Geology Reference
In-Depth Information
[ / x]
x=xs
/ x |
x=xs+
- / x |
x=xs-
= - {
l
F ' ( t) + F(t)} dx / T (1.117)
[ / x]
x=xs
= - {
l
F ' ( t) + F(t)} dx / T (1.118)
which, on substitution of our definition for / x in Equation 1.107, yields
[
l
u/ t + u]
x=xs
= - {
l
F ' ( t) + F(t)} dx / T (1.119)
Since F(t),
l
and are arbitrary, the derivative and undifferentiated terms lead
to separate identities, [ u/ t]
x=xs
= - F '(t) dx /T and [ u]
x=xs
= - F(t) dx /T. But
these are equivalent, the first being the derivative of the second; thus, we take
[u]
x=xs
= - F(t) dx / T (1.120)
without loss of generality. We return to our original equation for u(x,t),
Equation 1.104, and equivalently rewrite it as
2
u/ t
2
+
u/ t -T
2
u/ x
2
+
l
g
=
l
F(t) { (x-x
s
)- (x-x
s
)}
=
F(t)dx{ (x-x
s
)- (x-x
s
)}/(2 )
= F( t) dx '(x-x
s
)
= - T [u]
x=xs
'(x-x
s
)
(1.121)
Hence, if a jump [ u]
x=xs
in the dependent variable (as opposed to its spatial
derivative) is desired, an external dipole excitation -T [ u]
x=xs
'(x-x
s
) must be
applied, where '(x-x
s
) is the derivative of the delta function; here, the first
spatial derivative will be continuous and single-valued. The resulting equation,
as we will show, is easily solved using Laplace transforms. We have discussed
our delta function ideas having arguments in x, but it is clear that similar ideas
apply to delta function “impulse” loads with arguments in t (we will give an
example later). Then, the consequences would be expressed in terms of jumps
in u or the velocity u/ t.
Returning to our discussion of Figure 1.5, note how, when the
discontinuity [ u(t)]
x=xs
itself is prescribed, the formulation using (x,t) is
actually simpler. We show this by combining Equations 1.116 and 1.120,
l
2
/ t
2
+ / t - T
2
/ x
2
= -
l
gx + {
l
F ' ( t) dx + F( t) dx } (x-x
s
)
= -
l
gx - T {
l
[u]
x=xs
+ [u]
x=xs
} (x-x
s
) (1.122)
which contains a simple delta function and not its derivative '(x-x
s
). The
complementary transform renders the mathematical problem “force-like” or
“monopole-like,” since Equation 1.122 is similar in appearance to Equation
1.99. Force-like models are easily solved numerically, as we will demonstrate
in Chapter 4.
Search WWH ::
Custom Search