94 
THE HON. J. W. STRUTT ON THE THEORY OE RESONANCE. 
PART II. 
In order to complete the theory of resonators, it is necessary to determine the value 
of c, which occurs in all the results of Part I., for different forms of months. This we 
now proceed to do. Frequent use will be made of a principle which might be called 
that of minimum vis viva , and which it may be well to state clearly at the outset. 
Imagine a portion of incompressible fluid at rest within a closed surface to be sud- 
denly set in motion by an arbitrary normal velocity impressed on the surface, then the 
actual motion assumed by the fluid will have less vis viva than any other motion con- 
sistent with continuity and with the boundary conditions*. 
If u, v , w be the component velocities, and § the density at any point, 
the integration extending over the volume considered. The minimum vis viva corre- 
sponding to prescribed boundary conditions depends of course on § ; but if in any speci- 
fied case we conceive the value of § in some places diminished and nowhere increased, 
we may assert that the minimum vis viva is less than before ; for there will be a decrease if 
u, v, w remain unaltered, and therefore, a fortiori, when they have their actual values as 
determined by the minimum property. Conversely, an increase in will necessarily 
raise the value of the minimum vis viva. The introduction of a rigid obstacle into a 
stream will always cause an increase of vis viva ; for the new motion is one that might 
have existed before consistently with continuity, the fluid displaced by the obstacle re- 
maining at rest. Any kind of obstruction in the air-passages of a musical instrument 
will therefore be accompanied by a fall of the note in the musical scale. 
Long Tubes. 
The simplest case that can be considered consists of an opening in the form of a cylin- 
drical tube, so long in proportion to its diameter that the corrections for the ends may 
be neglected. If the length be L and area of section <r, the electrical resistance is 
L 
T 
and 
For a circular cylinder of radius E 
( 21 ) 
( 22 ) 
Simple Apertures. 
The next in order of simplicity is probably the case treated by Helmholtz, where the 
opening consists of a simple hole in the side of the reservoir, considered as indefinitely 
thin and approximately plane in the neighbourhood of the opening. The motion of the 
* Thomson and T ait’s ‘Natural Philosophy,’ p. 230. 
