TilK THEORY <)!<' Tlltt HKLMHOLT7, HKSONATOK. nwtli'iif/x nftlir liui/dl tiucicti/, A, Vol. xcn. pp. 2(Jf)---;275, !1!M5.'|
TiiK ideal form of 1 lelmholt/, resonator in a cavernous space, ahno.sl. enclosed y a thin, iiiiuiovahle wall, in which then; in a small perforation establishing communication between the interior and exterior gas. An a,pproxiinate icory, based upon the supposition that the, perforation is Hinall, and eon-•(jUently that the wave-length of the aerial vibration is great, is due to leliiiliult/,*, who arrived at definite results for perforatioiiH whose outline 'IK ireular t»r elliptic. A Himplific.d, and in some respects generalised, treatment as given in my [taper on " Resonance f." In the extreme case, of a wave-•ngth Hidlieietitty groat, the kinetic energy of the. vibration in thai, of the gats ear (he month an it moves in and out, much as an ineompreHNihlo llnid Hf^ht do, and the potential energy in that of the alun»Nt uniform coiupreHsionH ml ran-factioriH of the ga,H in the interior. The. latter in a question merely f* the volume *S' of the wvily and of th(« quantity of gas which has pawed, ut the calculalion of the kinetic, energy preHcntH difticultieH which have been uly partially overcome, In the URHO of wimple aporturen in the thin wall "e^ardetl nn plane), only circular and elliptic forma admit of complete treat-lent. The mathematical problem in tho satru; as that of finding tho (iloctro-:.nti(! inifiticiti/ of a thin conducting plato having tho form of tht* aperture, tid HUpptmed to be nituated in the open.
The project of a Htricter treatment of fcho problem, in tho case of a pherira) wall and an aperture of circular outline, has been in my mind more mn 40 yearn, partly with the hope of reaching a closer approximation, and artly bmuiHc Homo mathornaticianH have found tho former method nnnalis-ictory, or, at itny rate, difficult to follow. Tho present paper is on ordinary nen, lining the appropriate spherical (Legendro's) functions, much as in former one, "On tho Acoustic Shadow of a Sphere J."
8 CrelleJourn. Math, Vol. MTII. (1B60).
t Phil. Tram. Vol. GI.XI. p. 77 (1870); Heienttjla Papers, "Vol. i. p. 88. Also Theory of Sound,
J /Vi«. Tram, A, Vol. can. p. 87 (1904) ; Selent^e Papers, Vol. v. p. 141).On account of the magnitude of x we have only the one curvature to deal with. For this curvature