Page 301 - Rock Mechanics For Underground Mining

ENERGY TRANSMISSION IN ROCK

u y = f 2 (x − C s t) + F 2 (x + C s t)
u z = f 3 (x − C s t) + F 3 (x + C s t) (10.37)

As for the stress wave in a bar, each expression in equations 10.36 and 10.37 with ar-
gument (x − Ct) corresponds to a forward progressive wave, and each with argument
(x + Ct) to a backward progressive wave.
Equation 10.36 describes particle motion which is parallel to the direction of propa-
gation of the wave. Wave propagation occurs at a velocity C p , given by the expression
in equation 10.34. The waves are called P waves, or primary or longitudinal waves.
Equations 10.37 describe particle motion which is transverse to the direction of wave
propagation. Wave propagation occurs at a velocity C s , as defined in equations 10.35.
The waves are called S waves, or secondary or shear waves.
The expressions for P- and S-wave velocities indicate that, for = 0.25, C p /C s =
1.73. Natural sources of wave motion normally generate both P and S waves. The
higher velocity of the P wave means that it is received at some observation point
remote from a wave source earlier than an S wave, allowing field identification of P-
and S-wave velocities. It therefore offers the possibility of ready determination of the
in situ dynamic elastic properties of rock masses.
Transmission of P and S waves in a non-homogeneous medium is subject to internal
reflection, in the manner determined for the bar wave. For the case of normal incidence
on an interface between domains with different elastic properties, an incident P wave
generates transmitted and reflected P waves. For these waves, stresses and particle
velocities can be calculated from expressions similar to equations 10.19–10.22, except
that the ratio of characteristic impedances n p is given by

n p = 2 C p2 / 1 C p1
Similar considerations apply to an S wave, except that the relevant ratio of character-
istic impedances is now

n s = 2 C s2 / 1 C s1
Oblique incidence of P and S waves at an interface between dissimilar materials
results in more complicated interaction than for normal incidence. Considering an
incident P wave, transmitted and reflected P waves are generated in the usual way. In
addition, transmitted and reflected S waves (called PS waves) are produced, i.e. the
interface acts as an apparent source for S waves. Similar considerations apply to an
incident S wave, which gives rise to SP waves, in addition to the usual transmitted
and reflected waves. As might be expected from the complexity of the wave motion
induced at the interface, there is no simple expression for calculating the intensities
of any of the transmitted and reflected waves.

10.3.3 Spherical and cylindrical waves
The mechanics of plane waves provides a useful basis for understanding wave prop-
agation in a three-dimensional body, such as a rock mass. Several important sources
of wave motion in rock are either cylindrical or concentrated, i.e. having a low
length/diameter ratio. Due to their symmetry, spherical and cylindrical sources can
be expected to produce spherically and cylindrically divergent P waves.

283