Influence of Spectral Density Distribution on Wave Parameters

and Simulation in Time Domain

by
K.-F. Daemrich, S. Mai, N. Ohle , E. Tautenhain

Abstract
Design works, especially for non-linear wave related problems, require information on the
statistics of heights and periods of single waves in a wave train.
Whereas the RAYLEIGH-distribution is widely used and accepted for wave heights, period dis-
tributions have more variety. Apart from this, for more detailed investigations, statistical in-
formation on combinations of heights H and periods T can be helpful. Wave run-up e.g. is
related to T ⋅ H and therefore the significant wave run-up Ru2% should be calculated from
( )
T ⋅ H 2% , rather than from individual combinations of characteristic wave parameters as e.g.
H1/3 and Tm or Tp.
Such parameters are usually not analysed in measurements and they cannot be taken from the
widely used phase averaging numerical models like SWAN, which deliver only spectral in-
formation on the design sea state. Therefore, in this paper, time-series generated by linear
superposition are analysed, and the influence of the spectral shape (TMA-spectra, double peak
spectra) on the distributions of heights and periods is demonstrated. Furthermore wave run-up
at sea dikes is investigated with this method and the usefulness of characteristic wave parame-
ters in the design formula is discussed.

1. Introduction
Sea waves are irregular in time and space. The irregularity of the surface, from which all other
relevant features (as orbital velocities, pressures etc.) have to be derived, can be analyzed or
modelled either in time-domain or in frequency-domain.
Working in time-domain, the irregular wave train is seen as a sequence of individual waves,
characterized by heights H and periods T. Usually these individual wave parameters are de-
termined by zero-downcrossing definition. The statistics of the irregularity can be expressed
as distributions of the probabilities of the parameters H and/or T. Characteristic mean values
of these parameters (e.g. H1/3, TH1/3, Tm) are used for characterisation of the sea state and in
design procedures.
The irregular sea surface can also be seen as a composition or superposition of a number of
sinusoidal wave components with different frequencies (the directional spread is not treated in
this context). The decomposition of a time-series is performed by FOURIER-technique. The
irregularity is expressed in the frequency-domain as wave spectra, if necessary with related
phase information. Characteristic parameters can be calculated from the spectral moments
(e.g. Hm0, T0,2) or attributed to characteristic values of the spectrum (e.g. Tp).
Both methods of analysis are related to simulation methods for design.
In time-domain, in principle each individual wave with height H and period T is seen as a
regular wave with these parameters, although the shape of such an event is seldom symmetric
in nature. With the help of wave theories or empirical knowledge, any effects related to waves
can be calculated for individual wave events, and if necessary, a statistics of the result can be
created.
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 2
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

In frequency-domain, each wave component is treated as independent. The effect of each

component is formulated (on the basis of linear wave theory or empirical knowledge), and the
overall effect is expressed as a spectrum of the result.
Both methods are valuable and necessary for the various design works related to wave prob-
lems. The simulation method in time-domain is mainly used for more non-linear processes
(e.g. wave forces on structures, wave breaking, wave run-up at sloped structures). For more
linear processes (e.g. diffraction, refraction), the superposition method in frequency-domain is
the preferred tool.
The most common method today to get information on a design sea state is based on wave
forecasting in combination with phase averaged numerical modelling of shallow water effects.
Such models deliver only spectra of the sea state and/or characteristic spectral parameters.
Design methods in time-domain, however, require information on the statistics of heights and
periods. Either, for a maximum wave, a related period has to be determined, or the complete
statistics is needed.
The probability distribution of wave heights is well described in most cases by the RAYLEIGH-
distribution, which can be seen as the universal distribution. The distribution of wave periods
for standard spectra is generally narrower than the distribution of the heights. Under condi-
tions of sea and swell at the same time, or deformation of the spectra due to shallow water
effects, however, the period distribution might become broader. Insofar, a great uncertainty
exists, related to wave periods.
Apart from this, for more detailed investigations, distributions of combined parameters of
heights and periods can be helpful. Wave run-up e.g. is related to T ⋅ H and therefore the
( )
significant wave run-up Ru2% should be calculated from T ⋅ H 2% , rather than from individ-
ual combinations of characteristic wave parameters as e.g. H1/3 and Tm or Tp. There is only
limited information on such distributions and parameters.
To provide better information on wave period statistics or combined statistics of heights and
periods, it seems consequent to use the superposition method and generate time-series, from
which the requested parameters or distributions can be taken without any restrictions to stan-
dard types of spectra. This corresponds to the procedure of generating time-series for hydrau-
lic model tests and phase resolving numerical models.

2. Time-series generation by superposition of wave components

The generation of a time-series by superposition is demonstrated for the example of a JON-
SWAP-spectrum. Characteristic frequency-domain parameters are selected to be Hs = 4.0 m
with Tp = 8.0 sec. The density distribution is calculated by the formula

α ⋅ g2 ⎡ 5 ⎛ f ⎞ −4 ⎤ ⎡ − (f − f p )2 ⎤
S(f ) = ⋅ exp ⎢ − ⎜ ⎟ ⎥ ⋅ γ ⎢ exp
2 ⎥
(2π )4 ⋅ f 5 ⎢ 4 ⎜⎝ f p ⎟⎠ ⎥ ⎢
⎣ 2 ⋅ σ 2
⋅ f ⎥⎦
⎣ ⎦ p

with
γ = 3,3 σa = 0,07 (f < fp) σb = 0,09 (f ≥ fp)
and scaled by α to result in Hs = 4.0 m. Hs is calculated from the moment m0 = ∫S(f)⋅f0⋅df:
H s = 4 ⋅ m0

The spectral density distribution is shown in Fig. 2.1.

