THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
93 
In this form the interpretation is very simple, namely, that at the closed end the shape 
is of no consequence, and only the volume need be attended to. The air in this part of 
the pipe acts merely as a spring, its inertia not coming into play. A few measurements 
of this kind will be given in Part III. 
The overtones of resonators which have not long necks are usually very high. Within 
the body of the reservoir a nodal surface must be formed, and the air on the further 
side vibrates as if it was contained in a completely closed vessel. We may form an idea 
of the character of these vibrations from the case of a sphere, which may be easily 
worked out from the equations given by Professor Stokes in his paper “ On the Com- 
munication of Motion from a vibrating Sphere to a Gas”*. The most important vibra- 
tion within a sphere is that which is expressed by the term of the first order in Laplace’s 
series, and consists of a swaying of the air from side to side like that which takes place 
in a doubly closed pipe. I find that for this vibration 
radius : wave-length = *3313, 
so that the note is higher than that belonging to a doubly closed (or open) pipe of the 
length of the diameter of the sphere by about a musical fourth. We might realize this 
vibration experimentally by attaching to the sphere a neck of such length that it would 
by itself, when closed at one end, have the same resonant note as the sphere. 
Lateral Openings. 
In most wind instruments the gradations of pitch are attained by means of lateral 
openings, which may be closed at pleasure by the fingers or otherwise. The common 
crude theory supposes that a hole in the side of, say, a flute establishes so complete a 
communication between the interior and the surrounding atmosphere, that a loop or 
point of no condensation is produced immediately under it. It has long been known that 
this theory is inadequate, for it stands on the same level as the first approximation to 
the motion in an open pipe in which the inertia of the air outside the mouth is virtu- 
ally neglected. Without going at length into this question, I will merely indicate how 
an improvement in the treatment of it may be made. 
Let \J/„ \|q denote the velocity-potentials of the systems of plane waves on the two 
sides of the aperture, which we may suppose to be situated at the point ^=0. Then 
with our previous notation the conditions evidently are that when #=0, 
(20 c) 
the escape of energy from the tube being neglected. These equations determine the con- 
nexion between the two systems of waves in any case that may arise, and the working 
out is simple. The results are of no particular interest, unless it be for a comparison with 
experimental measurements, which, so Tar as I am aware, have not hitherto been made.] 
* Professor Stokes informs me that he had himself done this at the request of the Astronomer Royal. 
MDCCCLXXI. 
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