Geology Reference
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U
p
(x,s; (t)) = - |U
1
(x
s
,s)U
2
(x,s)-U
1
(x,s)U
2
(x
s
,s)|/
{T |U
1
(x
s
,s)U
2,x
(x
s
,s)-U
1,x
(x
s
,s)U
2
(x
s
,s)| }+ ...
(1.130)
whose time-domain inverse we denote u
p
(x,t). Over large times, the effects of
viscous dissipation will damp out all traces of the initial conditions u(x,0) and
u
t
(x,0), so that the “ ...” will no longer enter the solution. Then, the particular
solution u
p
(x,t) becomes the sole response of the impulse (t) when the effects
of gravity are neglected.
Now it is clear that had we carried out our analysis for a more general
time-dependent force F(t) on the right side of Equation 1.127, as opposed to (t),
we would have obtained instead the particular solution
U
p
(x,s;F(t)) = - F
*
(s) |U
1
(x
s
,s)U
2
(x,s)-U
1
(x,s)U
2
(x
s
,s)|/
{T |U
1
(x
s
,s)U
2,x
(x
s
,s)-U
1,x
(x
s
,s)U
2
(x
s
,s)| }
(1.131)
where F
*
(s) is the Laplace transform of F(t). Interestingly, we can rewrite
Equation 1.131 as the product of two Laplace transforms
U
p
(x,s;F(t)) = - F
*
(s) U
p
(x,s; (t))
(1.132)
where we have used the impulse response in Equation 1.130. We return to our
Convolution Theorem (see Equation 1.59) which states that the time domain
function corresponding to the product f
1
(s) f
2
(s) of two Laplace transforms f
1
(s)
and f
2
(s) with inverses f
1
(t) and f
2
(t) is the integral
t
0
f
1
( ) f
2
(t- ) d
(1.133)
Accordingly, the inverse solution u(x,t) to U
p
(x,s;F(t)) in Equation 1.132 must
take the form
t
0
F( ) u
p
{t- ; (t)} d
(1.134)
where u
p
, again, is the undamped time domain response due to an impulse.
This completes the proof: once u
p
(t) is known or available, the complete
transient response to a more general excitation F(t) can be written using
Equation 1.134. Of course, generating a true impulse experimentally is a
difficult challenge. As an exercise, the reader should repeat these steps using an
initial test waveform
other
than an impulse, and determine the superposition
integral analogous to Equation 1.134. The above conclusion is quite general,
and applies to all linear systems. It is not restricted to the wave operator,
although we chose that route to make the argument concrete. Observe that, from
the variation of parameters argument onwards,
any
linear differential operator
may have preceded the discussion.
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