Geology Reference
In-Depth Information
g(ct+x) = H{ct+x-L} - H{ct+x-(L+a)} (4.2.138)
In Equation 4.2.138, H( ) is the “Heaviside step function,” or simply “step
function,” equal to unity for > 0 and zero for < 0. Noting that L(H{t}) = 1/s,
we successively obtain
g( ) = H{ -L} -H{ -(L+a)}
(4.2.139)
g*(s) = e
-Ls
/s - e
-(L+a)s
/s
(4.2.140)
)
L
-1
[e
-Ls
/{s(s - /
)} - e
{-(L+a)s}
/(s(s - /
f( ) = g( ) + 2( /
)}] (4.2.141)
f(ct) = g(ct) -2[1- e
{ (ct-L)/ }
] H(ct-L)
+ 2[1 - e
{ (ct-L-a)/ }
] H(ct-L-a)
(4.2.142)
f(ct-x) = g(ct-x) -2[1- e
{ (ct-x-L)/
}
] H(ct-x-L)
+ 2[1 - e
{ (ct-x-L-a)/
}
] H(ct-x-L-a)
(4.2.143)
The complete solution obtained from the general wave formula u(x,t) = f(ct-x) +
g(ct+x) is therefore
u(x,t) = H{ct+x-L} - H{ct+x-(L+a)} + H{ct-x-L} - H{ct-x-(L+a)}
-2[1- e
{ (ct-x-L)/ }
] H(ct-x-L)
+ 2[1 - e
{ (ct-x-L-a)/
}
] H(ct-x-L-a)
(4.2.144)
Figure 4.2.10 qualitatively shows how elastic rock response transforms
rectangular pulses into distorted ones with exponentially shaped edges. If the
exact exponential and pulse length changes are available from computations, the
coefficient / in Equation 4.2.129 is known. As in the first example, reference
to the (hypothetical) rock-bit interaction database will define the formation type,
since the drillbit model is known. Because all incoming pulses will distort when
elastic effects are important, as Equation 4.2.137 clearly shows, it is important to
have a simple, well-characterized, incoming pulse so relative distortions are
easily observed and conveniently measured. But in general, there is nothing
special about a rectangular pulses, as any observable pulse readily suffices.
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