428 



of 

 condition at first glance appears to limit the solutions'^ (2) to cases 



for which the supplementary pressure p vanishes on the cylinder ends. 

 This limitation is actually not implied if only the terms for the axial 

 mode, 'A = 1, are retained in Equation (2), since in this approximation 

 the equations of motion and the orthogonality of the trigonometric func- 

 tions show that there is no interaction between the terms of Equations (2) 

 and terms in u of the form sin (if/^z/L), excited by supplementary pressure 

 on the cylinder ends. It is understood that terms for n = 1 are to be 

 omitted from the suras (2), since they correspond to viniform translation 

 perpendicular to the axis without deformation. 



Substitution of the series (2) into Equations (l) and use of 

 the orthogonality of the trigonometric functions yields for each Fourier 

 component a set of Equations of the form, 



LL t S**^ ft^Jt) Avf**) 



w. 



'? 



(3) 



a 



and 



The matrix elements A ? ' and C ^" '' are given in Appendix A. The solu- 

 tions of Equation (3) appropriate to vanishing values of the displacements 



