Wave and Current Interaction in the Ocean - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 333

applications other than water waves; for example, they can be used to model
acoustic wave motions in nonuniform winds, or MWD signal propagation in
ducts with slowly varying cross-sections.
Note that the kinematic ray equations corresponding to Equations 7.6 and
7.7 are
dk/dt =

r x (k,x,t) + U x k

(7.8)

and

dx/dt = U(x,t) + r k (k,x,t) (7.9)
where an obvious Doppler shift has been invoked. That is, to Equations 2-102
and 2-103, namely, dk/dt = - r x (k,x,t) and dx/dt = r k (k,x,t), we simply
replace r (k,x,t) by U(x,t)k + r (k,x,t). Equations 7.8 and 7.9, and then
Equations 7.6 and 7.7, may be integrated numerically as discussed in Chapter 2.
The function U(x,t) may be prescribed analytically, but it may also be specified
from known climactic conditions, say from meteorological data.
Continuing with our deep water example, where

r t (k,x,t) and r x (k,x,t)

both vanish, we clearly have

E/ t + ((U(x,t) +

r k (k))E)/ x =

= - 4 k 2 E -{k

r k (k)/ r }U x E

(7.10)

where k r k (k)/ r = 1/2 for pure gravity waves (T = 0), and 3/2 for pure
capillary waves (g = 0). But in general, k r k (k)/ r takes the value
(1+3 )/(2+2 ), where = Tk 2 / g is a dimensionless number characterizing
relative capillary-to-gravity forces. Equation 7.8 shows how these factors
modify U x , and through the sign of U x , how local flow acceleration or
deceleration affects wave energy growth. The right side of Equation 7.10 shows
that wave growth or decay depends on the relative combination of dissipation
and heterogeneity.

7.1.2 Waves in finite depth water.
Consider next the more difficult problem for “gravity waves” (that is,
ocean waves without capillary pressure effects) in water of finite depth h(x,t). It
can be shown that
r (k,x,t) = (gk tanh kh) 1/2 (7.11)
following Lamb (1945). The terms r t (k,x,t) and r x (k,x,t) in Equation 7.6 do
not vanish, since r depends explicitly upon x and t through h(x,t). Expanding
Equation 7.6, and simplifying with the mass conservation law
h/ t +

{U(x,t)h}/ x = 0 (7.12)

leads to