Wave and Current Interaction in the Ocean - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 334

E/ t + ((U(x,t) + r k (k,x,t))E)/ x =
= - 4 k 2 E -{1/2 + 2kh/sinh(2kh)}U x E (7.13)
In deep water, kh , and “2kh/sinh(2kh)” vanishes, thus leaving the “1/2”
obtained in the deep water example above. Equation 7.13 agrees with Longuet-
Higgins and Stewart (1964), who apply detailed perturbation methods.

7.2 Sources of Hydrodynamic Loading
In offshore applications, three types of loading are important: hydraulic
pressures imposed by mean currents (that is, U(x,t)), induced loads due to vortex
oscillations arising from unsteady flow separation, and finally, dynamic forces
that arise from wave interactions. The first two do not fall within the scope of
this topic. The last may be estimated in various ways. For example, direct
tabulations for E(x,t) and M(x,t) will provide useful statistical information on
local wave elevation. Numerical solution of the three-dimensional KWT initial
value problem formulated in Chapter 2, for example, may determine how storm
effects are likely to propagate in oceans with variable currents, and how
dissipation will affect different types of initial disturbances.
On the other hand, the results of wave impact are more accurately assessed
by following the net energy
x 2
(t) = E(x,t) dx (7.14)

x 1
between two rays. If the ray integration discussed in Chapter 2 is applied to
Equations 7.6 and 7.14, we have
x 2
d (t)/dt = E{2

i +{ r t (k,x,t)+U r x (k,x,t)-k r k (k,x,t)U x }/ r }dx (7.15)

x 1
and similarly for total wave momentum. Once k(x,t) and E(x,t) are determined,
Equation 7.15 can be integrated to provide potential impact information.

7.3 Instabilities Due to Heterogeneity
In Chapter 3, we showed that gravity waves in deep water satisfy the
dispersion relation
r (k) = (gk) 1/2 (7.16)
where we have retained the “ ” to describe two possible wave families
propagating in opposite directions. When a background flow with mean speed
U(x,t) exists, our dispersion relation is modified by a Doppler shift, so that
= U(x,t)k

(gk) 1/2 (7.17)