558 
a wave-packet consisting of waves of infinitely short wavelengths will be propagated not with infinite 
velocity but with a velocity equal to the velocity of extensional waves of Infinite wavelength. 
When the shearing, the rotation and the lateral displacements of the elements of the bar are 
taken into account, the differentia) equation takes the form given by Timoshenko 6 and by Prescott 
(loc. cit.) 
atu, tu, au ex2 ay 
oF Pt pat F(t ee) oy + bas al Wale (A.8) 
, x ot ax2at 2 c2 ate 
This equation differs from equation (A.7) by the terms involving th® non-dimensional parameter 
€ as a factor. 
tt can be shown that the phase and group velocities derived from equation (A.8) are given 
by the relationships 
By c2 2 
0 
ome os Met AL "4. € 
ce of 7792 
e, c 1 weet) elie (A.9) 
5 2,2 
sre a cae fh: 1478 (946 -2€ 4 
° ° — ll 
n c? 
° 
For a given value of a/A the first of these equations becomes a quadratic in c4/c2 ; the two roots 
are real, the larger giving a value of ¢ which exceeds cy except in the limiting case where a/A7 ©, 
whilst the smaller gives a value of ¢ which is always less than Coe Tne larger root does not appear 
to have any physical significance under ordinary conditions of experiment (c.f, Prescott, loc. cit.), 
and moreover the exact theory shows that cic, is a single=valued function of a/A; we shall 
therefore ignore this root, and, with this restriction, it follows that the values of cic, and ¢ It, 
given by equation (A.9) reduce to those given by equations (A.3) and (A.5) when a/A is small. When 
a/A. is large, c/c, and cy/c, approach the value 1/ /€ asymototically. 
Assuming that o = 0.29, the variation of c/¢, and c,/c, with a/A. given by equations (4.3), 
(A-5), (A-7) and (A.9) is shown graphically in Figures A.i. and A.2.; the values given by equations 
(a.3) and (A.5) are shown in the broken-Tine curve labelled "Flexural waves - Elementary Theory", 
those given by equation (A.7) in the chaim-dotted curve labelled "Flexural waves — Rayleigh Theory*, 
and those derived from equation (A.9) by the points marked by crosses. These diagrams also show 
the values of c/c. and c Icy deduced from the data given by Hudson; these values are given by the 
curve labelled "Flexural waves (exact theory)" and, for purposes of comparison, the values of c/e, 
and c_/c_, for extensional waves (ist mode, exact theory) have been included in the curves marked 
"Extensional Waves". The curves given in these diagrams show a number of points of interest. 
When a/~ is small, the elementary theory gives the correct result. As a/A increases from zero, 
the values of cic, and c Ic, given by the elementary theory and by the Rayleigh theory are greater 
than those alven by the exact theory. when a/A becomes large, the phase and group velocities become 
inflnite on the elementary theory and equal to c, on the Rayleigh theory, whereas on the exact theory, 
they become equal to Cg, the velocity of the Rayleigh surface waves. One interesting feature of 
the curves is the degree of agreement between the values of cic, and ¢ /c, given by the exact theory 
and by the Timoshenko theory, and even when the error is a maximum, i,e., when a/A is very large, 
the error of the Timoshenko theory is not excessive, since the limiting values of cic, and c Icy 
(when @ = 0.29) are 0.576% on the exact theory and 0.5906 on the Timoshenko theory. This feature of 
the Timoshenko theory justifies its use in dealing with problems, such as the determination of the 
frequency of lateral vibrations of bars, where the exact theory cannot be employed because of its 
complexity. 
FOr wesc 
@ S. Timoshenko, Phil. Mag., Vol. Ui, pe7¥e, (1921) 
