Vibration of an Elastic Spherical Shell. 189 



Putting sc=iy, the equation becomes 



tf-(V + n l ly)tf+$v-~v*)tf--(Zv-v l )y = 0. , (9) 



It should be noticed that when the outside atmosphere is 

 absent, p and consequently n' vanishes, and the period 

 equation reduces to 



a 1 b 2 — a 2 bi = 0, 



or .v 2 -(3v-v 2 )=0, 



which is the period equation of free vibration of a spherical 

 elastic shell *. This period is independent of the thickness. 



5. 



The elastic constants which appear as coefficients of the 

 period equation (9) are all positive, and since there are three 

 changes of sign, one of the roots, at least of this equation, 

 must be real and positive. Also, if the roots be all real, all 

 of them must be positive ; and in case the equation has a 

 pair of imaginary roots, let the roots be r, p + iq andp — iq, 

 where r is positive. Writing the equation in form F(?/) = Q,. 

 it is to be noticed that F(y) is negative if y = and positive 

 if y = (y+n l /y), n' and y being both positive. Hence the 

 positive root r lies between and (y-\-n'/y). Also r + 2p 

 = {y + n'/y) ; and since r< (y + n'/y), p must be positive. As 

 the roots are all positive (if real) or have their real parts 

 positive (in case of a complex pair), the terms corresponding 

 to them in the solution are of the form e~ crt '^ a and e~P ct 'y a * 

 Hence the motion is always damped. 

 If we write the equation in the form 



a oV z ~ 3«i?/ 2 + 2>a 2 y 2 — a 3 — 0, 



where a = l, «i= 3 (7 + ^/7)5 a 2 =^(3v — v 2 ) = 3 mand a B = my > . 

 these quantities being all positive, the discriminant of the 

 cubic is 



D(rti) = 4a 3 . a x 3 — 3a 2 2 • a>i 2 — &a 2 a s . a 1 -f- (a 3 2 + 4a 2 3 ). 



If the roots of the period equation are all real, T)(a-[) 

 should be negative. Now D(a 1 ) = 0, taken as a cubic in a ls 

 has one negative root, and if the other two roots are real 

 they are positive. From the curve of D(aj) it is evident 

 that this function will be negative for all values of a x lying 

 between these two positive roots (if they exist) and for no 

 other positive values of a x ; and if !)(«!) = has a pair of 

 imaginary roots, D(aj) will be negative for no positive value 

 of a x . Hence the original period equation will have all its 



* Love's ' Theory of Elasticity,' 2nd ed. p. 275. 



