Modeling of Coastal Waves and Hydrodynamics 27.1 Wind Wave Modeling 599
Part C | 27.1
waves in very shallow water. However, even outside
these cases, quick wave property calculations based on
the above will still provide a useful preliminary reference point for shallow water wind waves.
Linear wave transformation in shallow water includes the effects of shoaling, refraction and diffraction,
and can be expressed in the general form
H
H o
D K S K R K D ;
where H represents the wave height at some coastal location, H o the height at some offshore location, K S is the
shoaling factor, K R is the refraction factor, and K D is the
diffraction factor. Assuming no dissipation of energy
during propagation, the conservation of wave power between any two locations in the propagation path gives
the shoaling coefficient
K s D
r
n o L o
nL
;
where
n D
1
2
Ä
1 C
2kh
sinh .2kh/
:
This n value varies from 0.5 for a deep water wave to 1.0
for a shallow water wave. It is remarked that the basic
linear theory presented here does not include bottom reflection or bottom dissipation, and can thus be applied
between any two points regardless of the bathymetry
between these two points. However, the theory implicitly assumes inviscid propagation and a mildly sloping
bottom, and this must be satisfied at a minimum for the
shoaling estimation above to be physically appropriate.
Wave refraction is the process of wave crests bending due to propagation-normal variations in bathymetry.
Wave crests will bend into shallow water regions as the
propagation speed slows in such regions, with the refraction coefficient
K R D
r
cos ˛ o
cos ˛
;
where ˛ is the angle of incidence relative to the shoreline or some prescribed depth contour, and can be found
using Snell’s Law
sin ˛ D
c
c o
sin ˛ o :
This relation may only be used in simple bathymetries
with shore-parallel bottom contours. In more complex
regions, modified forms of Snell’s law can be used
to create wave rays, which were commonly used in
the twentieth century to show wave paths and outline possible areas of wave focusing and de-focusing.
These techniques have been displaced by numerical
methods, some of which are discussed in later sections
of this chapter. There is no general equation for the
diffraction coefficient. The engineer should reference
the numerous diffraction diagrams that can be found in
the Coastal Engineering Manual CEM) [27.3], which
plot K D surfaces for various combinations of breakwater configurations and incident wave directions.
Once the wave reaches the break point, the coastal
transformation equation given above can no longer be
used, and some other approach must be used to cap the
wave height. A common engineering approach is the
introduction of the breaker index, which is the ratio of
the breaking wave height to the local water depth. For
a given wave period and bathymetry slope, a breaker index can be estimated. Once the shoaling and refraction
analysis estimates a wave height to depth ratio that exceeds the breaker index, it is assumed the wave starts to
break; this is the break point. Breaking is then usually
assumed to be depth limited, meaning that the breaker
index remains constant as the water depth decreases
onshore. The difficulty in this approach is determining a reliable estimation of the breaker index. There
are numerous different equations for this index; the
most common are found in the CEM. For waves shoaling over a very steep slope, the breaker index can be
greater than 1.0, while for waves breaking over an essentially flat bottom, the breaker index may be lower
than 0.4 [27.4]. A commonly assumed value for the
breaker index found in engineering studies is 0.78; this
is derived from the maximum theoretical solitary wave
height, and its physical relevance to random coastal
wind wave breaking is questionable.
For studies that seek to use straightforward analysis
methods but also provide an estimate of the wave height
envelope across a cross-shore beach profile, the energy
flux method [27.5] is a useful tool. Here, it is assumed
that the spatial rate of change of wave energy flux F is
due to some dissipation function '
dF
dx
D D'.x/ :
The dissipation function is driven by breaking and is
proportional to the difference between the local wave
energy flux and some stable energy flux for the local
depth. Determination of a stable energy flux is, similarly to the determination of a breaker index, difficult
for a wide range of conditions. Commonly used models for the dissipation function include [27.6] and [27.7],
which are both established for random wave break-
Part C | 27.1
waves in very shallow water. However, even outside
these cases, quick wave property calculations based on
the above will still provide a useful preliminary reference point for shallow water wind waves.
Linear wave transformation in shallow water includes the effects of shoaling, refraction and diffraction,
and can be expressed in the general form
H
H o
D K S K R K D ;
where H represents the wave height at some coastal location, H o the height at some offshore location, K S is the
shoaling factor, K R is the refraction factor, and K D is the
diffraction factor. Assuming no dissipation of energy
during propagation, the conservation of wave power between any two locations in the propagation path gives
the shoaling coefficient
K s D
r
n o L o
nL
;
where
n D
1
2
Ä
1 C
2kh
sinh .2kh/
:
This n value varies from 0.5 for a deep water wave to 1.0
for a shallow water wave. It is remarked that the basic
linear theory presented here does not include bottom reflection or bottom dissipation, and can thus be applied
between any two points regardless of the bathymetry
between these two points. However, the theory implicitly assumes inviscid propagation and a mildly sloping
bottom, and this must be satisfied at a minimum for the
shoaling estimation above to be physically appropriate.
Wave refraction is the process of wave crests bending due to propagation-normal variations in bathymetry.
Wave crests will bend into shallow water regions as the
propagation speed slows in such regions, with the refraction coefficient
K R D
r
cos ˛ o
cos ˛
;
where ˛ is the angle of incidence relative to the shoreline or some prescribed depth contour, and can be found
using Snell’s Law
sin ˛ D
c
c o
sin ˛ o :
This relation may only be used in simple bathymetries
with shore-parallel bottom contours. In more complex
regions, modified forms of Snell’s law can be used
to create wave rays, which were commonly used in
the twentieth century to show wave paths and outline possible areas of wave focusing and de-focusing.
These techniques have been displaced by numerical
methods, some of which are discussed in later sections
of this chapter. There is no general equation for the
diffraction coefficient. The engineer should reference
the numerous diffraction diagrams that can be found in
the Coastal Engineering Manual CEM) [27.3], which
plot K D surfaces for various combinations of breakwater configurations and incident wave directions.
Once the wave reaches the break point, the coastal
transformation equation given above can no longer be
used, and some other approach must be used to cap the
wave height. A common engineering approach is the
introduction of the breaker index, which is the ratio of
the breaking wave height to the local water depth. For
a given wave period and bathymetry slope, a breaker index can be estimated. Once the shoaling and refraction
analysis estimates a wave height to depth ratio that exceeds the breaker index, it is assumed the wave starts to
break; this is the break point. Breaking is then usually
assumed to be depth limited, meaning that the breaker
index remains constant as the water depth decreases
onshore. The difficulty in this approach is determining a reliable estimation of the breaker index. There
are numerous different equations for this index; the
most common are found in the CEM. For waves shoaling over a very steep slope, the breaker index can be
greater than 1.0, while for waves breaking over an essentially flat bottom, the breaker index may be lower
than 0.4 [27.4]. A commonly assumed value for the
breaker index found in engineering studies is 0.78; this
is derived from the maximum theoretical solitary wave
height, and its physical relevance to random coastal
wind wave breaking is questionable.
For studies that seek to use straightforward analysis
methods but also provide an estimate of the wave height
envelope across a cross-shore beach profile, the energy
flux method [27.5] is a useful tool. Here, it is assumed
that the spatial rate of change of wave energy flux F is
due to some dissipation function '
dF
dx
D D'.x/ :
The dissipation function is driven by breaking and is
proportional to the difference between the local wave
energy flux and some stable energy flux for the local
depth. Determination of a stable energy flux is, similarly to the determination of a breaker index, difficult
for a wide range of conditions. Commonly used models for the dissipation function include [27.6] and [27.7],
which are both established for random wave break-