Geology Reference
In-Depth Information
2.2.3
Relating Q to standing wave decay.
We might have modeled the decay of a standing wave by introducing, as
we did in Chapter 1, the imaginary part
i
of the complex frequency given in
Equation 2.31, setting
A(t) = A
0
exp (
i
t)
(2.35)
If we equate timewise attenuation with its equivalent in space, Equations 2.34
and 2.35 suggest that we set
i
t = -
r
x/2cQ. But since x = ct, the expression
simplifies to
i
= -
r
/2Q; Equation 2.29 is, alternatively,
1/Q = - 2
i
/
r
(2.36)
where we must have |
i
/
r
| << 1 in order to rule out critically damped waves.
In summary, when the frequency
r
of the standing wave
and
Q are known, the
i
used in more general KWT analysis is simply
i
= -
r
/2Q.
2.2.4 Kinematic wave generalization.
We return to Equation 2.11 (that is, a
2
/ t + (C(k)a
2
)/ x = 0, where C(k)
is the “group velocity”
r
k
(k)), and examine the effect of adding 2
i
a
2
to the
right side,
a
2
/ t +
(C(k)a
2
)/ x = 2
i
a
2
(2.37)
Equation 2.37 was postulated by Landahl (1972), who derived it intuitively, so
that it satisfies the simple propagating and standing wave limits introduced
earlier. Chin (1976, 1980) formally obtained this result using multiple-scaling
methods, carrying out asymptotic expansions to higher order, e.g., see Equation
2.95. Note that Landahl' s equation appears in conservation form. It is
interpreted analogously to the mass conservation equation / t + ( q)/ x = 0
used in fluid mechanics, which balances timewise changes in density with
differences in the mass flux q, where q is the transport velocity.
Equation 2.37 states that timewise changes to an energy-like a
2
must be
balanced by differences in the energy flux C(k)a
2
, where the relevant transport
velocity, consistently with classical physics, is the group velocity. The right
side of Equation 2.37 is sink-like, representing distributed attenuation. Let us
consider two limits of Equation 2.37. In the propagation limit with / t = 0, the
equation (C(k)a
2
)/ x = 2
i
a
2
on division by C(k) leads to
da
2
/a
2
= 2
i
/C(k)
d
x
= - {
r
/C(k)Q}dx
(2.38)
where we have used 1/Q = - 2
i
/
r
. Direct integration yields the result log
e
a
2
=
- {
r
/C(k)Q}x, or a
2
exp {
-
(
r
/C(k)Q)x}. On taking square roots,
a(x)
exp {
-
r
x/2C(k)Q}
(2.39)
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