268-270] VIBRATIONS OF AIR IN SPHERICAL CAVITY. 489 



269. To determine the motion of a gas within a space 

 bounded by two concentric spheres, we require the complete 

 formula (9) of Art. 267. The only interesting case, however, is 

 where the two radii are nearly equal ; and this can be solved more 

 easily by an independent process*. 



In terms of polar coordinates r, 6, o>, the equation (v 2 + & 2 ) = becomes 



dr 2 r dr r 2 \_dfj, \ dp) 1 pPdca^J 



If, now, d(f)/dr=0 for r=a and r=b, where a and b are nearly equal, we may 

 neglect the radial motion altogether, so that the equation reduces to 



It appears, exactly as in Art. 191, that the only solutions which are finite over 

 the whole spherical surface are of the type 



0n (iii), 



where S n is a surface-harmonic of integral order n, and that the corresponding 

 values of k are given by 



In the gravest mode (71 = !), the gas sways to and fro across the equator 

 of the harmonic S lt being, in the extreme phases of the oscillation, condensed 

 at one pole and rarefied at the other. Since ka = *J% in this case, we have 

 for the equivalent wave-length A/2a= 2*221. 



In the next mode (w = 2), the type of the vibration depends on that of 

 the harmonic S z . If this be zonal, the equator is a node. The frequency is 

 determined by ka=j6, or X/2a = 1-283. 



270. We may next consider the propagation of waves outwards 

 from a spherical surface into an unlimited medium. 



If at the surface (r=a) we have a prescribed normal velocity 



f = S n .e iat (i), 



the appropriate solution of (v 2 + 2 )0=0 is 



tOto-)J ~~kr 



for this is included in the general formula (13) of Art. 207, and evidently 

 represents a system of waves travelling outwards f. 



* Lord Rayleigh, Theory of Sound, Art. 333. 



t This problem was solved, in a somewhat different manner, by Stokes, "On 

 the Communication of Vibrations from a Vibrating Body to a surrounding 

 Gas," Phil. Trans., 1868. 



