Geology Reference
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but the inverse of the remaining term is less obvious. However, it is clear that it
can be expressed as the product of simpler functions of s.
The “Convolution Theorem” provides direct inversions when a transform
can be factored into simpler transforms with known inverses. If f
1
(s) and f
2
(s)
represent Laplace transforms for F
1
(t) and F
2
(t), then the inverse of f
1
(s)f
2
(s),
denoted by
L
-1
{f
1
(s)f
2
(s)}, is simply
t
L
-1
{f
1
(s)f
2
(s)}=
F
1
(t- ) F
2
( ) d
(1.59)
0
Note that a function cannot be
arbitrarily
decomposed into the product of
any
two functions f
1
(s) and f
2
(s). From Equation 1.51, each of these functions must
vanish as s . If not, the functions F
1
(t) and F
2
(t) will not exist, and the
required inversions will not appear in any published tables.
We will take f
1
(s) = G(s) and f
2
(s) = 2Ts/{Mc
2
s
2
+ ( c+T)s +k}, noting
that f
2
( ) = 0 as required. Then, the Convolution Theorem yields
f( ) = -g( ) + (2T/Mc
2
)(a-b)
-1
g(
) {ae
a(
)
- be
b(
)
} d
(1.60)
0
where the constants “a” and “b” satisfy
a = {- ( c+T) - [( c+T)
2
- 4k
c
2
]}/2
c
2
(1.61)
b = {- ( c+T) + [( c+T)
2
- 4k
c
2
]}/2
c
2
(1.62)
Since g( ) is prescribed, f( ) is now known, and is easily obtained by
integration. By substituting the required dummy variables in ct+x and ct-x, we
construct the exact solution as
u(x,t) = g(ct+x) - g(ct-x)
ct-x
+ {2T/[
c
2
(a-b)]} g(
) {ae
a(ct-x- )
- be
b(ct-x- )
} d (1.63)
0
Note that M, , k, T, c and
l
do not appear individually in the final
solution, but implicitly through the parameters a and b. We also infer from the
exponential terms in Equation 1.63 that “ac” and “bc” have dimensions of
inverse time. Thus, (ac)
-1
and (bc)
-1
define time scales in addition to the period;
these are analogous to those in electrical RC circuits. In Equation 1.63, the
g(ct+x) term represents the known, prescribed incident wave emerging from
infinity, whereas the term -g(ct-x) represents the reflection at a rigid pinned
connection (this is so because the sum g(ct+x) - g(ct-x) vanishes at x = 0). The
last term is the “shape distortion” to the incoming wave due to the flexible
support; this consists of a phase shift and a shape change.
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