Problem 7.5a
In Figure 7.5a, a linear vertical source
(such as a column of explosives) of length
,
being a constant and
the wavelength, is activated at all points simultaneously at time
. Taking the initial waveform as
, show that the effect at point
is
|
|
(7.5a)
|
What is the array response?
Background
A distributed source can be thought of as an array. The array response is the ratio of the output of an array to the output when all of the elements are concentrated at the midpoint of the array.
Detonating cord is an explosive cord with a constant velocity of detonation; it is used to connect two charges in order to delay detonation of the second charge. Detonation of the first charge initiates detonation in the cord, which in turn detonates the second charge. By varying the length of cord, detonation of the second charge can be delayed a desired amount.
Solution
Although the explosive is exploded instantaneously, energy from different parts of the column arrive at
at different times because they must travel different distances. Denoting
by
, the total effect at
at time
will be
Figure 7.5a Geometry for linear source.
To integrate we must get a relation between
and
. We assume that
; then
|
|
(7.5b)
|
Thus,
Noting that
and
, this becomes
|
|
(7.5c)
|
where sinc
.
If the linear source is replaced by a concentrated source of equal strength at the center
, the effect at
would be
. Dividing the right-hand side of equation (7.5c) by this quantity, we get for the array response
|
|
(7.5d)
|
Problem 7.5b
An explosion initiated at the top of the explosive column
in Figure 7.5a travels down the column with velocity
. Show that the array response is
|
|
(7.5e)
|
where
is the velocity in the rocks. Under what circumstances does equation (7.5e) reduce to equation (7.5d)?
Solution
In part (a), the entire column
exploded at
. We now consider the case where the explosion starts at point
and travels down the column with velocity
, that is, the explosion starts at
at
, where
. Writing
,
,
the wave generated by the element dz arrives at
with the phase
Using equation (7.5b) the phase becomes
Assuming a harmonic wave function
, we can write
where
amplitude,
,
. Integrating, we obtain
If we locate the same amount of explosive at
and explode it at
, we get at
hence the array response is
Omitting the first factor, which is independent of
and
and hence is merely a scale factor, we have
But
so
|
|
(7.5f)
|
[The minus sign for sinc occurs here and not in equation (7.5d) because the direction of integration is opposite to that assumed in deriving equation (7.5d)]. For an instantaneous explosion the result is given by equation (7.5d), namely
Equation (7.5c) reduces to equation (7.5d) whenever
. For
, i.e., for rays traveling almost vertically downward, the required condition is that
. For most explosives,
km/s, so
should be no more than about 1.5 km/s, the velocity of water, for the two equations to give nearly the same result.
Problem 7.5c
Calculate the array response
for a column 10 m long, given that
m,
km/s,
km/s, and
,
,
,
.
Solution
thus
Table 7.5a. Array response as a function of direction.
|
|
|
|
|
|
|
0.99
|
30
|
0.094
|
0.094
|
1.00
|
60
|
0.38
|
0.37
|
0.97
|
90
|
0.49
|
0.47
|
0.98
|
Differences in directivity are negligible when the charge length is much smaller than the wavelength.
Problem 7.5d
If the column in part (c) is replaced by six charges, each 60 cm long and equally spaced to give a total length of 10 m, the charges being connected by spirals of detonating cord with detonation velocity 6.2 km/s, what length of detonating cord must be used between adjacent charges to achieve maximum directivity downward?
Solution
Let
be the length of detonating cord between successive charges; maximum directivity downward is achieved when the traveltime through the explosive column is the same as that in the adjacent rocks. In part (c) we were given
km/s,
km/s, so
, hence
m.
Problem 7.5e
What are the relative amplitudes (approximately) of the waves generated by the explosives in part (d) at angles
,
,
, and
when
m?
Solution
An approximate solution can be obtained by assuming that the average velocity
is equal to
; this means that the traveltime down through the 10 m column of explosives is the same as that for a wave in the adjacent 10 m of rock. In this case,
,
and
Table 7.5b. Relative amplitudes.
|
|
|
|
|
–0.79
|
–0.71
|
0.90
|
30
|
–0.39
|
–0.38
|
0.97
|
60
|
–0.11
|
–0.11
|
1.00
|
90
|
0.00
|
0.00
|
1.00
|
Continue reading
Also in this chapter
External links
find literature about Directivity of linear sources
|
|
|
|
|
|
|
|