Part A | 4.6
90 Part A Fundamentals
4.6 Transformation of Waves Approaching Land
As waves approach the shoreline they undergo significant transformation in response to the arbitrarily
varying bathymetry. Refraction and shoaling (as discussed above) will occur, but cannot be described by the
Snell’s law formulation. In addition, wave diffraction
(the flow of energy along a wave crest due to high local gradients of wave height, and most often associated
with waves encountering breakwaters) is also possible
over bathymetric shoals.
The mild slope equation, developed by Berkhoff [4.23], has been used to simulate wave propagation
over arbitrarily varying bathymetry
r r
CC g r
C k
2 CC g D 0 ;
(4.78)
where r refers to differential operations in the horizontal (x; y) directions. It can be shown that the equation
reduces to the Helmholtz equation for a flat bottom,
and that it can also be transformed into a set of coupled equations for wave refraction by neglecting curvature terms. The equation is elliptic, which requires
prespecification of the boundaries in advance. This is
straightforward for enclosed harbors but challenging for
open coasts, in which the location of wave breaking is
unknown in advance. This has been addressed in the
model’s elliptic form [4.24]. However, the model has
been transformed into a parabolic form [4.25], which
affords a more straightforward numerical solution.
Kirby [4.26] outlines several limitations of the parabolic
form and offers some measures for ameliorating these
limitations. The mild slope equation has been augmented with nonlinear interaction terms [4.20, 22, 27].
In the nearshore area, wave breaking is primarily
controlled by the proximity of the bottom. The wave
transformation process will force the wave to reach
a limiting wave height, beyond which it cannot sustain
its form. At this point the wave breaks. The breaking
process and resultant generation of white water in the
nearshore (the surf zone) is a major engine of nearshore
processes (current generation, rip currents, infragravity
waves, sediment transport, etc.)
The location of the initial breaking (or incipient
breaking) of the wave demarcates the outer edge of the
surf zone. It is important to identify conditions leading
to incipient breaking for several reasons. For example,
it is important in numerical modeling of wave propagation in the nearshore, since the wave breaking and dissipation terms are only activated within the surf zones,
so it is essential to know when they need to be active.
One of the first incipient breaking criteria was developed by McCowan [4.28], who adapted the deep
water solution of Stokes [4.10] to shallow water. Both
studies assumed that a wave broke when the particle
acceleration at the crest exceeds gravitational acceleration g, and showed that this condition resulted in an
included crest angle of 120
o . McCowan [4.28], however, also showed that, at this breaking condition
H b
h b
D 0:78 ;
(4.79)
or that the wave height reaches approximately 80% of
the water depth at breaking. This criterion was developed using the properties of a solitary wave over a flat
bottom, which would not generally fit the model of
oscillatory motions in the surf zone. In an alternative
treatment, Miche [4.29] developed a breaking criterion
connected with wave steepness
 H b
L b
Ã
D
1
7
tanh kh b :
(4.80)
Note that this criterion approaches the maximum steepness criterion of Stokes [4.10] in deep water (large kh)
and H b =h b D 0:9 in shallow water. A more complete development of the breaking criterion was developed by
Weggel [4.30], who examined results from a number of
laboratory experiments and determined that the bottom
slope played an important role in the initiation of breaking. The resulting criterion is
H b
h b
D b.m/ a.m/
H b
gT 2 ;
(4.81)
where
a.m/ D 43:8 .1 e19m/ ;
(4.82)
b.m/ D
1:56
.1 C e 19:5m /
:
(4.83)
This criterion reduces to (4.79) when the slope m D 0.
Equation (4.81) is implicit for the breaking wave height
H b , so iteration is required.
Once the wave has broken, a description of the decay of the wave energy in the surf zone is needed. In this
section, we discuss wave height decay mechanisms for
both monochromatic and random waves. In general, the
assumption of a spilling breaker is used; random wave
formulations are typically determined by marrying this
assumption to a probability distribution of some kind.
The equivalence between the energy decay in
a spilling breaking wave and that of a hydraulic jump
has been well established [4.31]. Horikawa and Kuo
[4.32] conducted laboratory tests on breaking waves
and determined the existence of a stable energy flux,
which defines a state at which waves no longer break.
