Vibration of an Elastic Spherical Shell. 191 



Then we have approximately 



y = '06, pc 2 /{\ + 2,/jL) = '2xlQ- 5 and i/ = l'32. 



The following table * shows approximately how the period 

 changes with e : — 



(thickness) 

 (radius) 



,_ pc 2 



r (positive 



root of period 



equation). 



p+iq (com- 

 plex root). 



Periodic time 

 =2iryalqc. 



(X+2/*)6- 



2X10" 3 



•001 



•06 



•008+1-5* 



■25 x a/c 



2X10" 4 



•01 



•06 



•08 ±1'5 i 



•25 X a/c 



2X10"° 



•1 



•06 



•83 +1-2* 



•31 X a/c 



4xl0" 6 



•5 



81 



•1.6, -1 





2xl0" 6 



1 



16-5 



•06 +-06i 



6-28 X a/c 



2xl0" 7 



10 



166-7 



•01 ±-02« 



18-8 xa/c 



Periodic time for 



a shell vibrating 



in vacuo =2tt- 



y{v(3- v )}-it 



t/c—"25xa/c 



In this table there is a value of e for which all the roots 

 -are real. Solving the equation D(a 1 ) = Q, in the present 

 case we find the two positive roots to be 3*0 and 1'0, the 

 negative root being —1*0. Putting l(ry + n / /ry) = l, 3 in 

 succession, we find the critical values of e to be *37 X 10~ 5 

 -and "1 x 10" 4 nearly. 



7. 



The period of free radial vibration of a thin spherical shell 

 in vacuo is known to be independent of its thickness. The 

 preceding analysis suggests that the ratio of the thickness of 

 the shell to the radius has an important part to play in the 

 determination of the frequency when the vibration is com- 

 municated to a surrounding atmosphere. The shell now 

 executes damped harmonic vibration attended with a dissi- 

 pative motion given by an exponential term (<? -c?Y Y a ). The 

 recovery of the shell from the strain corresponding to this 

 part of the displacement becomes quicker as the thinness 

 increases, and below the critical thickness the decay of this 

 part of motion is almost immediate. The period of vibration 

 will depend on the ratio of the thickness of the shell io its 

 radius, and may differ considerably from its free period in 

 vacuo. It should be observed that unless the shell be so big 



* The table is only intended to show in a general way how the various 

 quantities vary with thickness. The data at the disposal of the writer 

 do not permit computations correct up to the second place o\' decimals. 



