557 
-9- 
y= : w = ZT = yo =_2. 
KN T A 
The simplest theory of the flexural vibrations of a bar assumes that the displacement of an 
élement of the bar consists solely of translation parallel to oy; the differential equation may be 
written in the form 
4 2 
«2 8 Uy. + 3 We SEO. ce ween ety acee (A. 1) 
Ox ate 
For sinusoidal waves of unit amplitude, uy will be of the form 
2 
& 
use i (yx + wt) acp ASO .ode "ooo 900. 6086 (As 2) 
and, substituting in equation (A-1), it is easy to show that the phase velocity ¢ of sinusoidal 
flexural waves of wave-iength A. is given by the equation 
¢ = sye-K, = To % Noche ed creMiarilemel ccivabd cies ects (A.3) 
The corresponding value of the group velocity, Cg given by the equation 
¢ 
9 c a d (c/c,) 
F  aliatapai der ailliemenctsart Bote Dod | Book. eG (A.4), 
0 cy n d (ala) 
is here 
Cy = 2 = 27 CyalA BBO. SoM) Abo, 86a loud (A.5) 
This theory leads to a result which is physically absurd, namely tiiat a wave-packet consisting 
of waves of infinitely short wave-lengths will be propagated with an infinite velocity; it atso 
follows that a wave-packet consisting of waves of infinitely long wave-lengths will be propagated 
with an infinitesimally smal) velocity. 
When the elements of the oar are considered to undergo rotation (without distortion) in 
addition to lateral displacement, the differential equation (A.1) is modified to the form (due 
apparently to. Rayleigh) ° 
4 4 2 
2b ee eEu ou Ofu 
FS I NE A ir CS ee (A.6) 
x Bx2at 2 at2 
The effect of the rotation of the elements of the bar is represented by the term 
Ox 23 2 
Proceeding as before, it may be shown that the phase and group velocities, derived from 
this equation, are given by 
ic c 1 
° ° 
n Ea ciape ener tee Ae7 
c= H Coie ee a ( ) 
A AS 1+ 02 N2 
Ta* area? 
When = is small, the values of c and c_ given by this equation become equal to those given 
in equation (A, 6); where 5 is large, both c and Cg approach the value cy asymptotically, so that 
A WAVE seeee. 
cord Rayleigh, ("Theory of Sound" Vol. 1, p.294, London (1894). 
