Geology Reference
In-Depth Information
2.1.6 Nonuniform media.
Now consider wave motions in nonuniform, inhomogeneous, or
heterogeneous media, for example, the transverse vibrations of a guitar string
whose lineal mass density
l
varies slowly with x, or whose tension T varies
slowly with time, or both, in which case = T(t)/
l
(x)} k. Solutions to such
problems are almost always impossible to obtain analytically, at least, in any
simple usable form. The restriction to slow is natural, since fast variations in
properties may not support wave-like motions. We now determine what wave
crest conservation implies, when applied to dispersion relations of the form
=
(k,x,t)
(2.21)
where the dependence on x and t is weak. Again, we start with
k/ t +
(k,x,t)/ x = 0
(2.22)
Thus, it follows from the chain rule of calculus that
k/ t +
k
(k,x,t) k/ x = -
x
(k,x,t) (2.23)
As before, the total derivative dk satisfies dk/dt = k/ t + dx/dt k/ x. Hence,
dk/dt = -
x
(k,x,t)
(2.24)
provided that
dx/dt =
k
(k,x,t) (2.25)
If the dispersion relation = (k,x,t) is known, Equations 2.24 and 2.25 provide
two coupled nonlinear ordinary differential equations that completely determine
the wave kinematics once initial conditions on k and x are assigned. They show
that k(x,t) still propagates at group velocity, but now it varies slowly along a ray;
thus, trajectories in the x-t plane defined by Equation 2.25 will be curved and no
longer straight.
2.1.7
Example 2-4. Numerical integration.
These two equations are amenable to numerical integration. If initial
values k
i
and x
i
for k and x are specified at time t
i
, the propagated values at the
end of a time step t may be determined from the forward-differenced forms of
Equations 2.24 and 2.25, that is,
k
f
= k
i
-
x
(k
i
,x
i
,t
i
) t (2.26)
x
f
= x
i
+
k
(k
i
,x
i
,t
i
) t (2.27)
Equations 2.26 and 2.27, or their analogues in sophisticated schemes, can be
computed repeatedly in order to update k and x along a ray in time.
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