Page 298 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

different material properties, denoted by subscripts 1 and 2. The interface between
the component bars is welded. The forward, incident wave in medium 1 impinges on
the interface, and is partly transmitted, generating a forward wave in medium 2, and
partly reflected, generating a backward wave in medium 1. The quantities to be de-
termined are the relative intensities of longitudinal stress in the forward and reflected
waves.
Suppose the longitudinal stress and particle velocity in the forward incident wave
are defined by magnitudes 0 , V 0 , and the corresponding magnitudes in the transmitted
and reflected waves are given by t , V t and r , V r . The ratio of the characteristic
impedances of the two media is defined by

n = 2 C B2 / 1 C B1 = 2 C 2 / 1 C 1 (10.16)

The conditions to be satisfied at the interface between the bar components are conti-
nuity of longitudinal stress and displacement (and therefore particle velocity). These
continuity conditions are expressed by the equations
(10.17)
0 + r = t
(10.18)
V 0 + V r = V t
Introducing equations 10.14 and 10.15 to relate stresses and velocities for incident,
reflected and transmitted waves, equation 10.18 becomes

0 / 1 C 1 − r / 1 C 1 = t / 2 C 2
and introducing equation 10.17

0 / 1 C 1 − r / 1 C 1 = ( 0 + r )/ 2 C 2
On rearrangement, this yields

r = [(n − 1)/(n + 1)] 0 (10.19)

It is then readily shown that

t = [2n/(n + 1)] 0 (10.20)
Similarly, the relation between particle velocities may be shown to be

(10.21)
V r =−[(n − 1)/(n + 1)]V 0
(10.22)
V t = [2/(n + 1)]V 0
It is useful to explore, briefly, some of the consequences of these expressions
governing wave transmission and reflection. Suppose 0 is compressive. For the case
n > 1, the reflected wave is characterised by a compressive stress. For n < 1, the
reflected wave induces a tensile stress. Thus an important general point to note is
that internal reflections of a compressive wave in a medium may give rise to tensile
stresses. Preceding chapters have described the low tensile strength of rock masses,
and their inability to sustain tensile stress.

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