Page 302 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

In the description of the plane P wave, the differential equation of motion (equation
10.34) can be recast in the form

2
2 2
2
∂ u x /∂x = C ∂ u x /∂t 2 (10.38)
p
for which the general solution is of the type given in equation 10.36. The wave
equation, written in its most general form, may be expressed as
2
2
2
2
2
2
2
2
∂ /∂x + ∂ /∂y + ∂ /∂z = (1/C )∂ /∂t 2 (10.39)
where the nature of the variable is determined by the fundamental particle motion
associated with passage of the wave. ForaPwave, the parameter is the volumetric
strain, or dilatation, .
In spherical polar co-ordinates, the governing equation for a spherically divergent
P wave is shown by Kolsky (1963) to take the form
2
2
2
2
2

∂ (ru r )/∂r − 2ru r /r = 1/C ∂ (ru r )/∂t 2 (10.40)
p
where r is the spherical co-ordinate radius. The general solution of equation 10.40
can be readily verified to be of the form
2

u r = (1/r) f (r − C p t) − (1/r ) f (r − C p t) (10.41)
where f is some arbitrary function, and f its first derivative with respect to the

argument (r − C p t). The form of equation 10.41 indicates that remote from the wave
2
source (when 1/r 1/r ), the term in f predominates, and the spherical wave

solution is approximated by

u r = (1/r) f (r − C p t) (10.42)
As discussed for the bar wave, for an elastic medium the wave function is invariant
with respect to the local co-ordinate of the propagating wave, in this case (r − C p t).
Equation 10.42 therefore implies that for the spherically divergent wave
u r ∝ 1/r (10.43)
For a long cylindrical source, the wave equation takes the form

2
2
2
2

∂ r 1/2 u r /∂r − 3 r 1/2 u r /r 2 = 1/C ∂ r 1/2 u r /∂t 2 (10.44)
p
4
There appears to be no completely general solution to this equation. For the case
where r is large, equation 10.44 is approximated by
2
2
2
2

∂ r 1/2 u r /∂r = 1/C ∂ r 1/2 u r /∂t 2
p
which, by comparison with the one-dimensional wave equation 10.38 and its solution,
equation 10.36, has the solution
r 1/2 u r = f (r − C p t) (10.45)
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