1908 - 9 .] 
Flexural Vibrations of Thin Rods. 
393 
XXII. — Flexural Vibrations of Thin Rods. By George Green, M.A., 
B.Sc., Assistant to the Professor of Natural Philosophy in the 
University of Glasgow. Communicated by Professor Gray. 
(MS. received December ]4, 1908. Read January 18, 1909.) 
§ 1. The main object of this paper is to point out a method of applying 
hydrodynamic solutions already obtained to the solution of problems 
relating to flexural vibrations of thin elastic rods. 
The results found apply to rods not subjected to permanent tension, and 
vibrating so that one principal axis of each transverse section lies in the 
plane of vibration. The central line of the rod is assumed to remain 
unaltered in length ; and particles lying in a plane transverse section, when 
undisturbed, remain always in a plane normal to the central line. For 
convenience in what follows we may here derive the equation of motion of 
a rod subject to these conditions. Taking w as the area of each section, 
k 2 co its moment of inertia, q as Young’s modulus, dx as the length of the 
central line of a small portion, R its radius of curvature, and N as the total 
tangential force acting on the cross-section at x, we obtain the equation of 
angular motion of the element 
2 3 1 i \T 2 
(/K“(jl > — — - + Al = pK A (Ji> 
dx R dxdt' 2 
(i), 
where y is the vertical displacement of the element and p is its density. 
The equation of vertical motion is 
0X 0 2 // 
dx P M dt 2 
As pointed out by Lord Rayleigh,* whose notation we have adopted, terms 
depending on the angular motions of the sections of the bar may be 
neglected in the above equations ; accordingly, by eliminating N from 
(1) and (2) we obtain finally the equation 
a2 // , 272 di y 0 
W + Kh d^ = 0 
( 3 ), 
in which we have put for f and b 2 for 
dx 2 R p 
2. Consider now frictionless liquid in a straight canal with vertical 
* Theory of Sound , vol. i. 
