Part A | 4.4
86 Part A Fundamentals
so-called bound waves (forced waves that do not satisfy
the linear dispersion relation and, thus, can grow only
a limited amount) and resonant waves (forced waves
that do satisfy the linear dispersion relation and are thus
able to grow without bound as the forcing is secular).
The expansion of the surface boundary conditions in
powers of leads to products of unknowns that increase
as the order of the expansion increases. Thus, given
Á e
i
; e
i
;
(4.41)
(where D k x !t), higher-order terms lead to
Á
2
e
2i
; 1; e
2i
;
(4.42)
Á
3
e
3i
; e
i
; e
i
; e
3i
;
(4.43)
and so on, until the desired truncation. Terms that
result from this operation will remain bound to the
primary harmonic and not experience unbounded
growth. However, the Á
2 will also generate terms proportional to e
i
and e
i , and Á
3 will generate terms
proportional to e
2i , 1, and e
2i , or (in both cases)
terms proportional to the next lower-order solution.
Solution of the resulting boundary value problem will
generate terms that will grow linearly in either space or
time without bounds (secular behavior). To ameliorate
this, the method of multiple scales is often used to
treat these problems. The application of this method
allows the potentially secular behavior to be split into
motions with different degrees of variation. In many
cases, the fast degree of variation is associated with
the waveform itself (the periodic motion) and the slow
degree of variation with another property of the wave
(for example, the amplitude of the wave form), which
can then be linked to variations in some external field
(e.g., the bathymetry).
The formalism in the previous section is most relevant to the case where a single wave component gives
rise to harmonics, all of which can evolve during propagation to comprise a changing wave form. A coordinate
system moving with the group velocity of the wave is
defined; the resulting coordinates are defined as (time
coordinate defining the slow temporal variation) and
(space coordinate defining the slow spatial variation).
Carrying this analysis to third order will lead to the cubic nonlinear Schrödinger equations [4.11]
i
@A
@@
1
2
@
2
!
@k 2
@
2 A
@@ 2 C ˇjAj
2 A C A D 0 ; (4.44)
where
ˇ D
!k
2
2
D C
g
2 k
2
2!.C 2
g gh/
Â
1 C
kC g
2! cosh
2 kh
à 2
;
(4.45)
and
D kS../
Â
1 C
kC g
2! cosh
2 kh
Ã
;
(4.46)
where S../ is an integration constant, and
D D
cosh 4kh C 8 2tanh
2 kh
8 sinh
4 kh
:
(4.47)
However, in most cases in the ocean, a description of
wave propagation via the harmonics of a single wave
is inapplicable, and we must turn toward a description
comprised of a summation of waves of different frequencies and directions with random phases
D
1
X
nD1
i
2
B n
cosh k n .h C z/
cosh k n h
e
i.knx!ntCn/
Cc:c: ;
(4.48)
where B n is the (complex) amplitude of the n-th component of and n is a random phase associated with
this component. The corresponding expressions for the
free surface elevation is
Á D
1
X
nD1
A n
2
e
i.knx!ntCn/
C c:c: ;
(4.49)
where c:c: refers to complex conjugate.
Substituting these expressions into the second-order
boundary value problem will lead to expressions for the
second-order contributions to the random sea, as shown
by Sharma and Dean [4.12]
D
1
X
nD1
ig
2! n
A n
cosh k n .h C z/
cosh k n h
e
i.knx!ntCn/
C
1
X
iD1
1
X
jD1
ig
2 A i A j D
C
ij
8! i ! j .! i C ! j /
jk i C k j j.h C z/
cosh jk i C k j jh
e
i. i C j/
C
1
X
iD1
1
X
jD1
ig
2 A i A j D
ij
8! i ! j .! i ! j /
jk i k j j.h C z/
cosh jk i k j jh
e
i. i j/ C c:c: ;
(4.50)
where i and j are two arbitrary frequency components
in the spectrum. The coefficients D
C
ij and D
ij are
D
C
ij D
.
p
R i C
p
R j /
p
R j .k
2
i R
2
i /C
p
R i .k
2
j R
2
j /
C2.
p
R i C
p
R j /
2
.k i k j R i R j /
.
p
R i C
p
R j / 2 k
C
ij tanhk
C
ij h
;
(4.51)
86 Part A Fundamentals
so-called bound waves (forced waves that do not satisfy
the linear dispersion relation and, thus, can grow only
a limited amount) and resonant waves (forced waves
that do satisfy the linear dispersion relation and are thus
able to grow without bound as the forcing is secular).
