The period of a spherical resonator with a circular aperture

In a paper on “The Theory of the Helmholtz Resonator,” Lord Rayleigh has carried the determination of the wave-length of the fundamental aërial vibration in a spherical vessel with a small circular perforation to a second approximation, obtaining the result λ = π√(2S/a ) . {1-9/10a /2πc +...}, (1) wherec is the radius of the sphere,a the small radius of the aperture, and S the volume of the sphere. To obtain this value he assumes a form for the normal velocity U over the aperture and adjusts it so as to lead to agreement, to a corresponding approximation, in the values of the velocity potential derived therefrom over the aperture inside and outside, so as to provide for the condition of continuity of pressure at the opening. It is the object of this note to carry the result, by a slightly different assumption for U, to a higher approximation.