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which can be conveniently reexpressed as the sum of two contributions
G( s ) = - { ( c+ )/( c- )} F(s)
- {{1 - ( c+ )/( c- )}/( c- )} {1/{s - (- /( c- ))}} F(s) (4.2.47)
The first line represents a reflection identical in form to the original
incoming wave, except for possible phase and amplitude changes. The second
line embodies more subtle effects, which are best explored by inverting G(s) and
returning to the time domain. Now, “{1/{s -(- /( c- ))}} F(s)” can be viewed
as the product of two transforms. The curly-bracketed term is a valid transform
because it vanishes as s tends to infinity; its inverse is the exponential exp (-
/( c- )). The inverse of F(s) is just f( ), so that the inverse of the product in
the time domain follows from the convolution integral
g( ) = - {( c+ )/( c- )}f( )
-
{{1 - ( c+ )/( c-
)}/( c-
)}
f( ) e
- ( - )/( c-
)
d
(4.2.48)
In this general form, the subtleties are still unclear. To understand axial
displacement effects induced by drillbit rotation, let us consider a sinusoidal
wave incident upon x = 0, having the Fourier wave component
f( ) = A sin
/c
(4.2.49)
The result in Equation 4.2.48 specializes to
g( ) = - A {( c+ )/( c-
)} sin
/c
(4.2.50)
- A {{1 - ( c+ )/( c-
)}/{( c-
){( /( c-
)
2
+ (
/c)
2
)}}
/c) e
-
/( c-
)
}
X {( /( c-
sin
/c - (
/c) cos
/c + (
The solution containing both incoming and outgoing waves is obtained by
construction, replacing the dummy variable in f( ) and g( ) by ct+x and ct-x as
required in u(x,t) = f(ct+x) + g(ct-x). Thus, we have the entire solution
u(x,t) = A sin
(ct+x)/c - A {( c+ )/( c-
)} sin
(ct-x)/c
(4.2.51)
- A {{1 - ( c+ )/( c- )}/{( c- ){( /( c- )
2
+ ( /c)
2
)}}
x {( /( c- sin (ct-x)/c -( /c) cos (ct-x)/c +( /c) e
- (ct-x)/( c- )
}
The outgoing wave contains expected phase changes to the incoming sinusoidal
input due to elastic-resistive boundary interactions, but it also includes a
nonharmonic DC contribution whose average over integer periods does
not
vanish. At the x = 0, this DC level can be written as
DC
{u(0,t)} = -A {{1 -( c+ )/( c-
)
2
+(
/c)
2
)}}
)}/{( c-
){( /( c-
x {(
/c) e
- ct/( c-
)
}
(4.2.52)
where
DC
{u(0,t)}
is
the
non-periodic penetration depth
that the drill bit
descends with.
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