556 
-8- 
Appendix. The propagation of a pulse of flexural waves in a cylindrical bar. 
If the forces applied to the pressure end of a bar are unsymmetrical with respect to the 
axis of the bar, the forces are equivalent to a longitudina force along the axis, together witha 
couple; the longitudinal force gives rise to a stress pulse composed of longitudinal or extensional 
waves and the couple to a pulse consisting of flexurat or transverse waves. Since a cylindrical 
condenser unit can respond to flexural waves, it is important to find whether an oscillogram, such 
as that shown in Figure 2, is affected by this type of wave, and this appendix deals with the 
propagation of a flexural wave pulse from the standpoint of the method of stationary phase, following 
the lines of the discussion for extensional wave putses. 
The differential equations, which have been used for describing flexural vibrations in a bar, 
fall into two classes, - the exact equations, due to Pochhammer and to Chree, derived from the general 
equations of the theory of elasticity, and the simpler less exact equations, derived in a more 
elementary manner from a consideration of the stresses im the bar. The former have been summarised 
by Love", and-calculations, based on these equations giving the phase velocity of flexural waves for 
different wavelengths, have been published recently by Hudson ¢. 
The simpler equations, and the assumptions on which they are based have been sunmarised by 
Timoshenko %; more recently, a very thorough discussion of the problem has been given by Prescott x. 
Considering a cylindrical bar of radius a, let the axis of the bar (assumed straight when in 
equilibrium) be taken as the axis of x, and let the direction of’ the displacement of the bar in flexural 
or transverse vibration be taken as the axis of y. tt will be assumed that the bar is uniform and 
that the displacements are small. 
Let uy = flexural displacement at time t at a cross-section of abscissa, x. 
p = density of the material of the bar. 
—E = Young's modulus of the materéal of the bar, 
4 = modulus of rigidity of the material of the bar. 
@ = Poisson's ratio of the material of the bar. 
c, =f/= = velocity of extensional waves of infinite wavelength in the bar. 
i) 
c, C, = phase and group velocities respectively of waves of wavelength A 
in the bar. 
kK +25 radius of gyration of the cross=section of the bar about an axis 
through the centre of gravity, perpendicular to the xy ptane. 
R = a nonedimensional constant, dépending on the shape of the cross~ 
section of the bar. (For a circular cross-section, R= 10/9).« 
€ = REfu= 2R (1 +0) 
A, T = wavelength and period of a flexural wave. 
Y = ccvvce 
* A.E.H. Love "Mathematical Theory of Elasticity", art. 201 (uth Ed.) Cambridge (1934). 
@ G.£, Hudson, Phys. Rev., Vol. 63, p.46, (1943). 
*  S. Timoshenko, “Vibration Probiems in Engineering", Arts. 40-43 New York, (1929). 
me Je Prescott, Phil. Mag., Vol. 33, p.703, (1942). 
