Wave and Current Interaction in the Ocean - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 335

In the limit when U(x,t) depends on x only, and dynamically steady waves
having a constant frequency

0 are considered, Equation 7.17 leads directly to

U(x)k + s(gk) 1/2 =

0 > 0

(7.18)

where s = 1 (see Equation 2-116). That is, k 1/2 (x) can be explicitly solved,
using the quadratic formula for second-order algebraic equations.

The consequences of Equation 7.18, the result of an innocuous “Uk”
background term, may be surprising, but they describe the “kinematic barrier”
well known to oceanographers (Chin, 1976, 1979, 1980b, 1981) - an analogous
“kinematic barrier” was uncovered in Chapter 4, where it was shown how
variations in axial loading along a drillstring can trap violent lateral vibrations.
The present discussion proceeds generally if we introduce the nondimensional
variables U * and k * defined by
U * = 4 U

0 /g

(7.19)

and

(k * ) 1/2 = (gk) 1/2 /2

0

(7.20)

Thus, Equation 7.18 can be written in the concise form

Uk * + 2 s k *1/2 = 1 (7.21)

Now, we introduce the corresponding dimensionless group velocity C g *
C g * = U * + s k *-1/2 (7.22)
Equations 7.21 and 7.22 can be rearranged to form U * = (1- 2sk *1/2 )/k and
C g * = U * + s k *-1/2 = 1 / k * - s/k *1/2 . A complete family of solutions is produced
by varying the parameter k * throughout a range of positive values. The
solutions so obtained are shown in Figure 7.1, where C g * is plotted against U * .
Zeros in the group velocity C g * , as discussed earlier in this topic, identify areas
where energy propagation ceases. The resulting trapping of wave energy
implies high amplitude wave motions. We next discuss Figure 7.1.
First, we note that along ABCDE, the segments ABCD and DE are derived
using s = +1 and -1, respectively; the corresponding (dimensionless)
wavenumber solutions along each of these curves are directly indicated. Now,
consider a wave approaching C from A. As the wave nears C, the group
velocity decreases to zero, thus producing singular values in the wave energy
density E, while the dimensionless wavenumber increases to unity. In the
velocity range -1 < U * < 0, the medium supports two sets of waves, one with
positive and the other with negative group velocity. Thus, the wave approaching
C from A may reflect from the focus along the second branch, but it then