Geology Reference
In-Depth Information
Consider first the case when F(t) is the timewise unit delta function (t), so that
2 u/ t 2 +
u/ t - T 2 u/ x 2 + l g = (t) (x-x s )
(1.124)
l
Since we are concerned with initial value problems, we take the time Laplace
transform of this impulsively loaded equation. We multiply throughout by e -st
and integrate with respect to time from t = 0 to . We define the Laplace
transform of u(x,t), omitting the limits of integration for brevity, by
U(x,s) = e -st u(x,t) dt
(1.125)
Then, Equation 1.124 becomes
l {s 2 U(x,s) -su(x,0) -u t (x,0)} + {sU(x,s)-u(x,0)} - T U xx (x,s) + l g/s
=
e -st
(t) (x-x s ) dt = (x-x s )
(1.126)
or
U xx (x,s) - {( l s 2 + s)/T}U
= - (x-x s )/T - {( l s+ )/T}u(x,0) - ( l /T)u t (x,0) + ( l g/T)/s
= - (x-x s )/T - ...
(1.127)
Equation 1.127 is an ordinary differential equation for U(x,s) in x. It can
also be solved by transform methods, depending on the type of spatial domain.
If the problem is defined over - < x < , the Fourier transform can be used; if
it is defined on x > 0 only, the Laplace transform in x can be invoked. It is not
necessary to actually solve Equation 1.127 in order to obtain general results.
We simply assume that the solution to the “homogeneous equation” U xx (x,s) -
{( l s 2 + s)/T}U = 0 can be obtained using standard methods, as U 1 (x,s) and
U 2 (x,s). Then, the method of “variation of parameters” from the theory of
second-order ordinary differential equations (e.g., see Leighton (1967))
guarantees that a “particular solution” for Equation 1.127 exists in the form
x
U p (x,s; (t)) = - { ... } ( -x s )/T d
(1.128)
x o
where the (t) in U p (x,s; (t)) emphasizes the fact that Equation 1.128 applies to
the impulsive excitation (t) only, “...” refers to terms associated with initial
conditions, and the curly bracket denotes
= |U 1 ( ,s)U 2 (x,s)-U 1 (x,s)U 2 ( ,s)|/
|U 1 ( ,s)U 2,x ( ,s)-U 1,x ( ,s)U 2 ( ,s)|
(1.129)
Using the “sifting property” of the delta function (see Equation 1.101), Equation
1.128 simplifies to
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