88 LORD BAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



attracts attention is the comparatively slight deviation from uniformity in the 

 intensities in different directions. Even when the circumference of the sphere 

 amounts to twice the wave-length, there is scarcely anything to be called a sound 

 shadow. But what is, perhaps, still more unexpected is that in the first two cases 

 the intensity behind the sphere [/*= !] exceeds that in a transverse direction 

 [ju, = Oj. This result depends mainly on the preponderance of the term of the first 

 order, which vanishes with /JL. The order of the more important terms increases with 

 kc ; when kc is 2, the principal term is of the second order. 



" Up to a certain point the augmentation of the sphere will increase the total energy 

 emitted, because a simple source emits twice as much energy when close to a rigid 

 plane as when entirely in the open. Within the limits of the table this effect masks 

 the obstruction due to an increasing sphere, so that when /j. = 1, the intensity is 

 greater when the circumference is twice the wave-length than when it is half the 

 wave-length, the source itself remaining constant.'' 



The solution of the problem when kc is very great cannot be obtained by this 

 method, but it is. to be expected that when (U, = 1 the intensity will be quadrupled, 

 as when the sphere becomes a plane, and that when p, is negative the intensity will 

 tend to vanish. It is of interest. to trace somewhat more closely the approach to this 

 state of things to treat, for example, the case of kc = 10.* In every case where it 

 can l)e carried out the solution has a double interest, since in virtue of the law of 

 reciprocity it applies when the source and point of observation are interchanged, thus 

 giving the intensity at a point on the sphere due to a source situated at a great 

 distance. 



But before proceeding to consider a higher value of kc, it will be well to 

 supplement the information already given under the head of kc = 2. The original 

 calculation was limited to the principal values of /A, corresponding to the poles and 

 the equator, under the impression that results for other values of p would show 

 nothing distinctive. The first suggestion to the contrary was from experiment. In 

 observing the shadow of a sphere, by listening through a tube whose open end was 

 presented to the sphere, it was found that the somewhat distant soui-ce was more 

 loudly heard at the anti-pole (//,= ]) than at points 40 or 50 therefrom. This 

 is analogous to POISSON'S experiment, where a bright point is seen in the centre of the 

 shadow of a circular disc an experiment easily imitated acoustically! and it may 

 be generally explained in the same manner. This led to further calculations for 

 values of //, between and 1, giving numbers in harmony with observation. The 

 complete results for this case (kc = 2) are recorded in the annexed table. In 

 obtaining them, terms of LEGENDBE'S series, up to and including P fi , were retained. 

 The angles 6 are those whose cosine is /z. 



* See RAYLEIGH, 'Proc. Roy. Soc.,' vol. 72, p. 40; also MACDONALD, vol. 71, p. 251 ; vol. 72, p. 59 ; 

 PoiNCAiiri, vol. 72, p. 42. 



t 'Phil. Mag.,' vol. 9, p. 278, 1880; 'Scientific Papers,' vol. 1, p, 472. 



