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CX II. The Problem of tlte Wliisperirtg Gallery. 

 By Lord Rayleigh, O.M., F.R.S.* 



HPHE phenomena of the whispering gallery, of which 

 JL there is a good and accessible example in St. Paul's 

 cathedral,- indicate that sonorous vibrations have a tendency 

 to cling to a concave su'rface. They may be reproduced 

 upon a moderate scale by the use of sounds of very high 

 pitch (wave-length = 2 cm'.), such as are excited by a bird- 

 call, the percipient being a high pressure sensitive flame f. 

 Especially remarkable is the narrowness of the obstacle, held 

 close to the concave surface, which is competent to intercept 

 most of the effect. 



The explanation is not difficult to understand in a general 

 way, and in ' Theory of Sound,' § 287, I have given a cal- 

 culation based upon the methods employed in geometrical 

 optics. I have often wished to illustrate the matter further 

 on distinctively wave principles, but only recently have re- 

 cognized that most of what I sought lay as it were under my 

 nose. The mathematical solution in question is well known 

 and very simple in form, although the reduction to numbers, 

 iri the special circumstances,' presents certain difficulties. 



Consider the expression in plane polar coordinates (r } 6) 



yjr n = J n (kr) cos (kat — n0), . . . (1) 



applicable to sound in two dimensions, -ty denoting velocity- 

 potential ; or again to the transverse vibrations of a stretched 

 membrane, in which case yjr represents the displacement at 

 any point J. Here a denotes the velocity of propagation, 

 k = 2irj\, where \ is the wave-length of straight waves of 

 the given frequency, n is any integer, and J n is the BessePs 

 function usually so denoted. The waves travel circtim- 

 ferentially, everything being reproduced when and t 

 receive suitable proportional increments, For the present 

 purpose we suppose that there are a large number of waves 

 round the circumference, so that n is great. 



As a function of r, yjr is proportional to J n (Jcr), When 

 z is great enough, J n (z), as we know, becomes oscillatory 

 and admits of an infinite number of roots. In the case of 

 the membrane held at the boundary any one of these roots 

 might be taken as the value of &R, where R is the radius of 

 the boundary. But for our purpose we suppose that /cR is 



* Communicated bv the Author. 

 t Proc. Roy. Inst. Jan. 15,- 1904. 

 % ' Theory of Sound/ §§ 201, 339, 



Phil Mag. S. 6. Vol. 20. No. 120, Dec. 1910. 3 U 