The expansion of the surface boundary conditions in
powers of leads to products of unknowns that increase
as the order of the expansion increases. Thus, given
Á e
i
; e
i
;
(4.41)
(where D k x !t), higher-order terms lead to
Á
2
e
2i
; 1; e
2i
;
(4.42)
Á
3
e
3i
; e
i
; e
i
; e
3i
;
(4.43)
and so on, until the desired truncation. Terms that
result from this operation will remain bound to the
primary harmonic and not experience unbounded
growth. However, the Á
2 will also generate terms proportional to e
i
and e
i , and Á
3 will generate terms
proportional to e
2i , 1, and e
2i , or (in both cases)
terms proportional to the next lower-order solution.
Solution of the resulting boundary value problem will
generate terms that will grow linearly in either space or
time without bounds (secular behavior). To ameliorate
this, the method of multiple scales is often used to
treat these problems. The application of this method
allows the potentially secular behavior to be split into
motions with different degrees of variation. In many
cases, the fast degree of variation is associated with
the waveform itself (the periodic motion) and the slow
degree of variation with another property of the wave
(for example, the amplitude of the wave form), which
can then be linked to variations in some external field
(e.g., the bathymetry).
The formalism in the previous section is most relevant to the case where a single wave component gives
rise to harmonics, all of which can evolve during propagation to comprise a changing wave form. A coordinate
system moving with the group velocity of the wave is
defined; the resulting coordinates are defined as (time
coordinate defining the slow temporal variation) and
(space coordinate defining the slow spatial variation).
Carrying this analysis to third order will lead to the cubic nonlinear Schrödinger equations [4.11]
i
@A
@@
1
2
@
2
!
@k 2
@
2 A
@@ 2 C ˇjAj
2 A C A D 0 ; (4.44)
where
ˇ D
!k
2
2
D C
g
2 k
2
2!.C 2
g gh/
Â
1 C
kC g
2! cosh
2 kh
à 2
;
(4.45)
and
D kS../
Â
1 C
kC g
2! cosh
2 kh
Ã
;
(4.46)
where S../ is an integration constant, and
D D
cosh 4kh C 8 2tanh
2 kh
8 sinh
4 kh
:
(4.47)
However, in most cases in the ocean, a description of
wave propagation via the harmonics of a single wave
is inapplicable, and we must turn toward a description
comprised of a summation of waves of different frequencies and directions with random phases
D
1
X
nD1
i
2
B n
cosh k n .h C z/
cosh k n h
e
i.knx!ntCn/
Cc:c: ;
(4.48)
where B n is the (complex) amplitude of the n-th component of and n is a random phase associated with
this component. The corresponding expressions for the
free surface elevation is
Á D
1
X
nD1
A n
2
e
i.knx!ntCn/
C c:c: ;
(4.49)
where c:c: refers to complex conjugate.
Substituting these expressions into the second-order
boundary value problem will lead to expressions for the
second-order contributions to the random sea, as shown
by Sharma and Dean [4.12]
D
1
X
nD1
ig
2! n
A n
cosh k n .h C z/
cosh k n h
e
i.knx!ntCn/
C
1
X
iD1
1
X
jD1
ig
2 A i A j D
C
ij
8! i ! j .! i C ! j /
jk i C k j j.h C z/
cosh jk i C k j jh
e
i. i C j/
C
1
X
iD1
1
X
jD1
ig
2 A i A j D
ij
8! i ! j .! i ! j /
jk i k j j.h C z/
cosh jk i k j jh
e
i. i j/ C c:c: ;
(4.50)
where i and j are two arbitrary frequency components
in the spectrum. The coefficients D
C
ij and D
ij are
D
C
ij D
.
p
R i C
p
R j /
p
R j .k
2
i R
2
i /C
p
R i .k
2
j R
2
j /
C2.
p
R i C
p
R j /
2
.k i k j R i R j /
.
p
R i C
p
R j / 2 k
C
ij tanhk
C
ij h
;
(4.51)