The theory of the Helmholtz resonator

Abstract
The ideal form of Helmholtz resonator is a cavernous space, almost enclosed by a thin, immovable wall, in which there is a small perforation establishing a communication between the interior and exterior gas. An approximate theory, based upon the supposition that the perforations is small, and consequently that the wave-length of the aërial vibration is great, is due to Helmholtz who arrived at definite results for perforations whose outline is circular or elliptic. A simplified, and in some respects generalised, treatment was given in my paper on "Resonance." In the extreme case of a wave-length sufficiently great, the kinetic energy of the vibration is that of the gas near the mouth as it moves in and out, much as an incompressible fluid might do, and the potential energy is that of the almost uniform compressions and rarefactions of the gas in the interior. The latter is a question morels of the volume S of the cavity and of the quantity of gas which has passed, but the calculation of the kinetic energy presents difficulties which have been only partially overcome. In the case of simple apertures in the thin wall (regarded as plane), only circular and elliptic forms admit of complete treatment. The mathematical problem is the same as that of finding the electrostatic capacity of a thin conducting plate having the form of the aperture, and supposed to be situated in the open. The project of a stricter treatment of the problem, in the case of a spherical wall and an aperture of circular outline, has been in my mind more than 40 years, partly with the hope of reaching a closer approximation, and partly because some mathematicians have found the former method unsatisfactory, or, at any rate, difficult to follow. The present paper is on ordinary lines, using tire appropriate spherical (Legendre's) functions, much as in a former one, " On the Acoustic shadow of a Sphere.

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