JEE306: Hydrodynamics of Offshore Systems
Wave Spectrum Calculation Assignment: Com paring Pierson and Moskowitz
with JONSWAP models
Alex Fuglsang
155229
06/08/2014
Abstract
This report studies the JONSWAP and PM wave spectrum models. The differences of the two models are analysed theoretically
using MATLAB and through literature. Various simulations of each spectrum are provided to compare different wind velocities
and fetch lengths. These simulations are compared to spectral densities from measured in previous experiments. Using the
theoretical wave spectrums, estimations are made of various resulting wave parameters including significant wave height and
mean wave period.
Nomenclature
Symbol
Description
Units
F
Fetch length
m
f
Wave frequency
Hz
fp
Peak wave frequency
Hz
g
Gravity
m/s2
H
Wave height
m
S
Wave spectrum density
m2/Hz
T
Wave period
s
W
Wind Velocity
m/s
1 Introduction
The ability to predict sea states and the energy that waves
can impart onto offshore structures and vessels is the
reasoning behind studies such as the Pierson-Moskowitz and
Joint North Sea Wave Observation Project studied here.
Figure 1-1: Pierson-Moskowitz wave spectral density (Stewart, 2006).
The Joint North Sea Wave Observation Project (JONSWAP)
was conducted 160 km off the west coast of Denmark in the
North Sea during 1969 (Hasselmann, 1973). This study
found that the wave spectrum is never fully developed, even
over large distances and time. Due to this, fetch was
introduced into the theoretical model to better fit the
measured data. Figure 1-2 displays JONSWAP spectral
density against frequency for different fetch distances. Fetch
is a particularly important factor for fetch limited seas such
as those in the North Sea.
The Pierson-Moskowitz study on wave spectral density was
conducted in 1964 in the Atlantic Ocean (Stewart, 2006).
This study assumed that if a constant wind blows over a
large area over a long time period a fully developed sea will
form; that is the waves come into equilibrium with the wind.
This study was performed in the Northern Atlantic, with
large fetch distances, and fetch was not considered in the
Pierson-Moskowitz (PM) equation.
This study produced the plot of wave spectral density with
respect to frequency (Hz) for various wind speeds in Figure
1-1, below.
1
In this equation peak frequency (fp), α and γ have fetch (F)
as a variable, as equations(1.6),(1.7) and (1.8) show,
respectively:
0.22
3.5g gF
W W2
fp
0.33
gF
0.076
7
(1.6)
(1.7)
W2
0.143
gF
(1.8)
W2
To calculate a, equation (1.9) is used.
f
Figure 1-2: JONSWAP Spectral density for different fetch distances
(Stewart, 2006).
2
a
Theory
Typical wave spectra are determined through equations such
as the one below (equation(1.1)). A and B are constants, and
f is frequency in hertz.
A
S (f)
f5
B
e
f4
(1.1)
The PM equation uses wind speed (W) measured at 19.5m
above the sea surface;
g2
S (f )
4
2
0.74
e
f
g
2 Wf
2 2 fp2
e
5
fp
0.09 when f
fp
From the spectral density equations (1.2) and (1.5) other
wave parameters such as spectral moments (m0,1,2…), mean
wave period (T1), mean zero-crossing period (Tz),
significant wave height (Hm0 or Hs) and maximum wave
height (Hmax) can be calculated.
mn
f n S (f )df
(1.10)
m0
(1.11)
0
T1
Where α = 0.0081.
Tm 01
Tz
From the PM spectrum significant wave height and peak
wave frequency can be calculated using equations (1.3)
and(1.4):
Hm0
0.21W 2
g
(1.3)
fp
0.87g
2W
(1.4)
1.25
S (f )
2
4
e
f5
Hs
Hm 0
H max
m1
m0
m2
4 m0
1.86Hs
(1.12)
(1.13)
(1.14)
3 Methodology and Results
The JONSWAP spectrum modifies the PM model to include
the effects of fetch. This is seen by the addition of gamma in
equation(1.5)
g2
(1.9)
0.07 when f
4
(1.2)
2
fp
fp
f
4
a
(1.5)
MATLAB is utilised to process the equations listed above.
Wind velocity for both PM and JONSWAP are entered as
user defined variables from a menu. Fetch is selected from a
second variable menu for the JONSWAP spectrum.
Frequency is varied from 0 to 0.5 Hz for both models.
