Geology Reference
In-Depth Information
where '(x-x
s
) = { (x-x
s
) - (x-x
s
) }/dx and dx = 2 > 0 is small. Kanwal
(1983) discusses in detail the “unit dipole” '(x); Hudson (1980) discusses
“three-dimensional dipoles” and their use in earthquake seismology. Next, we
simplify the result so that
(
l
u/ t + u )/ t - (T u/ x -
l
gx + F(t) (x-x
s
) dx)/ x = 0 (1.106)
Now introduce a new dependent variable (x,t) defined by the transformations
/ x =
u/ t + u
(1.107)
l
and
/ t = T u/ x -
l
gx + F(t)
(x-x
s
) dx
(1.108)
If we differentiate these with respect to x and t, respectively, we have
2
/ x
2
=
l
2
u/ t x + u/ x (1.109)
2
/ t
2
= T
2
u/ x t + F '(t) (x-x
s
) dx (1.110)
noting that the body force term disappears. Next, multiply the first equation by
T and the second equation by
l
, to obtain
T
2
/ x
2
= T
l
2
u/ t x + T
u/ x
(1.111)
2
/ t
2
= T
l
2
u/ x t +
l
F ' ( t)
(x-x
s
) dx
(1.112)
l
Subtraction leads to
2
/ t
2
- T
2
/ x
2
=
l
F ' ( t)
(x-x
s
) dx - T
u/ x
(1.113)
l
From the definition
/ t = T u/ x -
l
gx + F(t)
(x-x
s
) dx in Equation
1.108, we have
- T
u/ x = -
l
gx + F( t)
(x-x
s
) dx -
/ t
(1.114)
which can be used to eliminate the - T
u/ x term in Equation 1.113. Thus, the
governing equation satisfied by (x,t) is
2
/ t
2
- T
2
/ x
2
=
l
l
F ' ( t)
(x-x
s
) dx -
l
gx + F( t)
(x-x
s
) dx -
/ t
(1.115)
or
2
/ t
2
+ / t - T
2
/ x
2
= -
l
gx +
l
F ' ( t)
l
(x-x
s
) dx + F( t)
(x-x
s
) dx
= -
l
gx + {
l
F ' ( t) + F( t) } dx
(x-x
s
)
(1.116)
Drawing upon the arguments leading to Equation 1.103, the force-like quantity
{
l
F ' ( t) + F(t)} dx must be responsible for a discontinuity [ / x] in the first
spatial derivative of . In particular, using Equation 1.103, we have
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