69 
VI. On Spherical Radiation and Vibrations in Conical Pipes. 
By E. H. Barton, JJ.Sc, F.R.S.E., Professor of Experi- 
mental Physics, University College, Nottingham*. 
IT is well known that the vibrations in parallel pipes may 
be treated by plane waves and elementary methods. 
'When, however, a change is made from parallel to conical 
pipes the waves cease to be plane, and the method hitherto 
available is powerless to deal with the phenomena. Yet the 
importance of the subject is at least equal to that of parallel 
pipes, since the brass instruments in musical use are 
conical or quasi-conical, and also the oboe, bassoons, and 
English horn. Thus, apart from a knowledge of spherical 
radiation and its application to such pipes, the student is left 
without a clue to the phenomena occurring. He is accord- 
ingly somewhat at a loss to understand why a conical pipe, 
closed at the vertex and open at the base, should have the 
same pitch and the same complete series of harmonic tones as 
a parallel pipe open at both ends. Whereas, a parallel pipe 
if closed at one end falls in pitch about an octave and loses 
all the evenly-numbered partials. 
The mathematical aspect of the matter is of course treated 
with great generality and elegance in the classical treatises 
(see, for example, Rayleigh's ' Theory of Sound,' vol. ii. 
chaps, xi., xii., & xiv.). But the use in such treatises of the 
velocity potential as the dependent variable, slight as this 
obstacle is, may prove sufficient to prevent some readers from 
assimilating the articles in question. If, however, following 
Kiemannt, we take for the dependent variable the so-called 
condensation, which is a more familiar conception, the analysis 
is somewhat simplified and the whole problem is solved by 
methods within the range of every one familiar with the 
elements of the calculus. It seems desirable, therefore, that 
the physical student should be provided with a treatment 
intermediate between the recondite mathematical treatises on 
the one hand, and the mere statement of the musical facts 
respecting conical pipes on the other hand. This plan, already 
found useful in dealing with such students, is here given in the 
hope it may thus prove of service to others. Another matter 
which seems very puzzling, in the simple statement without 
proof, is the fact that in a conical pipe the anti-nodes remain 
in equidistant positions as for a parallel pipe, but that the 
nodes are all shifted, some considerably and others slightly. 
* Communicated by the Author. 
f Partielle Differ entialgleichung en und deren Anicendung auf physi- 
kalische Fragen. 1882. Sixth part. 
