Part A | 4.6
92 Part A Fundamentals
There is ample evidence for the existence of very
long period of motion in the nearshore. These motions
are generally thought to be largely the result of time
variation of wave heights and breaking locations [4.42],
though contributions from nonlinear wave–wave interactions [4.43] (see also Bowen and Guza [4.44]) may
also be responsible. These long waves (often termed infragravity waves) have periods on the order of minutes
and play a significant role in the evolution of the beach
face.
The basic equations governing these long wave motions can be derived from the uniform depth equations
of motion; the details can be found in [4.45]. The result
can be expressed as an inhomogeneous wave equation
@
2
Á
@t 2
@
@x
Â
gh O
@Á
@x
Ã
@
@y
Â
gh O
@Á
@y
Ã
D F ; (4.92)
where h O is a representative water depth and Á is a mean
(wave-averaged) free surface elevation. The term F represents forcing of the long wave motion.
In many cases [4.46], the long wave motion in
the nearshore results from the transformation of long
wave energy from offshore; the long wave propagates
as a free wave. In this case, F D 0 in (4.92). The resulting motion in the nearshore is called an edge wave. The
mean free surface elevation is assumed to have a longshore periodicity
Á D A.x/e
i.kyy˙!t/
;
(4.93)
where k y D k sin  , or the longshore component of the
wave number; the wave approach direction  is assumed to be from shore normal. Substituting (4.93) into
(4.92), and further assuming a plane beach, leads to
a solution for A.x/ [4.47]
A.x/ D a n e
kyx L n .2k y x/ ;
(4.94)
where the subscript n denotes a mode of the solution
(matching the number of zero crossings of the crossshore structure of the edge wave) with amplitude a n .
The cross-shore structure is given by L n .2k y x/, which
is the Laguerre polynomial of order n. The associated
dispersion relation for this motion is
!
2
D gk y sin Œ.2n C 1/ˇ ;
(4.95)
where ˇ is the angle of the beach from horizontal.
Since there is no dissipation, the edge wave will fully
reflect from the shoreline. For n D 0, all edge waves
will propagate in the longshore direction, with crests
perpendicular to the shoreline. For n > 0, however, the
edge wave is affected by refraction; the wave reflects
seaward from the shoreline, refracts to the point where
the crests are perpendicular to the shoreline, then continues refracting back toward the shoreline. Such an
edge wave is said to be trapped. On the other hand,
a long wave which is not refracted back toward shore
is said to be leaky. The phenomenon of surf beat [4.48,
49] is considered a leaky mode.
The discussion of edge waves in the previous section is useful as an introduction to nearshore long
waves. However, because the motion is free, these long
wave frequencies must be present at reasonable energy
levels in the incident wave spectrum. The more significant source of long wave energy in the nearshore and
surf zones comes from forced motion [4.46]. In this
case, the forcing term F in (4.92) would be nonzero.
If we dictate that the forcing is (as for nearshore
circulation) dependent on the gradients of radiation
stress [4.50], then the term F becomes
F D
1
@
2 S xx
@x 2 ;
(4.96)
where the analysis is limited to processes in the crossshore direction only (longshore derivatives are zero in
(4.92)). In a wave group, the changes in radiation stress
are assumed to be due to the spatial variations of the
wave heights within the group. Using this assumption
and limiting the domain to a constant water depth h O
gives the following solution for the mean sea surface
Á D D
S xx .x; t/
.gh O c g /
;
(4.97)
where it can be seen that the mean sea surface is in antiphase with the gradient of radiation stress. Besides the
limitation to constant depth, this solution is problematic
in that Á becomes quite large as the group velocity of the
short waves in the group begins to approach the shallow
water asymptote.
Later approaches sought to remedy these issues
and move the area of interest closer to the surf zone.
Symonds et al. [4.42] represented the wave group as
a prescribed wave height variation about a mean amplitude and propagating over a sloping bottom, then used
a constant breaking height-to-depth ratio breaking index inside the surf zone. The varying wave height led to
a moving breakpoint, which served as a varying boundary condition for the generation of the long wave. The
assumption of a constant breaking index was used to
constrain the wave height in the surf zone, but destroyed
any remaining group structure therein, which was later
shown to be at least partially incorrect [4.51]. Foda and
Mei [4.52] and Schäffer and Svendsen [4.53] perused
an alternative formulation, in which the breakpoint was
fixed but allowed for the group structure to remain in
the surf zone.
