488 WAVES OF EXPANSION. [CHAP. X 



In this, the gravest of all the normal modes, the air sways to and fro much 

 in the same manner as in a doubly-closed pipe. In the case of any one of 

 the higher roots, the roots of lower order give the positions of the spherical 

 nodes (d<l>/dr=0). For the further discussion of the problem we must refer to 

 the investigation by Lord Rayleigh*. 



2. To find the motion of the enclosed air due to a prescribed normal 

 motion of the boundary, say 



we have, <t>=Aty n (kr)r n S n .e i<rt .............................. (ix), 



with the condition A (katyn (ka) + ^n (ka)}a n ~ 1 = I , 



and therefore <fr = z , //t^ i n \- a (-V&- &* .......... ( x )' 



katyn(ka)+nty n (kd) \aj 



This expression becomes infinite, as we should expect, whenever Tea is a root 

 of (ii) ; i.e. whenever the period of the imposed vibration coincides with that 

 of one of the natural periods, of the same spherical-harmonic type. 



By putting a=0 we pass to the case of an incompressible fluid. The 

 formula (x) then reduces to 



as in Art. 90. It is important to notice that the same result holds approximately, 

 even in the case of a gas, whenever ka is small, i.e. whenever the wave-length 

 (27T/&) corresponding to the actual period is large compared with the circum- 

 ference of the sphere. This is otherwise evident from the mere form of the 

 fundamental equation, Art. 266 (1), since as k diminishes the equation tends 

 more and more to the form v 2 </>=0 appropriate to an incompressible fluid t. 



* " On the Vibrations of a Gas contained within a Rigid Spherical Envelope," 

 Proc. Lond. Math. Soc., t. iv., p. 93 (1872) ; Theory of Sound, Art. 331. 



t In the transverse oscillations of the air contained in a cylindrica vessel we 



have 



where Vi 2 =^ 2 /^ 2 + ^ 2 /^ 2 - In the case of a circular section, transforming to polar 

 coordinates r, 0, we have, for the free oscillations, 



with k determined by 



//(fe)*0, 



a being the radius. The nature of the results will be understood from Art. 187, 

 where the mathematical problem is identical. The figures on pp. 308, 309 shew the 

 forms of the lines of equal pressure, to which the motion of the particle is ortho- 

 gonal, in two of the more important modes. The problem is fully discussed in 

 Lord Rayleigh's Theory of Sound, Art. 339. 



