Geology Reference
In-Depth Information
In wave propagation, we often start with localized initial disturbances
containing many wavelength components. The goal is to determine how
wavenumber and amplitude fields evolve in space and time. Knowing how the
disturbance (or information) packet disperses is vital to reassembling it, a
process important to transmission line applications in communications.
2.3.1 Example 2-5. General initial value problem in uniform media.
From calculus, the total differential dk satisfies dk = k/ t dt + k/ x dx,
which on division by dt yields dk/dt = k/ t + dx/dt k/ x. On comparison with
Equation 2.43, we have
dk/dt = 0
(2.44)
provided that
dx/dt = r k (k) (2.45)
Equation 2.44 indicates that k is constant along a ray. Equation 2.45, taking
advantage of this fact, states that the group velocity is likewise constant; this
implies straight line trajectories in the characteristic x-t plane.
The latter result, applicable to homogeneous media only, allows us to write
down the integral of Equation 2.45 immediately as
x - r k (k) t = constant (2.46)
The fact that k(x,t) is constant along rays can, therefore, be concisely expressed
through the functional statement
k(x,t) = f{x - r k (k) t} (2.47)
Equation 2.47 represents the general solution to Equation 2-42, a nonlinear first-
order partial differential equation governing k(x,t), for the initial value problem
k(x,0) = f(x). To see why, we simply set t = 0 in Equation 2.47 and find that
k(x,0) = f(x) (2.48)
Thus, when the starting wavenumber distribution k(x,0) = f(x) is specified and
the function r (k) is known, Equation 2.47 is the general solution to the initial
value problem solving Equations 2.42 and 2.48.
We have shown that characteristics remain straight in homogeneous media,
since each initial group velocity along a ray remains unaffected and constant
during the propagation. But this is not to say that all of the initial velocities
r k (k) equal the same constant. This raises the possibility that our
characteristics can intersect within a finite time (in Equation 1.1, the
characteristics within each wave family always remain parallel). Since we have
demonstrated above that k(x,t) must remain constant along a ray, the intersection
of rays therefore implies that more than one value of k exists at a point, thereby
forming “kinematic shocks.”
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