) 
In comparing this with (5), it is necessary to introduce the value of c, which is 
Jj 
(5) will accordingly give the same result as (19) if one-third of the contents of the neck 
be included in S. The first overtone, which is often produced by blowing in preference 
to the fundamental note, corresponds approximately to the length L of a tube open at 
both ends, modified to an extent which may be inferred from (18) by the finiteness of S. 
The number of vibrations is given by 
n: 
a 
1 _L Q 
e\ L w 2 S 
( 20 ) 
[The application of (20) is rather limited, because, in order that the condensation 
within S may be uniform as has been supposed, the linear dimension of S must be con- 
siderably less than the quarter wave-length ; while, on the other hand, the method of ap- 
proximation by which (20) is obtained from (18) requires that S should be large in 
comparison with QL. 
A slight modification of (18) is useful in finding the pitch of pipes which are cylin- 
drical through most of their length, but at the closed end expand into a bulb S of no 
great capacity. The only change required is to understand by L the length of the pipe 
down to the place where the enlargement begins with a correction for the outer end. 
Or if L denote the length of the tube simply, we have 
tan lc{h-\-a) = ^r, 
and 11 approximately. 
If S be very small we may derive from (20 a) 
( l+ “ + q) 
(20 a) 
n— 
(20 h) 
