Radiation and Vibrations in Conical Pipes. 
79 
Here we consider the slant length R o£ our open cone as a 
constant and N" to vary in accordance with the numbers on 
the right side of the equation. Now, on dividing equation 
(35) by (36) we eliminate the frequency N and obtain the 
required relations between r and R, viz. : — 
e 1 e, or e 2 <9„ e 2 or e z e^ e* e* or e x e u e s , # 3 , e± or e 5 
2 ' 3 
Oi, o» 0* 4 , e 5 or e % 
or 
4 ' 5 
0\9 @2-> #3» ^45 #o> #6 or #7 
&C, 
I 
37) 
It is necessary to cross combine the right sides of (35) and 
(36) in this way to obtain all the values sought. The deno- 
minator of any one of the fractions on the right of (37) 
shows the order of the tone being emitted by the pipe. The 
various values of r/R obtained by taking the various 0's in 
the numerator of that fraction locate the nodes for the tone in 
question. The series of d's in each numerator is finite, being- 
limited by the obvious fact that r/R cannot exceed unity. 
The first few nodal positions are given in Table I. They are 
also exhibited graphically together with the positions of the 
Tajble I. — Nodal Positions of an Open Cone. 
Order m of 
Natural Tone ; 
i. e. denominator 
of fraction on 
right side of 
equation (37). 
Nodal Positions ; 
i. e., values of r/~R=Q/m in equation (37). 
1 







07152 
0-4768 
0-3576 
0-2861 
0-2384 
0-2043 
0-8197 
0-6148 
0-4918 
0-4098 
0-3513 
0-8677 
06942 
0-5785 
04958 
0-8949 
0-7458 
0-6392 
0-9136 
0-7831 
0-9263 
2 
3 
4 
5 
6 
7 
Values of 0's in 
numerators of 
fractions on right 
side of equation (37) 
!• 
e 2 
1-4303 
S3 
2-4590 
04 
3-4709 
4-4747 
o 6 
5-4818 
0r 
6-4844 
