GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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the rays from a given location in shallow water to deeper
water in a fan of directions. The corresponding relationship
between the shallow water direction of the ray and
its deep water direction provides the wave height estimate
in shallow water (Dorrestein, 1960). This technique
avoids the caustic problem by an inherent smoothing
over spectral directions.
With either of the above techniques it is relatively
simple to calculate the change in wave height for a
harmonic wave. A proper calculation for a random,
short-crested sea then proceeds with the combination of
many such harmonic waves and the inclusion of wave
generation and dissipation. The spectrum of such a
discrete spectral model is more conveniently formulated
in wavenumber (k) space. This has the advantage that the
energy density, moving along a wave ray with
the group velocity, is constant (Dorrestein, 1960).
Generation and dissipation can be added as sources
so that the energy balance moving along the wave ray
with the group velocity cg is (for stationary and nonstationary
conditions):
d E(k)
= S(k,u,h) . (7.9)
d t
This is exploited in the shallow water model of
Cavaleri and Rizzoli (1981). However, non-linear effects
such as bottom friction and non-linear wave-wave interactions
are not readily included in such a Lagrangian
technique. An alternative is the Eulerian approach where
it is relatively simple to include these effects. This technique
is fairly conventional in deep water, as described in
Section 5.3. However, in shallow water the propagation
of waves needs to be supplemented with refraction. The
inclusion of refraction in an Eulerian discrete spectral
model is conceptually not trivial. In the above wave ray
approach, a curving wave ray implies that the direction of
wave propagation changes while travelling along the ray.
In other words, the energy transport continuously
changes direction while travelling towards the coast. This
can be conceived as the energy travelling through the
geographic area and (simultaneously) from one direction
to another. The speed of directional change cθ, while
travelling along the wave ray with the group velocity, is
obtained from the above generalized Snell’s law:
(7.10)
To include refraction in the Eulerian model, propagation
through geographic x — and y — space
(accounting for rectilinear propagation with shoaling) is
supplemented with propagation through directional
space:
∂ E( ω,θ )
∂ t
E( ω,θ )
+ ∂ c θ
+ ∂ c x
[ ]
∂θ
c θ = – c g
c
[ E( ω,θ ) ]
∂ x
∂c
∂n .
+ ∂ c y
= S( ω,θ,u,h ).
GUIDE TO WAVE ANALYSIS AND FORECASTING
[ E( ω,θ ) ]
∂ y
(7.11)
In addition to the above propagation adaptations, the
source term in the energy balance requires some adaptations
in shallow water too. The deep water expressions of
wind generation, whitecapping and quadruplet wave-wave
interactions need to be adapted to account for the depth
dependence of the phase speed of the waves (WAMDI
Group, 1988; Komen et al., 1994) and some physical
processes need to be added. The most important of these
are bottom friction (e.g. Shemdin et al., 1980), bottom
induced wave breaking (e.g. Battjes and Janssen, 1978)
and triad wave-wave interactions (e.g. Madsen and
Sorensen, 1993). When such adaptations are implemented,
the above example of a confused wave ray pattern
becomes quite manageable (Figure 7.2).
7.4 Diffraction
The above representation of wave propagation is based
on the assumption that locally (i.e., within a small region
of a few wavelengths) the individual wave components
behave as if the wave field is constant. This is usually a
good approximation in the open sea. However, near the
coast, that is not always the case. Across the edges of
sheltered areas (i.e. across the “shadow” line behind
obstacles such as islands, headlands, rocks, reefs and
breakwaters), rapid changes in wave height occur and
the assumption of a locally constant wave field no longer
holds. Such large variations may also occur in the
absence of obstacles. For instance, the cumulative effects
of refraction in areas with irregular bathometry may also
cause locally large variations.
In the open sea, diffraction effects are usually
ignored, even if caustics occur. This is usually permissible
because the randomness and short-crestedness of
the waves will spatially mix any caustics all over the
geographic area thus diffusing the diffraction effects.
This is also true for many situations near the coast, even
behind obstacles (Booij et al., 1992).
However, close behind obstacles (i.e. within a few
wavelengths) the randomness and short-crestedness of
the waves do not dominate. Moreover, swell is fairly
regular and long-crested, so the short-crestedness and the
randomness are less effective in diffusing diffraction
effects. The need for including diffraction in wave
models is therefore limited to small regions behind
obstacles and to swell-type conditions and mostly
concerns the wave field inside sheltered areas such
as behind breakwaters and inside harbours (see, for
example, Figure 7.3).
For harmonic, unidirectional incident waves, a variety
of diffraction models is available. The most
illustrative model is due to Sommerfeld (1896) who
computed the penetration of unidirectional harmonic
waves into the area behind a semi-infinite screen in a
constant medium (see also Chapter 5 of Mei, 1989).
Translated to water waves, this means that the water
depth should be constant and the screen is interpreted as
a narrow breakwater (less than a wavelength in width).
Sommerfeld considers both incoming waves and