192 Type of Vibration of an Elastic Spherical Shell. 



that its radius is comparable to c, the damping coefficient is 

 very large and there is hardly any vibration. "With regard 

 to the motion of the medium, the energy o£ the shell is 

 communicated through it by means of progressive waves of 

 damped harmonic type and a motion given by an exponential 

 term, which is the counterpart of a similar motion noticed 

 above. Near the shell this part is gradually damped 

 according to exponential law, and at a short distance 

 outwards it becomes insensible, but comes into prominence 

 again near the surface of discontinuity of the advancing: wave. 

 As Love * has observed, "these waves of exponential type 

 accompany the waves of damped harmonic type as they travel 

 outwards, and serve to establish continually the front of the 

 advancing wave." 



It is to be noted that for shells of the same material but 

 of different sizes, if the ratio of the thickness to the radius be 

 constant, the period equation is unaltered, and the periods 

 and the moduli of 'decay of vibration would be proportional 

 to the radii of the shells. An examination of the preceding 

 table shows that the behaviour of a shell undergoes some 

 change as the critical thicknesses are reached. Thus, for a 

 shell of given radius as the thickness decreases, the period of 

 vibration increases, and the damping coefficient at first 

 increases till the upper critical thickness is reached, but 

 below the lower critical limit there is a steady fall in the 

 value of the coefficient. It will also appear from the table 

 that when e is greater than and is considerably removed from 

 its upper critical value, the period of vibration of the shell 

 is almost equal to the free period in vacuo, but as e approaches 

 the upper critical limit, the period becomes greater, and this 

 deviation continually increases in the same direction as e 

 passes the lower critical limit and continues to become 

 smaller. 



8. 



It only remains to be shown that with the three roots of 

 the period equation it is possible to complete the solution 

 of the problem of vibration. Let us write the solutions in 

 the form 



4 r d sin A,* -p d cosh s rl ikct ~ 

 8=1 L dli s r li s r dli 8 r h s r J 



3 n . 



. 2^»gt*,(tf-r+a) 



r i r 

 when (ct + a)>r and is zero when (ct + a)<r. 

 * Loc. cit. p. 112. 



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