The growth of the infragravity waves after this
initial generation must now be considered. Elgar
92 Part A Fundamentals
There is ample evidence for the existence of very
long period of motion in the nearshore. These motions
are generally thought to be largely the result of time
variation of wave heights and breaking locations [4.42],
though contributions from nonlinear wave–wave interactions [4.43] (see also Bowen and Guza [4.44]) may
also be responsible. These long waves (often termed infragravity waves) have periods on the order of minutes
and play a significant role in the evolution of the beach
face.
The basic equations governing these long wave motions can be derived from the uniform depth equations
of motion; the details can be found in [4.45]. The result
can be expressed as an inhomogeneous wave equation
@
2
Á
@t 2
@
@x
Â
gh O
@Á
@x
Ã
@
@y
Â
gh O
@Á
@y
Ã
D F ; (4.92)
where h O is a representative water depth and Á is a mean
(wave-averaged) free surface elevation. The term F represents forcing of the long wave motion.
In many cases [4.46], the long wave motion in
the nearshore results from the transformation of long
wave energy from offshore; the long wave propagates
as a free wave. In this case, F D 0 in (4.92). The resulting motion in the nearshore is called an edge wave. The
mean free surface elevation is assumed to have a longshore periodicity
Á D A.x/e
i.kyy˙!t/
;
(4.93)
where k y D k sin  , or the longshore component of the
wave number; the wave approach direction  is assumed to be from shore normal. Substituting (4.93) into
(4.92), and further assuming a plane beach, leads to
a solution for A.x/ [4.47]
A.x/ D a n e
kyx L n .2k y x/ ;
(4.94)
where the subscript n denotes a mode of the solution
(matching the number of zero crossings of the crossshore structure of the edge wave) with amplitude a n .
The cross-shore structure is given by L n .2k y x/, which
is the Laguerre polynomial of order n. The associated
dispersion relation for this motion is
!
2
D gk y sin Œ.2n C 1/ˇ ;
(4.95)
where ˇ is the angle of the beach from horizontal.
Since there is no dissipation, the edge wave will fully
reflect from the shoreline. For n D 0, all edge waves
will propagate in the longshore direction, with crests
perpendicular to the shoreline. For n > 0, however, the
edge wave is affected by refraction; the wave reflects
seaward from the shoreline, refracts to the point where
the crests are perpendicular to the shoreline, then continues refracting back toward the shoreline. Such an
edge wave is said to be trapped. On the other hand,
a long wave which is not refracted back toward shore
is said to be leaky. The phenomenon of surf beat [4.48,
49] is considered a leaky mode.
The discussion of edge waves in the previous section is useful as an introduction to nearshore long
waves. However, because the motion is free, these long
wave frequencies must be present at reasonable energy
levels in the incident wave spectrum. The more significant source of long wave energy in the nearshore and
surf zones comes from forced motion [4.46]. In this
case, the forcing term F in (4.92) would be nonzero.
If we dictate that the forcing is (as for nearshore
circulation) dependent on the gradients of radiation
stress [4.50], then the term F becomes
F D
1
@
2 S xx
@x 2 ;
(4.96)
where the analysis is limited to processes in the crossshore direction only (longshore derivatives are zero in
(4.92)). In a wave group, the changes in radiation stress
are assumed to be due to the spatial variations of the
wave heights within the group. Using this assumption
and limiting the domain to a constant water depth h O
gives the following solution for the mean sea surface
Á D D
S xx .x; t/
.gh O c g /
;
(4.97)
where it can be seen that the mean sea surface is in antiphase with the gradient of radiation stress. Besides the
limitation to constant depth, this solution is problematic
in that Á becomes quite large as the group velocity of the
short waves in the group begins to approach the shallow
water asymptote.
Later approaches sought to remedy these issues
and move the area of interest closer to the surf zone.
Symonds et al. [4.42] represented the wave group as
a prescribed wave height variation about a mean amplitude and propagating over a sloping bottom, then used
a constant breaking height-to-depth ratio breaking index inside the surf zone. The varying wave height led to
a moving breakpoint, which served as a varying boundary condition for the generation of the long wave. The
assumption of a constant breaking index was used to
constrain the wave height in the surf zone, but destroyed
any remaining group structure therein, which was later
shown to be at least partially incorrect [4.51]. Foda and
Mei [4.52] and Schäffer and Svendsen [4.53] perused
an alternative formulation, in which the breakpoint was
fixed but allowed for the group structure to remain in
the surf zone.
The growth of the infragravity waves after this
initial generation must now be considered. Elgar