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Figure 3.18 The λ L - γ graph. Circles indicate data from Tables 3.2 and 3.3 . Stars show conversion
of the finite-depth γ according to (3.39) - (3.42) . Figure is reproduced from Liu & Babanin ( 2004 )
(copyright of Copernicus Publications on behalf of the European Geosciences Union)
While the clear distinction between finite-depth and deep-water conditions is not unex-
pected, it presents a tangible challenge for an analytical and physically sound interpretation
of these results. Previously, we postulated that the limiting downward acceleration is deter-
mined by the balance of the gravitational and centrifugal forces. Therefore, the difference
between the two curves should be possible to explain if we take into account the difference
in centrifugal accelerations for deep water ( a deep ) and finite depth ( a shallow ) waves of the
same frequency
ω
. Conspicuously, this can be obtained if we consider what happens at the
wave crest.
We expect that the wave surface breaks once it can no longer sustain itself for some
reason and we surmise that, whenever it happens, the downward acceleration at the sur-
face must have exceeded some threshold level we wish to resolve. Effective downward
acceleration of the real physical particles at the crest is the difference between gravita-
tional acceleration and centrifugal acceleration caused by the motion of the particle along
its orbit. The latter is given by
2 R , where R is the radius of curvature of the motion of a
ω
particle on the breaking crest:
2 R deep
a deep
a shallow =
ω
R deep
R shallow .
2 R shallow =
(3.39)
ω
Here, R deep and R shallow are now radii of curvature at the highest points of wave orbits
configured as circles in deep water and ellipsoids in finite depth. The curvature of the
two-dimensional curve defined by y
=
y
(
x
)
is
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