Geology Reference
In-Depth Information
more generally. “Noether's Theorem” (Courant and Hilbert, 1989; Gelfand and
Fomin, 1963) describes energy and momentum concepts from a theoretical
physics point of view. Essentially, energy is associated with a variational
principle that is invariant with respect to translations in time, and similarly,
momentum is associated with analogous translations in space.
Then, using multiple-scaling arguments such as those introduced in
Equation 2.87, it is possible to show that the wave energy density E(x,t) and the
wave momentum density M(x,t) satisfy
E/ t + (
r
k
(k,x,t)E)/ x = E{2
i
+
r
t
(k,x,t)/
r
} (2.104)
M/ t + (
r
k
(k,x,t)M)/ x = M{2
i
-
r
x
(k,x,t)/k} (2.105)
following, again, our convention where (k) =
r
(k,x,t) + i
i
(k,x,t) describes
the uniform wave (simple dispersion relations are given elsewhere in this topic).
Let us consider Equation 2.104, where E(x,t) contains both kinetic and
potential energy contributions. The left side appears in conservation form. If
the right side were zero, then E/ t + (
r
k
(k,x,t)E)/ x = 0 shows that timewise
increases in wave energy density are balanced by differences in energy flux,
with group velocity
r
k
(k,x,t) being the energy transport velocity. Since the
simplified equation for E in the above paragraph takes the same form as
Equation 2.11, the result analogous to Equation 2.20 states that the total energy
between two group velocity rays is constant. The right side of Equation 2.104,
which is proportional to E, represents a distributed energy source or sink. It
models damping or instability through the imaginary frequency
i
, but the
presence of
r
t
(k,x,t)/
r
indicates that time-dependent heterogeneities, e.g.,
time-varying tension on a violin string or oil movement in a petroleum reservoir,
will affect energy in or loss from the system.
Note that
i
and
r
t
(k,x,t)/
r
both contain k(x,t) in their arguments; thus,
the use of Equation 2.104 requires the solution of Equations 2.102 and 2.103
beforehand. Similar comments apply to Equation 2.105 for wave momentum
density, the important emphasis being the
r
x
(k,x,t)/k at the right. Momentum
sources arise from dissipative effects and from the effects of spatial
heterogeneities (e.g., the variable mass on a guitar string, or in water waves, the
slope of an underlying beach). Changes to energy and momentum density arise
from three effects: heterogeneities in the propagation medium, dissipation and
geometric spreading, all of which are included in Equations 2.104 and 2.105.
Finally, in nonuniform media, E and M are different from the a
2
used for waves
propagating in uniform homogeneous media. Both E and M are proportional to
a
2
, and all may be used as qualitative measures of stability; however, E, M and
a
2
are physically different, behave differently and have different units.
Search WWH ::
Custom Search