Wave Action Flux Conservation Equation
For waves propagating on a current, it is the wave action flux that is
conserved rather than the wave energy flux. Wave action, defined as the
wave energy divided by the relative angular frequency, was originally
introduced by Bretherton and Garrett (1969). Jonsson, Skovgaard, and Wang
(1970) employed a similar concept for water waves. The conservation
equation for steady conditions derived by Jonsson and Christoffersen (1984),
including energy dissipation produced by wave breaking and bottom friction,
is employed here. Thus, to calculate the wave transformation across the
profile, the following conservation equation is solved:
d ECga cos β
PD + Pf
=
(1)
ωr
ωr
dx
where
E = wave energy (linear wave theory employed here)
Cga = absolute wave group speed
β = wave ray direction
relative wave period)
PD and Pf
= wave energy dissipation due to wave breaking
and bottom friction, respectively
x = cross-shore coordinate pointing offshore
The energy dissipation due to bottom friction is typically small compared to
following.
As discussed in the following paragraphs, wave action (and energy) is
conserved along the wave rays, typically differing from the wave orthogonals
that describe the direction in which the wave fronts move.
Wave Kinematics
Consider waves propagating on a steady current having a magnitude U
and direction δ (see Figure 1 for a definition sketch of the current and wave
angles used here; overbar denotes a vector). The waves propagate at an angle
α yielding the following absolute phase speed (Ca) for the waves,
Ca = Cr + U cos(δ - α)
(2)
where Cr = relative phase speed. The current is taken positive if it is in the
direction of the wave propagation (following current) and negative if it is
against the wave propagation (opposing current). This definition is intuitive
and conventional, even though it means that a positive current will flow in
the opposite direction to the x-axis according to Figure 1. Also, in the
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Chapter 3 Wave Model
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