Governing Equations
Interaction of waves with currents is considered in a reference frame moving
with the current. Wave parameters in this frame are denoted with the subscript r,
for being "relative" to the current, and parameters in the nonmoving reference
frame are subscripted a, for "absolute." The wave dispersion relationship is
given in the moving reference frame as (Jonsson 1990 and others):
ω r2 = gk tanh kd
(1)
where
ω = angular frequency
k = wave number
d = water depth
In the absolute frame of reference, the dispersion equation is:
ω a = ω r + kU cos(δ - α )
(2)
where
U = current speed
δ = direction of the current relative to a reference frame (the x-axis, here)
α = wave orthogonal direction (normal to the wave crest) (Figure 2)
The wave number is solved by substituting Equation 1 into Equation 2 and
iteratively solving for k. The wave number and wavelength (L=(2π)/k) are the
same in both reference frames.
Solutions for refraction and shoaling also require wave celerities, C, and
group celerities, Cg, in both reference frames. In the reference frame relative to
the current,
ωr
Cr =
(3)
k
Cgr = 0.5Cr 1 +
2kd
(4)
sinh 2kd
6
Chapter 2 Governing Equations and Numerical Discretization
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