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NOTE ON THE THEORY OF THE DOUBLE RESONATOR.
[Philosophical Magazine, Vol. xxxvi. pp. 231—234, 1918.]
IN my book on the Theory of Sound* I have considered the case of a double resonator (Fig. 1), where two reservoirs of volumes S, S' communicate
with each other and with the external atmosphere by narrow passages or necks. If we were to treat SS' as a single reservoir and apply the usual formula, we should be led to an erroneous result ; for that formula is founded on the assumption that within the reservoir the inertia of the air may be left out of account, whereas it is evident that the energy of the motion through the connecting passage may be as great as through the two others. However, an investigation on the same general plan meets the case perfectly. Denoting by X1} X2, Xs the total transfers of fluid through the three passages, we have for the kinetic energy the expression
and for the potential energy
Here p denotes the density of the fluid, a the velocity of sound, while d, c2, c3 may be interpreted as the electrical conductivities of the passages. Thus for
* § 310, first edition 1878, second edition 1896, Macmillan. Also Phil. Trans. 1870 ; Scientific Papers, Vol. i. p. 41.