scribes the evolution of the complex amplitude B of each free wave in the spectrum due to four-wave
interaction in a mild slope | ∇
h
| ≤ O
2
space do- main, which satisfies the near-resonant condition:
2 1
2 3
1 2
3
0, O
ω ω ω ω ε
+ − −
= + − −
≤
k k
k k
, 5 The mode-coupled discrete Zakharov equation can
be written as
{ }
{ }
2
2 2
,
, ,
, ,
, 2
, ,
, ,
, ,
exp 2
, ,
, exp
j p
q
j n
p q
g h
j j
j j
j j
j n
j n
n j
n j j
j p
q j
p q
p q j
j p
q j
n p
q n
p q
n p q j
j n
p q
i B
T B
B T
B B
T B B B
i x
T B B B
i x
= +
+ =
+
≠ ≠
≠
∇ = +
+ ×
− − −
+ ×
− + − −
∑ ∑
∑
k k
k
k k
k k
c k k k k
k k k k k k k k
k k
k k k k k
k k
k k
, , 1, 2,...
n p q N
=
,
6 The set of mode-coupled nonlinear complex or-
dinary differential equations is solved using the fourth-order Runge-Kutta method. When calculating
the kernel in Eq. 4, we have introduced Stokes’ corrections to remove near-resonance singularities.
Nevertheless, Eq. 4 is invalid for water of very shallow depth; the equation requires that the disper-
sion remains sufficiently strong see Agnon
21
. The first-order free surface elevation
ηx,t is related to the quantity B and computed through
{ }
1 2
1 ,
2 2
, exp
t
t g
B t
i
ω
ω η
π
•
∞ −∞
−
⎛ ⎞
= ⎜
⎟ ⎝
⎠ ⎡
⎤ + ∗
⎣ ⎦
∫
x k
k x
k k
, 7
2 Wave group structure
The structure of wave groups can be quantitatively described using a wave envelope. The wave enve-
lopes of various frequency bands can be calculated using a Hilbert transform. If the sea surface elevation
ηt is a stationary random function of time, then the Hilbert transform
ξt is given by 1
t t
P t
x
η ξ
π
∞ −∞
= −
∫
, 8
Fig.1 Definition of wave group structures.
where P indicates the Cauchy value. With the Hilbert transform
ξt of the function ηt, the analytic func- tion is given as
exp{ }
S t t
i t
A t i
t η
ξ ϕ
= +
= , 9
The wave envelope At can then be obtained by
1 2 2
2
A t t
t
η ξ
⎡ ⎤
= +
⎣ ⎦ , 10
The envelope At is always symmetrical with respect to the t-axis, as
ηt is composed of only first-order free waves. Only the fundamental frequency band
0.5f
p
~ 1.5f
p
, which produces free waves only and does not include the bound waves, is considered and
calculated. The amplitude A
ave
denotes the average value of
the envelope amplitude see Fig.1. The zero-up cross method relative to A
ave
is used to determine the wave group period Tg. The wave group amplitudes
Ag
mean
and Ag
max
denote the average and maximum of the envelopes, respectively. The wave group period
Tg
mean
is the average value of Tg and Tg
max
corre- sponds to the period of the wave group containing
Ag
max
.
3. NUMERICAL SIMULATIONS
The wave conditions for the numerical simulation, characterized by the peak period T
p
, relative water depth k
p
h, wave amplitude a
ω,θ and principal
direction
θ were defined for input in the nonlinear
wave interaction modeling. The principal wave di- rection
θ = 0 was used for all the simulations except
the field data. The wave model requires initial con- dition information, describing the initial state of the
sea. In this study, the initial sea state was described as a Gaussian spectrum in the form
135
2 2
2 2
, exp
2 2
2
i
m S
ω θ ω
θ
ω ω θ
ω θ πσ σ
σ σ
⎡ ⎤
− ⎢
⎥ =
− −
⎢ ⎥
⎣ ⎦
, 11 where m
is the zero-th moment of the spectrum,
ω
is the angular frequency, and
σ
ω
and σ
θ
are standard deviations for frequency and direction, respectively.
By taking a finite range of frequency
ω
min
,
ω
max
and direction
θ
min
,
θ
max
, the initial amplitude a
ω,θ
= 2S
ω,θdωdθ
12
was determined for calculating the complex amplitude B, which is obtained by
1 2
2 ,
exp g
B a
i
π ω θ
φ ω
⎛ ⎞
= ⎜ ⎟ ⎝
⎠ , 12
where
φ
is the random phase. The wave steepness ak
p
= 0.07 ~ 0.2 a and k
p
be- ing the carrier wave amplitude and number were
used for simulation. The relative water depths, de- noted by k
p
h, in deep water and intermediate water depth are equal to 5.0 and 1.0, respectively, and the
relative water depth on the sloping beach is 1.0 ≥ k
p
h ≥ 0.5, with slope calculation k
p
hi = k
p
26 – 13iz
2
, where z is the number of segments and i = 1,2,3,….z.
At intermediate water depth k
p
h = 1.0, we are not considering the adjustment of the spectrum as the
effect of water depth, as in the Wallops spectrum. We just assume that the same shape of the Gaussian
spectrum is used in deep water and at intermediate water depth.
Regarding the number of wave components of the directional spectrum, Japan Meteorological Agency
JMA has improved their operational wave model, so that 400 components have become 900 compo-
nents. The directional components of wave spectrum become fine and can express isotropic and smooth
spreading, unlike the previous model
22
. As the pur- pose of this study is to investigate the transformation
of wave groups, which strongly depends on the res- onance of the wave components, a larger number of
wave components were used in this simulation. At constant depth, directional spectra were simulated
with 1550 components, which consisted of 50 com- ponents of frequency and 31 directional components.
However, in sloping cases the directional spectra were simulated with 50 frequency components and
21 directional components. Additionally, refraction effects on sloping cases were calculated based on
linear theory. The directional spectra were normal- ized by the peak of the initial directional spectrum
S
f
p
,
θ
p
. Finally, evolution of wave groups as a result of the directional spectrum was compared with uni-
directional simulation, which consisted of 100 fre- quency components.
The Runge-Kutta method, which solves a differ- ential equation numerically, gives the integration of
the spatial evolution of the nonlinear waves.
4. RESULTS AND DISCUSSION
