348 Longitudinal Vibrations in Solid and Hollow Cylinders. 



in my earlier papers for solid cylinders ; the r 2 term repre- 

 senting one of the additions I deem it necessary to make to 

 Lord Ray lei gh's assumed type of vibration. 



The paraboloidal form of cross section, met with except at 

 nodes or when cos kt vanishes, seems to me an interesting 

 feature of the longitudinal type of vibrations. Possibly, 

 observations on light reflected from a polished terminal face 

 might lead a skilled experimentalist to interesting conclusions 

 as to the value of n. 



It should, however, be borne in mind that, inasmuch as the 



terminal condition zr = is not exactly satisfied by the above 

 solution in fixed-free vibrations, there may be a slight departure 

 from the theoretical form in the immediate neighbourhood of 

 a free end. 



§15. The result (43) is true irrespective of the relative 

 magnitudes of b and a. If h/a be very small, the correctional 

 term is the same as for a solid cylinder of the same external 

 radius. If, on the other hand, b/a be very nearly unity, or 

 the cylinder take the form of a thin- walled tube, we have 



Jc=p(E/p)Hl-hP*vW) (46) 



The correctional term is here twice as great as in a solid 

 cvlinder of radius a. 



* § 16. An experimental investigation into the influence of 

 the shape and dimensions of the cross-section on the frequency 

 of longitudinal vibrations is certainly desirable. In com- 

 paring the results of such an investigation with the theoretical 

 results here determined, several considerations must, however, 

 be borne in mind. 



Statical and dynamical elastic moduli are to some extent 

 different, so that the value of E occurring in (2) or (3) is not 

 that directly measured by statical experiments*. In other 

 words the difference between the observed frequency of the 

 fundamental vibration in a fixed-free bar, and the frequency 

 calculated from the ordinary formula 



if E be determined directly by statical experiments, is not to 

 be wholly attributed to the defect of the ordinary first 

 approximation formula. Again, it must be remembered that 

 E varies t, often to a very considerable extent, in material 

 nominally the same ; so that the difference of pitch observed 



* See Lord Kelvin's Encyclopaedia article on Elasticity, § 75, or 

 Todhunter and Pearson's ' History,' vol. iii. art. 1751. 



t For the effects of possible variation in the material throughout the 

 bar, see the Phil. Mag. for Feb. 1886, pp. 81-100. 



