Flexural Vibrations of Thin Rods. 
399 
1908-9.] 
§ 9. The diagrams of figs. 1 and 2 are taken from Lord Kelvin’s paper 
referred to in § 8, and they are reproduced without change of the lettering 
applicable to them as water-wave diagrams. To make them correspond 
exactly to the flexural-waves problem solved by equation (12), we must 
reduce the ordinates in both figures in the ratio J 2 : 1 ; then in fig. 1 
replace t by x on each of the seven curves, and take ordinates as represent- 
ing displacement and abscissas as representing time. As a water-waves 
diagram fig. 2 represents the vertical displacement of the water at point x — % 
from t — 0 to t = oo ; as a flexural-waves diagram it represents the shape of the 
right-hand half of the rod, x = 0 to x — oo , at time t = 2, corresponding to the 
initial configuration given by equation (13). 
These curves are useful chiefly as illustrations of the propagation of waves 
in dispersive media from a given initial disturbance confined in the main to 
the neighbourhood of the origin. They show clearly the distinctive features 
of wave-propagation in the two cases where the wave-velocity varies directly 
as the square root of the wave-length and inversely as the wave-length 
respectively, for an infinite succession of regular sinusoidal waves. It is 
interesting to observe that in both cases the wave-disturbance is ultimately 
spread throughout the entire medium, but that in the case of water-waves 
the wave-length of the disturbance at any time increases continuously, and 
in the case of flexural-waves it diminishes continuously, as we pass outwards 
from the middle point of the initial disturbance. In the case of water-waves 
also, Lord Kelvin’s investigations show that each individual wave lengthens 
and increases in speed as it advances, while in the case of flexural-waves 
each wave lengthens and diminishes in speed, as we shall see by equations 
(15) and (16) below for the solution we have chosen for illustration. 
§ 10. At a moderate time after the motion has commenced, Tinay be put 
equal to — in equation (12), and the argument of the cosine varies then 
xH 
only with so that it is easy for us to trace the outward progress of 
any particular maximum or zero of displacement along the bar. Thus the 
position of any zero is determined by an equation of the form 
xH 
4 n + 3 
C — : r 7 r 
4k6T 2 4 
and the velocity of the zero is given by 
( 11 ), 
dx 8>Kbct — x 2 . 2 Kbc 
dt 2 tx ' x 
■ ( 15 )- 
When t is large, so that we may put 
t_ 
rp 
1 
V 
equation (14) enables us to 
