38 Mr. R. H. M. Bosanqiiet on the Theory of Sound. 



IT 



centre of ttH^ ; then, if 8^=2779'"^, r^=. — y^. The reflected wave 



of a cylinder. (Helmholtz makes the correction =; 2; R; 



where R is the radius of the open end. Lord Rayleigh has 

 shown that this is a lower limit, and regards the real value as 

 about '82 R.) Assuming that the divergence depends chiefly 

 on the surface values, represent the circular end of the cylinder 

 ttR^ by Sq, a hemisphere of equal surface, described about the 



R 



v/2' 



reenters this hemisphere as if reflected from a distance = r^ 

 beyond its surface ; so that the correction to the length of the 



R TT 



pipe would be — -^ == '707 R, as aorainst the value -7 R and 



•82 R above mentioned. 



The transformation from the circle to the hemisphere of 

 equal area is only admissible as an approximation ; but for 

 this purpose it is valuable, as the whole reasoning admits of 

 being stated very shortly. 



The problem of spherical divergence is interesting as illus- 

 trating the precision of ideas attaching to the present method, 

 compared with that based on electrical analogies ; for, so far 

 as I am aware, it is not possible to obtain, or even to imagine, 

 any strictly corresponding result in the electrical theory. 

 Consider the case of the divergence of electricity from a sphe- 

 rical surface through a conducting medium indefinitely ex- 

 tended. Then, according to the analogy formerly applied, 

 the resistance to the diverging current furnishes the quantity 

 analogous to the retardation in the sound-problem. This re- 

 sistance is easily found, from the consideration that it is pro- 

 portional to the thickness, and inversely as the surface, of each 

 successive spherical layer. And 



f 



dr 1 



1 



47^''-^ 47r? 



'0 



Similarly for a hemisphere, total external resistance 



2^r; 



This result has been interpreted by supposing a cylindrical 

 tube, of radius r^, to be capped with a hemisphere, from which 

 the divergence takes place. If I be the length of the tube 

 equivalent to the resistance of divergence, a the section of the 

 tube, 



. . 11 



resistance = - = -^ — ? 



