498 Lard Rayleigh on the Influence of Obstacles 



curnstances the flow of gas round the obstacles follows the 

 same law as that of electricity, and the kinetic energy of the 

 motion is at once given by the expressions already obtained. 

 In fact the kinetic energy corresponding to a given total flow 

 is increased by the obstacles in the same proportion as the 

 electrical resistances of the original problem, so that the 

 influence of the obstacles is taken into account if we suppose 

 that the density of the gas is increased in the above ratio 

 of resistances. In the case of cylinders in square order 

 (35), the ratio is approximately (1+j>)/(1— p), and in the 

 case of spheres in cubic order bv (68) it is approximately 



But this is not the only effect of the obstacles which we 

 must take into account in considering the velocity of pro- 

 pagation. The potential energy also undergoes a change. 

 The space available for compression or rarefaction is now 

 (1— p) only instead of 1 ; and in this proportion is increased 

 the potential energy corresponding to a given accumulation 

 of gas*. For cylindrical obstruction the square of the velocity 

 of propagation is thus altered in the ratio 



so that if fjL denote the refractive index, referred to that of 

 the unobstructed medium as unity, we find 



or 



2 -l)//; = constant, (70) 



which shows that a medium thus constituted would follow 

 Newton's law as to the relation between refraction and 

 density of obstructing matter. The same law (70) obtains 

 also in the case of spherical obstacles ; but reckoned abso- 

 lutely the effect of spheres is only that of cylinders of halved 

 density. It must be remembered, however, that while the 

 velocity in the last case is the same in all directions, in the 

 case of cylinders it is otherwise. For waves propagated 

 parallel to the cylinders the velocity is uninfluenced by their 

 presence. The medium containing the cylinders has there- 

 fore some of the properties which we are accustomed to 

 associate with double refraction, although here the refraction 

 is necessarily single. To this point we shall presently return, 

 but in the meantime it may be well to apply the formulas to 

 the more general case where the obstacles have the pro- 

 perties of fluid, with finite density and compressibility. 



* < Theory of Sound,' § 303. 