The wind and fetch variables are run through equations (1.2)
- (1.9) to obtain wave spectrums for JONSWAP and PM
equations. These spectrums are compared in the plots in
Figure 3-1 and Figure 3-2 on the next page. The conditions
for these plots are wind velocity of 10 m/s and 200 km and
50 km fetch respectively.
Equations (1.3) and (1.4) are not used in this program,
instead equations (1.10) to (1.14) are utilised for calculating
the wave parameters listed in Table 3-1.
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4 Discussion
As Figure 3-1clearly displays, at an appropriate fetch
distance (in this case 200 km) both the JONSWAP and PM
spectrum densities reach their maximum at the same
frequency.
When fetch is relatively low; less than 100km, the maximum
wave spectral density of the JONSWAP model is lower than
that of the PM spectrum (Figure 3-2). As the fetch is
increased the JONSWAP spectrum density overtakes the
PM as shown in Figure 3-1. The frequency of the
JONSWAP waves also increases at lower fetches, indicating
shorter period, less developed wave states.
Figure 3-1: JONSWAP spectrum compared to PM spectrum at 10m/s
wind speed and 200km fetch
In Figure 4-1 the original data recorded in the 1964 PM
study is compared to the theoretical models. There are some
small variations in wind velocity and irregularities in the
plot (as expected). However the overall size, shape and peak
spectral density correlate very strongly. This verifies the
accuracy of the MATLAB program performing these
calculations.
Figure 3-2: Comparing JONSWAP at 50 km fetch with PM
By integrating S(f) from zero to infinity as described in
equation(1.10), each plot can be used to determine the
respective spectral moment. This spectral moment is then
used to develop the wave parameters listed in Table 3-1,
below.
Table 3-1: Predicted wave parameters
10 m/s
JONSWAP
spectrum at 10m/s
and 200km
Hs (m)
2.13
2.59
Hmax (m)
3.98
4.82
T1 (s)
5.64
6.06
Tz (s)
5.19
5.60
m0 (m)
0.284
0.419
PM Spectrum at
Parameter
As well as the comparison graph in Figure 3-1 and the
results listed above, this MATLAB program takes the wave
velocity and fetch variables inputted by the user and
displays individual plots of each spectrum.
Figure 4-1: Comparing the PM spectral model with experimental data
(Stewart, 2006).
Figure 4-2 on the next page is used to compare the
JONSWAP model produced by the MATLAB program with
the data collected by Hasselmann et all in 1974.
Similar to the PM model, the JONSWAP plots that are
produced by this MATLAB program have strong correlation
with the data obtained experimentally. A constant velocity
of 18 m/s is used to obtain this correlation. It is also
assumed that the station numbers are close to the fetch
3
distance of each station. There are too many assumptions for
this wave spectrum to be considered accurate, however the
peak spectrum densities are similar, and the relationship
with the PM spectrum is encouraging, and similar to what
was expected. For a more accurate variation more
comparison data is required.
6 References
Alves, J. H. (2003). Revisiting the Pierson–Moskowitz
Asymptotic Limits for Fully Developed Wind
Waves. Journal of Physical Oceanography, 13011323.
Hasselmann, K. (1973). Measurements of Wind-Wave
Growth and Swell Decay during the Joint North
Sea Wave Project (JONSWAP). Hamburg:
Deutsches Hydrographisches Institut.
Kibblewhite, K. C. (1989). An Examination of FetchLimited Wave Growth off the West Coast of New
Zealand by a Comparison with the JONSWAP
Results. Journal of Physical Oceanography, 12781296.
Stewart, R. H. (2006, November 15). Ocean-Wave Spectra.
Retrieved
from
OceanWorld:
http://oceanworld.tamu.edu/resources/ocng_textboo
k/chapter16/chapter16_04.htm
Figure 4-2: Comparing theoretical JONSWAP with recorded data
(Hasselmann, 1973).
5 Conclusion
As expected, the JONSWAP spectrum returns far lower
spectrum densities for low fetch distances than the PM
model. This reflects the greater accuracy that adjusting the
wave spectrum equation for fetch provides; it is impossible
for a fully developed sea to occur in areas with limited fetch.
The PM equation returns less error as the fetch increases,
with the peak wave spectral densities for the two models
converging at approximately 100 km of fetch.
Fetch is a major factor in the propergation in waves, and to
ignore it is a major over simplification in calculating
important paremeters such as wave height, wave period and
frequency and spectral density.
By comparing the results delivered by the MATLAB
program with the previous studies and by comparing the two
models, it can be verified that this program provides a
reasonable representation of the JONSWAP and PM wave
spectrum models.
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