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5.2 WATER WAVE MECHANICS 97
Table 5.2 Breaking Wave Characteristics and the Surf Similarity Parameter
ζ→ ≈0.1 0.5 1.0 2.0 3.0 4.0 5.0
Breaker Collapsing/ None
Type Spilling Plunging Surging (Reflection)
κ 0.8 1.0 1.1 1.2
N 6–7 2–3 1–2 <1 <1
r 10−3 10−2 0.1 0.4 0.8
κ = breaking index; N = number of waves in surf zone; r = reflection from beach.
Source: Battjes (1974).
defined as the ratio of the beach slope, tanβ, to the square root of the deep water
wave steepness, by the following expression:
ζ = tanβ/
√
H0/L0 (5.17)
His results are shown in Table 5.2, which shows the breaker type, the breaking index,
the number of waves in the surf zone, and the reflection coefficient from a beach as
a function of the surf similarity parameter.
At first, simple theoreticalmodelswere proposed to predict breaking. Theoretical
studies of solitary waves (a single wave of elevation caused, for example, by the
displacement of a wavemaker in one direction only) in constant-depth water showed
that the wave breaks when its height exceeds approximately 0.78 of the water depth.
This led to the widespread use of the so-called spilling breaker assumption that the
wave height within the surf zone is a linear function of the local water depth H = κh,
where κ , the breaker index, is on the order of 0.8. Later experiments with periodic
waves pointed out that the bottom slope was important as well, leading to elaborate
empirical models for breaking (e.g., Weggel 1972). The spilling breaker assumption,
however, always leads to a linear dependency of wave height with water depth. In the
laboratory, thewave height is often seen to decreasemore rapidly at the breaking line
than farther landward. In the field, on the other hand, Thornton and Guza (1982),
showed that the root-mean-square wave height in the surf zone (on their mildly
sloping beach) was reasonably represented by Hrms = 0.42h.
Dally, Dean, and Dalrymple (1985) developed a wave-breaking model based on
the concept of a stable wave height within the surf zone for a given water depth.
This model has two height thresholds, each of which depends on the water depth.
As waves shoal up to the highest threshold (a breaking criteria), breaking com-
mences. Breaking continues until the wave height decreases to the lower threshold
(a stable wave height). This stable wave height concept appears in experiments by
Horikawa and Kuo (1966), which involved creating a breaking wave on a slope.
This breaking wave then propagates into a constant depth region. Measurements
of the wave height along the wave tank showed that the waves approach a stable
(broken) wave height of H = �h, where � is about 0.35–0.40 in the constant depth
region.
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98 WAVES AND WAVE-INDUCED HYDRODYNAMICS
The Dally et al. model, valid landward of the location of initial breaking, is ex-
pressed in terms of the conservation of energy equation
∂ECg
∂x
= −K
h
[ECg − (ECg)s], (5.18)
where K is an empirical constant equal to approximately 0.17. For shallow water, the
energy flux can be reduced to
ECg = 18ρgH
2
√
gh,
and thus the preceding equation relates the wave height H to the water depth h.
For planar beaches, where the depth is given by h = mx , an analytic solution can be
obtained,
H
Hb
=
√√√√[( h
Hb
)( K
m
− 12 )
(1+ α)− α
(
h
Hb
)2]
, (5.19)
where m is the beach slope and
α = K�
2
m
( 5
2 − Km
) (Hb
Hb
)2
For the special case of K/m = 5/2, a different solution is necessary:
H
Hb
=
(
h
Hb
)√[
1− β ln
(
h
Hb
)]
(5.20)
Here,
β = 5
2
�2
(
Hb
Hb
)2
The range of solutions to Eqs. (5.19) and (5.20) are shown in Figure 5.3. Note that
for K/m > 3, the wave heights can be much less than predicted by a spilling wave
assumption (which coincides approximately with the K/m = 3 curve), whereas for
Figure 5.3 Wave height variation predicted across
a planar beach (Dally et al. 1985, copyright by the
American Geophysical Union).
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5.2 WATER WAVE MECHANICS 99
Figure 5.4 Comparisons of analytic solution of Dally et al. (1985,
copyright by the American Geophysical Union) with laboratory data
of Horikawa and Kuo (1966).
steeper beaches, K/m < 3, the wave heights are larger. Comparisons with Horikawa
and Kuo data for waves breaking on a planar beach are shown in Figure 5.4.
For more complicated beach profiles, numerical solutions to Eq. (5.18) are used.
Dally (1990) extended this model to include a realistic surf zone by shoaling a distri-
bution of wave heights rather than a monochromatic wave train.
Wave breaking is one of the most difficult hydrodynamics problems. The highly
nonlinear and turbulent nature of the flow field has prevented the development of a
detailed model of wave breakers, which has spurred the development of macroscale
models.
Peregrine and Svendsen (1978) developed a wake model for turbulent bores,
arguing that, from a frame of referencemovingwith thewave, the turbulence spreads
from the toe of the bore into the region beneath the wave like a wake develops.
A wave-breaking model for realistic wave fields was proposed by Battjes and
Janssen (1978), who also utilized the conservation of energy equation, but the loss of
wave energy in the surf zone was represented by the analogy of a turbulent hydraulic
jump for the wave bore in the surf zone. Further, the random nature of the wave
field was incorporated by breaking only the largest waves in the distribution of wave
heights at a point. In the field, thismodelwas extended byThornton andGuza (1983),
who were able to predict the root-mean-square wave height from the shoaling zone
to inside the surf zone to within 9 percent.
Svendsen (1984)developed the rollermodel,which is basedon thebore–hydraulic
jump model. The roller is a recirculating body of water surfing on the front face of
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100 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Figure 5.5 Schematic of vertically descending eddies with the arrow show-
ing the direction of breaker travel (from Nadaoka 1986).
the wave after breaking is initiated. This roller has mass and momentum that must
be accounted for in the governing equations. He showed that agreement between
theory and setup measurements was better with the roller model than a breaking-
index model.
Measurements of breaking waves in the laboratory have provided valuable in-
sights into the nature of the breaking process. Nadaoka, Hino, and Koyano (1989)
have shown that spilling breakers produce what they refer to as “obliquely descend-
ing eddies,” which are near-vertical vortices that remain stationary after the breaker
passes. These eddies descend to the bottom, pulling bubbles down into the water
column. Figure 5.5 shows a schematic of the eddies they observed. The role of the
eddies in nearshore mixing processes (both of mass and momentum) is as yet un-
known; further, the generation mechanism, which implies the rotation of horizontal
vorticity due to the roller on the face of a spilling breaker, into near-vertical vorticity,
is at yet unknown. However, Nadaoka, Ueno, and Igarashi (1988) observed these
eddies in the field and showed that they are an important mechanism for the suspen-
sion of sediment and that the bubbles, drawn into the vortices, provide a buoyancy
that creates an upwelling of sediment after the wave passage.
5.2.5 MEAN WAVE QUANTITIES
Associated with the passage of periodic waves are some useful quantities found by
averaging in time over the wave period. For example, there is a mean transport of
water toward the shoreline, the mass transport, which is not predicted by the linear
Airy theory, which assumes that each water particle under a waveform is traveling
in a closed elliptical orbit. We define the mass transport as
M = 1
t2 − t1
∫ t2
t1
∫ η
−h