Page 297 - Rock Mechanics For Underground Mining

ENERGY TRANSMISSION IN ROCK

co-ordinate direction, i.e. a backward progressive wave. Each of the functions f 1 and
f 2 is individually a solution to the wave equation, and since the constitutive behaviour
of the system is linear, any linear combination of f 1 and f 2 also satisfies the governing
equation.
During the propagation of the elastic wave, represented by equation 10.10, along a
bar, each particle executes transient motion about its equilibrium position. The tran-
sient velocity, V, of a particle is associated with a transient state of stress, xx , which
is superimposed on any static stresses existing in the bar. For uniaxial longitudinal
stress and using Hooke’s Law, dynamic stresses and strains are related by

xx = Eε xx =−E∂u x /∂x

or, from equation 10.10

xx =−E[ f (x − C B t) + f (x + C B t)] (10.12)
2
1
Transient particle velocity is defined by
˙ u x = V = ∂u x /∂t
or, from equation 10.10

V = (−C B ) f (x − C B t) + C B f (x + C B t) (10.13)

1 2
Considering the forward progressive wave, the relevant components of equations
10.12 and 10.13, together with equation 10.11, yield
V = C B xx /E = xx / C B

or
xx = C B V (10.14)

Thus the dynamic longitudinal stress induced at a point by passage of a wave is directly
proportional to the transient particle velocity at the point. In equation 10.14, the
quantity C B is called the characteristic impedance of the medium. For the backward
wave, it is readily shown that
xx =− C B V (10.15)

A case of some practical interest involves a forward wave propagating in a com-
posite bar, as indicated in Figure 10.7. The bar consists of two components, with

Figure 10.7 Geometry describing
longitudinal wave transmission and
reflection in a two-component bar.

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