559 
For the purpose of discussing the propagation of a pulse of flexura) vibrations, consider 
a bar which is stressed initially so that the flexural displacement is zero everywhere except at a 
certain cross-section where it is infinite; this cross-section will be taken as the origin of x, 
If the stress is released at time t' = 0, the period T_ of the dominant group in the disturbance at 
a cross-section of abscissa x at time t' can be deduced by group-velocity methods, It is convenient 
to take the non-dimensional ratio t'/4T, as the independent variable and the non-dimensional ratio 
T (S as the dependent variable, ey and aie being equal to the times taken by an extensional wave 
of infinite wave-length to traverse the distances a and x respectively. Since 
T 
p cS t* fo 
eA od Tite) ey PC BOC OcG N  COnenS aoeer As 10 
16 AG 47 tase 
O° fg 
the values of these non-dimensional variables can be derived from the curves of Figures A.1. and A.2. 
The (T /T,» t'/4T,) Curves deduced in this way are shown In Figure A.3. According to the 
elementary theory, ¥,/Ta = Npra?, t/t, ==. ; ‘the (T/T. t'/4T,) curve is thus a parabola 
aa = : 
passing through tne point t'/4T, = 0 and symmetrical about the axis of Th/Ts passing through this 
point. This implies, as has already been pointed out, that wave-packets consisting of infinitely 
short waves arrive at the cross-section of abscissa x at the instant at which the disturbance departs 
from the origin. AS a/A decreases, the values of T/T, and BU Al both increase, and for large 
values of a/A (outside the range of Figure A.3) the curve given by the elementary theory coincides 
with the curves given by the more exact theories. 
According to the curve given by the Rayleigh theory wave=packets consisting of infinitely 
short flexural waves take the same time to travel through a given distance as wave-packets composed 
of infinitely long extensional waves. As T fit increases (or a/A decreases) t'/4T, decreases yntil 
it reaches a minimum value at t'/4T, = 0,92, when T Ts is about 3 and a/A is about 0.8; beyond this 
point t/a, decreases as T/T, increases. Thus, according to this theory, when an infinitely intense 
flexural disturbance initially concentrated at the origin is released, the first components to arrive 
at a given point do so at time t' = 0.92 tof 2 their period is about 31, and their wavelength is about 
1.25a. In the interval between the arrival of these components and time t' = To/2s two groups of 
different wavelengths and period arrive simultaneously at each instant; finally when t* exceeds T,/2 
only one group arrives at a given instant, the period and the wavelength of the aroup increasing as t’ 
increases, 
Considering the results of the exact theory, it is clear that the faster components, originating 
in an infinitely intense disturbance at the origin, arrive at the cross-section of abscissa x at time 
t' = 1.568 T,/2: the period of these components is 5.29 T, and their wavelength is equal to 2.72, 
Between this value of t' and t* = 1,735 To/2 two groups of different wavelengths and periods arrive 
simultaneously at each value of t'; at t' = 1,735 T,/2 the Rayleigh surface waves and waves of period 
17.3 Tie arrive simultaneously. When t' exceeds 1.735 To/2s only one group will arrive at each instant. 
It is clear that the elementary theory and the Rayleigh theory give results which are very wide 
of the truth, and any solutions based on these theories of problems of the action of transient flexural 
stresses on bars are unlikely to be accurate unless it happens that the disturbances are such that 
waves of short-wavelength are unimportant. Again, it is interesting to notice that the results derived 
from the Timoshenko theory are in excellent agreement with those deduced from the exact theory. 
As -far as the present experiment is concerned, the main interest of the curves of Figure A.3. is 
that they show that, on the exact theory, no flexural displacements wil) occur at the cross-section of 
abscissa x until t* > 1,568 T,/2, i.e. t' > 1.568 x/Co; remembering that the record in a given 
experiment begins when extensional displacements of infinite wavelengths arrive at the cross—section, 
ie. when t* = T,/2 = x/Cos it follows that displacements due to flexural waves will not appear on the 
record until a time 0.568 x/c, after the beginning of the record. For the bar used in this experiment, 
X = 670.5 cm, C, = 5.26 x 108 cm. /sec., So that 0.568 x/C, = 0.725 millisecond, which is outside the 
range of time covered by the record. 