This concept was further developed into a general description of wave evolution in the surf zone [4.33], with
the hypothesis that the change in energy flux in the surf
90 Part A Fundamentals
4.6 Transformation of Waves Approaching Land
As waves approach the shoreline they undergo significant transformation in response to the arbitrarily
varying bathymetry. Refraction and shoaling (as discussed above) will occur, but cannot be described by the
Snell’s law formulation. In addition, wave diffraction
(the flow of energy along a wave crest due to high local gradients of wave height, and most often associated
with waves encountering breakwaters) is also possible
over bathymetric shoals.
The mild slope equation, developed by Berkhoff [4.23], has been used to simulate wave propagation
over arbitrarily varying bathymetry
r r
CC g r
C k
2 CC g D 0 ;
(4.78)
where r refers to differential operations in the horizontal (x; y) directions. It can be shown that the equation
reduces to the Helmholtz equation for a flat bottom,
and that it can also be transformed into a set of coupled equations for wave refraction by neglecting curvature terms. The equation is elliptic, which requires
prespecification of the boundaries in advance. This is
straightforward for enclosed harbors but challenging for
open coasts, in which the location of wave breaking is
unknown in advance. This has been addressed in the
model’s elliptic form [4.24]. However, the model has
been transformed into a parabolic form [4.25], which
affords a more straightforward numerical solution.
Kirby [4.26] outlines several limitations of the parabolic
form and offers some measures for ameliorating these
limitations. The mild slope equation has been augmented with nonlinear interaction terms [4.20, 22, 27].
In the nearshore area, wave breaking is primarily
controlled by the proximity of the bottom. The wave
transformation process will force the wave to reach
a limiting wave height, beyond which it cannot sustain
its form. At this point the wave breaks. The breaking
process and resultant generation of white water in the
nearshore (the surf zone) is a major engine of nearshore
processes (current generation, rip currents, infragravity
waves, sediment transport, etc.)
The location of the initial breaking (or incipient
breaking) of the wave demarcates the outer edge of the
surf zone. It is important to identify conditions leading
to incipient breaking for several reasons. For example,
it is important in numerical modeling of wave propagation in the nearshore, since the wave breaking and dissipation terms are only activated within the surf zones,
so it is essential to know when they need to be active.
One of the first incipient breaking criteria was developed by McCowan [4.28], who adapted the deep
water solution of Stokes [4.10] to shallow water. Both
studies assumed that a wave broke when the particle
acceleration at the crest exceeds gravitational acceleration g, and showed that this condition resulted in an
included crest angle of 120
o . McCowan [4.28], however, also showed that, at this breaking condition
H b
h b
D 0:78 ;
(4.79)
or that the wave height reaches approximately 80% of
the water depth at breaking. This criterion was developed using the properties of a solitary wave over a flat
bottom, which would not generally fit the model of
oscillatory motions in the surf zone. In an alternative
treatment, Miche [4.29] developed a breaking criterion
connected with wave steepness
 H b
L b
Ã
D
1
7
tanh kh b :
(4.80)
Note that this criterion approaches the maximum steepness criterion of Stokes [4.10] in deep water (large kh)
and H b =h b D 0:9 in shallow water. A more complete development of the breaking criterion was developed by
Weggel [4.30], who examined results from a number of
laboratory experiments and determined that the bottom
slope played an important role in the initiation of breaking. The resulting criterion is
H b
h b
D b.m/ a.m/
H b
gT 2 ;
(4.81)
where
a.m/ D 43:8 .1 e19m/ ;
(4.82)
b.m/ D
1:56
.1 C e 19:5m /
:
(4.83)
This criterion reduces to (4.79) when the slope m D 0.
Equation (4.81) is implicit for the breaking wave height
H b , so iteration is required.
Once the wave has broken, a description of the decay of the wave energy in the surf zone is needed. In this
section, we discuss wave height decay mechanisms for
both monochromatic and random waves. In general, the
assumption of a spilling breaker is used; random wave
formulations are typically determined by marrying this
assumption to a probability distribution of some kind.
The equivalence between the energy decay in
a spilling breaking wave and that of a hydraulic jump
has been well established [4.31]. Horikawa and Kuo
[4.32] conducted laboratory tests on breaking waves
and determined the existence of a stable energy flux,
which defines a state at which waves no longer break.
This concept was further developed into a general description of wave evolution in the surf zone [4.33], with
the hypothesis that the change in energy flux in the surf