1915] THE THEORY OF THE HELMHOLTZ RESONATOR
so that (54) becomes
375
V27rc
whence
= 7T -\/3
.(60)
k "V \2irc,
This gives the reduction of amplitude after one vibration. The decay is least when R is small relatively to c, although it is then estimated for a longer time.
The value found in (60) differs a little from that given in Theory of Sound, § 311, where the aperture is supposed to be surrounded by an infinite flange, the effect of which is to favour the propagation of energy away from the resonator.
So far we have supposed the boundary of the aperture to be circular. A comparison with the corresponding process in Theory of Sound, § 306 (after Helmholtz), shows that to the degree of approximation here attained the results may be extended to an elliptic aperture provided we replace R by
where J?a denotes the semi-axis major of the ellipse, e the eccentricity, and F the symbol of the complete elliptic function of the first order. It is there further shown that for any form of aperture not too elongated, the truth is approximately represented if we take <\7(<r/7r) instead of the radius R of the circle, where a- denotes the area of aperture.
It would be of interest to ascertain the electric capacity of a disc of nearly circular outline to the next approximation involving the square of BR, the deviation of the radius in direction w from the mean value. If BR = an cos nw, a: would not appear, and the effect of a2 is known from the solution for the ellipse. For other values of n further investigation is required.
In the case of the ellipse elongated apertures are not excluded, provided of course that the longer diameter is small enough in comparison with the diameter of the sphere. When e is nearly equal to unity,
- log
_1 g
.(62)
R2 being the semi-axis minor. The pitch of the resonator is now comparatively independent of the small diameter of the ellipse, the large diameter being given.. ^ #-t-&cos</> ( Q.