Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 65

(k)

r (k)

i

i (k)

(2.71)

We assume that
i (k)/ r (k) O( ) (2.72)
where << 1 is a small dimensionless number (e.g., one proportional to Q -1 in
Equation 2.36).
This flexibility allows nontrivial wave motions that are not highly or
critically damped. Here, k is a real wavenumber, and

r (k) and i (k) are the

real and imaginary parts of the complex frequency (k) respectively. The

general Fourier superposition integral can be written as

F(x,t) = B(k) e i(kx-

t) dk

(2.73)

where the limits (- , ) of integration are omitted for clarity. Equation 2.73
extends the simpler integral of Equation 2.3 in order to model dissipative effects.
In any particular problem, the function B(k) would be proportional to the
Fourier transform of the initial condition; this function exists, of course, when
the initial disturbances are suitably confined.

2.4.2 The general (x,t) equation.
If we now focus our attention at the neighborhood of a (real) center
wavenumber k = k 0 , with a corresponding (real) center frequency

r 0 , we can

certainly write

F(x,t) = (x,t) exp i(k 0 x -

r 0 t)

(2.74)

Equation 2.74 represents the first step of many that may not be obvious to the
reader. Its strategic role was clear only after intensive thought, upon which
Equations 2.94 and 2.95 would emerge. The reader who wishes to proceed
directly to those results may do so without break in continuity. In this explicit
(and still exact) representation, F(x,t) contains a purely periodic part, together
with a complex amplitude function

(x,t) that incorporates dissipative effects.

The latter function identically satisfies

(x,t) = B(k) exp i[ (k-k 0 )x- ( -

r 0 )t] dk = G(k) dk

(2.75)

where G(k) has been introduced to simplify our notation.
Now, the complex dispersion relation given above holds for every
component wave solution. Taylor expansion about centered properties gives
nk (k 0 )(k-k 0 ) n /n!

(2.76)

where the summation is taken from n = 0 to . Applying the operator

r 0 )t] dk

B(k) exp i[ (k-k 0 )x- ( - (2.77)