Geology Reference
In-Depth Information
Equations 6.32 and 6.33 yield dx/dt = c(x,y,z) k
x
{k
x
2
+ k
y
2
+ k
z
2
}
-1/2
or
{k
x
2
+ k
y
2
+ k
z
2
}
1/2
/c dx/dt = k
x
(6.34)
Now, on combining Equations 6.31 and 6.33, we obtain the relationship dk
x
/dt =
- c
x
(x,y,z){k
x
2
+ k
y
2
+ k
z
2
}
1/2
. This result can be rewritten as
1/c dk
x
/dt = - {c
x
(x,y,z)/c}{k
x
2
+ k
y
2
+ k
z
2
}
1/2
(6.35)
Next, we substitute Equation 6.34 for k
x
into Equation 6.35, to obtain
1/c d/dt {k
x
2
+ k
y
2
+ k
z
2
}
1/2
/c dx/dt
= - { c
x
(x,y,z)/c}{k
x
2
+ k
y
2
+ k
z
2
}
1/2
(6.36)
For dynamically steady problems, the expression in Equation 6.33 is equal to the
constant excitation frequency
0
, that is, c(x,y,z){k
x
2
+ k
y
2
+ k
z
2
}
1/2
=
0
, so
{k
x
2
+ k
y
2
+ k
z
2
}
1/2
=
0
/c(x,y,z)
(6.37)
Along a ray trajectory, we have
c(x,y,z) dt = ds (6.38)
where ds is a differential length. Substitution of Equations 6.37 and 6.38 into
Equation 6.36 leads to
d/ds {1/c(x,y,z)} dx/ds = -c
x
(x,y,z)/c
2
(6.39)
which is exactly the same as the condition d/ds {c
-1
dx/ds} + c
-2
c
x
= 0 given by
Equation 6.30 (the frequency
0
importantly drops out entirely).
We have
shown, this time using our kinematic wave theory formalism, that wave motions
satisfying Equation 6.37 also follow Fermat's Principle.
In general, wave motions need not follow minimization rules, since these
apply to restrictive classes of problems such as that governed by Equation 6.1.
However, kinematic wave theory will always embody the correct physics
relevant to a particular formulation, and consistently reproduce these rules, if
warranted. That is,
there is nothing in Fermat's Principle that is not already
contained in the wave conservation formalism developed in Chapter 2.
Again,
we emphasize that, like Equation 6.1, the assumptions underlying Equations
6.31 and 6.32 preclude wave dissipation; hence, Fermat' s Principle does
not
apply over large space and time scales when the cumulative effects of small
dissipation due to oil and gas bearing formations become important. We will
develop an alternative model later.
6.4 Modeling Wave Dissipation
To motivate our discussion of dissipation modeling, and to accentuate the
limitations of conventional analysis, it is useful to review some elementary
examples where exact results can be obtained.
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