Vibration of an Elastic Spherical Shell. 



187 



ratio (velocity of wave in the medium)/(velocity of wave of 

 dilatation), 7, then h=ky. Now let us assume the solutions 



u= \j 



d>= - e 

 r 



d sin hr 



dhr 



k(ct-r-\-b) 



*i = D. 



sin h 



hr 



-Jkct 



a fir 



I cos hr 



3 Jirl 

 \r~\ 



ikct 



S-. 



(7) 



Substituting in the equations, 

 relations : 



we 



get 



ike j A 

 ike |A- 

 0. + 2/*)[. 



<iAa 

 ihb 



sin Aa 



ha 

 sin A6 



B 



6? cos Aa 



hb 



+ B 



AA 



d 2 



alha 

 sin A a 



2X 



a (_ 



d(ha) 2 ' ha 

 r A rf sin Aa 



ha 



cos A6 



~AF 



+ BA 



]-D 



the following 

 sin &a 1 



a 12 



dka 

 ik\ 



+ V 



cos Aa" 



&a 



(X + 



!/*) f~AA 



dha ' ha dha ' 



d 2 sin hb 



] 



-~ a cos na~\ . 7 T . 



d{]ia) 2 ' Aa 

 d cos Aa" 



rf(A6; s 



2\ r 4 a* sin A6 



A6 

 + B 



+ BA 



ha 

 d 2 



sin #a 

 ka 



K») 



cos AZ> 



] 



a\A6) 2 * A6 

 cos hb~\ ike 



dhb' hb 



■]- 



P T C. 



Eliminating A, B, C, D from these four equations, we get 

 the frequency equation in the form of a determinant of the 

 fourth order. 



4. 



We shall confine our attention to the case of a thin 

 spherical shell, the interior of which is unoccupied by any 

 gaseous matter. If the shell be thin we can write a + da for 

 b, and putting e for the ratio da/a, which is necessarily small, 

 w r e have the following modifications of the conditions (8). 



The first relation is absent ; the third is unaltered except 

 that the right-hand side vanishes, and in virtue of this the 

 fourth relation becomes 



A[(X + V)/^ 



Uha) z 



n ha 



a 



T,r h lk) m, (Z 3 cos A 



d(h 



U( , 



_ d /l d sin ha\~\ , 



2X- r ( --77-. , da 



da\a dha ha /J. 



ha d /l d^ cos A(A ~j 



a ~' da\a dha ' ha /J 



= —ikcp{l—da/a)Q a. 



hr 



