BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 265 



obstacles the loss of energy to the primary waves is proportional to the surface of the 

 obstructing sphere. There is, it will be noticed, a distinct similarity between these . 

 results and those obtained above in the case of cylindrical obstacles. 



The expression (7) has been evaluated in a number of different cases, and the result^ 

 are arranged on p. 266 in tabular form. K denotes the expression (7) or the ratio of 

 the lost energy to that incident upon the obstacle, and A. represents the wave-length 

 (measured in centimetres) of the incident sound. 



9. Application of the above to the Problem of a Large Number of Spherical 

 Obstacles. Let us consider now the loss of energy to the primary waves when these 

 are incident upon a large number of spherical obstacles. We shall suppose that there 

 are n small spheres per c.cm. ; the validity of our argument will depend on the 

 volume occupied by the obstacles being small compared with the total volume. 

 Consequently f nira 3 must be a small fraction. 



At a distance r from the centre of any one of these spherical particles, great 

 compared with the wave-length of the incident sound, the secondary waves due to 

 that particle will be sensibly irrotational, and will be given very approximately in all 

 cases by 



*, = A,,/;, (/ 



Since hr is great, we may write 





and fa takes the form 



fa = (Ao/ 



which, along the course of the primary waves (/A = 1), reduces to 



(1). 



Consider now the spheres whicli occupy a thin stratum d.c perpendicular to the 

 course of the primary waves. Let P be any point in this stratum, and let O be the 

 point where the vibration is to be estimated at a great distance from the stratum.* 



If AP = z, the element of volume is 2irx.dx.dz, and consequently the number 

 of spherical particles in it is 2-rrnz.dx.dz. Also, if OP = r, AO = x, then 

 r 2 = x? + z 2 and rdr = zdz. 



Now by (1) the resultant at O of all the secondary vibrations which issue from the 

 stratum is given by 



2irn dx [" (Ao/T 1 - A^-'i) e~' hr dr. 



J -i 



Remembering that the angle AOP is to be regarded as very small, we see that the 



* See figure, p. 254. 

 VOL. CCX. A. 2 M 



