Mechanics of Ocean Waves 4.5 Shallow Water Wave Theories 87
Part A | 4.5
D
ij D
.
p
R i
p
R j /
p
R j .k
2
i R
2
i /
p
R i .k
2
j R
2
j /
C2.
p
R i
p
R j /
2
.k i k j C R i R j /
.
p
R i
p
R j / 2 k
ij tanhk
ij h
;
(4.52)
where
k
˙
ij D jk i ˙ k j j
(4.53)
R n D
!
2
n
g
:
(4.54)
The associated expression for the free surface elevation
is
Á D
1
X
nD1
A n
2
e
i.knx!ntCn/
C
1
X
iD1
1
X
jD1
1
8
A i A j
"
D
C
ij k i k j C R i R j
p
R i R j
C R i C R j
#
e
i. i C j/
C
1
X
iD1
1
X
jD1
1
8
A i A j
"
D
ij k i k j R i R j
p
R i R j
C R i C R j
#
e
i. i j/ :
(4.55)
The terms i C j and i j denote the sum and difference interactions, respectively. Of particular importance
is the difference interaction i j , a much longer wave
than those wave components responsible for its generation and which is considered to have a role in harbor
oscillation and seiching.
The waves generated by this second-order mechanism, as mentioned above, will not grow without limit,
as they do not satisfy the linear dispersion relation and,
thus, are not free waves. Resonant interactions, which
generate waves that do satisfy the linear dispersion relation, are a particular subset of interactions. It can be
shown [4.13] that no resonant interactions appear at second order (save for one trivial set), and one must go
to third order to determine these interactions. The application of multiple scale techniques to address these
resonances was first performed by Benney [4.14], and
the resulting evolution equations are quite complicated;
however, the resonance conditions can be written generically as follows
l C m p D n ;
(4.56)
where l, m, and p are indices that represent wave
components which interact with n . These quartet interaction terms have been approximated for wind wave
generation models [4.15] and are a primary engine
for energy transfer within the spectrum in wind wave
generation.
4.5 Shallow Water Wave Theories
In contrast to deep water, there is a distinct difference
in the scale of variability between horizontal and vertical motions in shallow water (small kh). The horizontal
motions are scaled similarly to deep water, but the vertical motions for small kh are
w D
@
@z
D
gkH
2!
sinh k.z C h/
cosh kh
sin.kx !t/
gHk
2
.z C h/
!
O.1/ ;
(4.57)
since sinh k.z C h/ k.z C h/ for small kh. Using the
shallow water asymptote of the linear dispersion relation and rearranging the shallow water asymptote of the
vertical velocity w
w
gHk
2
.z C h/
!
D
gak
2 h
!
1 C
z
h
Á
D
a
h
!
k
kh
1 C
z
h
Á
D ıı
!
k
Á Á
1 C
z
h
Á
:
(4.58)
As is the case with in deep water, two small parameters (ı D a=h and D kh) are evident and are used for
scaling. However, in this case, the vertical motions are
much smaller than the horizontal motions. The shallow water asymptote of the linear dispersion relation,
in addition to the scale difference between horizontal
and vertical motions, is now applied to the boundary
value problem. Unlike the deep water case, the governing equation here is altered
@
2
@z 2 C
2 @
2
@x 2 D 0 h Ä z Ä Á ;
(4.59)
and the disparity between horizontal and vertical motions is clear. The bottom boundary condition remains
unchanged. The dynamic free surface boundary condition becomes
2
Â
gÁ C
@
@t
Ã
C
ı
2
2
 @
@x
à 2
C
 @
@z
à 2
!
D 0
on z D Á ;
(4.60)
Part A | 4.5
D
ij D
.
p
R i
p
R j /
p
R j .k
2
i R
2
i /
p
R i .k
2
j R
2
j /
C2.
p
R i
p
R j /
2
.k i k j C R i R j /
.
p
R i
p
R j / 2 k
ij tanhk
ij h
;
(4.52)
where
k
˙
ij D jk i ˙ k j j
(4.53)
R n D
!
2
n
g
:
(4.54)
The associated expression for the free surface elevation
is
Á D
1
X
nD1
A n
2
e
i.knx!ntCn/
C
1
X
iD1
1
X
jD1
1
8
A i A j
"
D
C
ij k i k j C R i R j
p
R i R j
C R i C R j
#
e
i. i C j/
C
1
X
iD1
1
X
jD1
1
8
A i A j
"
D
ij k i k j R i R j
p
R i R j
C R i C R j
#
e
i. i j/ :
(4.55)
The terms i C j and i j denote the sum and difference interactions, respectively. Of particular importance
is the difference interaction i j , a much longer wave
than those wave components responsible for its generation and which is considered to have a role in harbor
oscillation and seiching.
The waves generated by this second-order mechanism, as mentioned above, will not grow without limit,
as they do not satisfy the linear dispersion relation and,
thus, are not free waves. Resonant interactions, which
generate waves that do satisfy the linear dispersion relation, are a particular subset of interactions. It can be
shown [4.13] that no resonant interactions appear at second order (save for one trivial set), and one must go
to third order to determine these interactions. The application of multiple scale techniques to address these
resonances was first performed by Benney [4.14], and
the resulting evolution equations are quite complicated;
however, the resonance conditions can be written generically as follows
l C m p D n ;
(4.56)
where l, m, and p are indices that represent wave
components which interact with n . These quartet interaction terms have been approximated for wind wave
generation models [4.15] and are a primary engine
for energy transfer within the spectrum in wind wave
generation.
4.5 Shallow Water Wave Theories
In contrast to deep water, there is a distinct difference
in the scale of variability between horizontal and vertical motions in shallow water (small kh). The horizontal
motions are scaled similarly to deep water, but the vertical motions for small kh are
w D
@
@z
D
gkH
2!
sinh k.z C h/
cosh kh
sin.kx !t/
gHk
2
.z C h/
!
O.1/ ;
(4.57)
since sinh k.z C h/ k.z C h/ for small kh. Using the
shallow water asymptote of the linear dispersion relation and rearranging the shallow water asymptote of the
vertical velocity w
w
gHk
2
.z C h/
!
D
gak
2 h
!
1 C
z
h
Á
D
a
h
!
k
kh
1 C
z
h
Á
D ıı
!
k
Á Á
1 C
z
h
Á
:
(4.58)
As is the case with in deep water, two small parameters (ı D a=h and D kh) are evident and are used for
scaling. However, in this case, the vertical motions are
much smaller than the horizontal motions. The shallow water asymptote of the linear dispersion relation,
in addition to the scale difference between horizontal
and vertical motions, is now applied to the boundary
value problem. Unlike the deep water case, the governing equation here is altered
@
2
@z 2 C
2 @
2
@x 2 D 0 h Ä z Ä Á ;
(4.59)
and the disparity between horizontal and vertical motions is clear. The bottom boundary condition remains
unchanged. The dynamic free surface boundary condition becomes
2
Â
gÁ C
@
@t
Ã
C
ı
2
2
 @
@x
à 2
C
 @
@z
à 2
!
D 0
on z D Á ;
(4.60)