2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 3
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig. 2.1 JONSWAP-spectrum (Hs = 4.0 m, Tp = 8.0 sec)

The superposition requires discrete wave components, which are calculated from the spectral
density distribution:
a (f ) = S(f ) ⋅ 2 ⋅ ∆f .

For that a ∆f has to be selected, which controls the length (periodicity) T0 of the time-series to
be generated:
T0 = 1/∆f.
For the example ∆f was chosen to be ∆f = 0.004167 Hz, which generates a time-series of
240 sec (30 peak periods). The plot of the amplitudes is given in Fig. 2.2.

Fig. 2.2: Discrete spectrum of amplitudes

A phase angle information, which is selected to be a random value in the range ±π is attrib-
uted to each component. Different seeds of random phase angles result in different time-
series. The phase angles selected for this example are shown in Fig. 2.3. Finally the time-
series, which is generated under these conditions, is shown in Fig. 2.4. From this time-series
individual values of wave heights and periods can be calculated according to zero-
downcrossing definition.
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 4
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig. 2.3: Phase angles (random)

Fig. 2.4: Time-series related to the discrete amplitude spectrum (Fig. 2.2) and the selected
phase angles (Fig. 2.3)

3. Wave height and period distributions for JONSWAP- and TMA-spectra

The necessary calculations related to the superposition model for time-series generation and
analysis of the wave parameters according to zero-crossing definition were performed in
MATLAB.
To demonstrate the results, time-series with characteristic parameters Hm0 = 4 m and
Tp = 8 sec with a duration of 341⋅Tp = 45.5 min were selected exemplarily. The realization
with a peak enhancement factor of 3.3 (mean for JONSWAP) and a certain random number seed
(state 400) resulted in a time-series with 416 individual wave events. The scatter diagram of
the combinations of heights H and periods T is shown in Fig. 3.1.

Fig. 3.1: Scatter diagram of heights and periods

2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 5
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

The data show the typical range of periods for the various wave heights. Up to about the mean
wave height, the range of periods is about proportional to the wave heights. For larger waves
the range of periods decreases, and the highest waves tend to have periods in the order of the
mean period or the peak period.
The distribution of the wave heights is shown in Fig. 3. 2. For comparison, the RAYLEIGH-
distribution is given, which does not fit perfectly, but relatively good.

Fig. 3.2: Distributions of wave heights

The distributions of wave periods is shown in Fig. 3.3. Again the RAYLEIGH-distribution is
given for comparison. It can be clearly stated, that this distribution does not at all fit to the
data.

Fig. 3.3: Distributions of wave periods

To show the influences of spectral shapes, wave height and period distributions for three JON-
SWAP-spectra with different peak enhancement factors γ = 1, 3.3 and 7 are compared (a peak
enhancement factor of γ = 1 corresponds to the PIERSON-MOSKOWITZ shape). Furthermore a
TMA-spectrum for the same characteristic wave parameters in a water depth d = 10 m
(d/Lp = 0.1) is included. All spectra have the same length of the time-series, however, contain
different numbers of waves. To make the data comparable, results are presented as relative
wave heights H/Hm and T/Tm (Hm and Tm are the mean wave heights and periods).
In Fig. 3.4 and 3.5 height and period distributions of the various spectra are plotted, together
with the RAYLEIGH-distribution as reference.
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 6
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig 3.4: Distributions of wave heights for Fig. 3.5: Distributions of wave periods for
various spectra various spectra
The distributions of the wave heights are almost equal. That confirms the RAYLEIGH-
distribution to be reasonable for those conditions. The distributions of the wave periods are
narrower for JONSWAP-spectra with higher peak enhancement factors. The distribution of the
wave periods of the TMA-spectrum is wider than the corresponding JONSWAP-spectrum.

4. Influences of different random seeds and length of the time-series

The above results are produced from time-series of about 400 waves. All time-series were
generated from the same set of random phases. To assess the influence of the randomness and
the statistical stability, for the mean JONSWAP-spectrum (γ = 3.3) two more time-series with
different random phases have been generated. Whereas the distributions of the wave heights
were almost identical, the distributions of periods came out to have clear deviations around a
mean trend (Fig. 4.1). Using time-series of twice the length with about 800 wave events im-
proved the situation (Fig. 4.2), but only time-series of 4 times the length (about 1600 waves)
resulted in an acceptable course of the period distribution. (Fig. 4.3).
For comparison, in Fig. 4.3 data from a set of 10753 wave events are plotted to demonstrate
that a stable shape of the distribution is reached.

Fig. 4.1: Influence of number of waves and Fig. 4.2: Influence of number of waves and
random phase seed on the period distribution random phase seed on the period distribution
(about 400 waves) (about 800 waves)
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 7
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig. 4.3: : Influence of number of waves and random phase seed on the period distribution
(about 1600 waves)

5. Double-peak spectra
In coastal locations, especially in relative shallow water, design spectra may not be of stan-
dard type JONSWAP or TMA. To demonstrate the influence of non-standard wave spectra, two
double-peak spectra are investigated exemplarily.
The spectra are superposed from two JONSWAP-spectra. For the reference spectrum (spec-
trum 1) the peak period is kept Tp = 8 sec (peak frequency fp = 0.125 Hz) as before. For the
second spectrum, the peak is selected to be fp2 = 0.5⋅fp1 for the first case, and fp2 = 1.5⋅fp1 for
the second case. In both cases the energy of the secondary spectrum is selected to be 50 % of
the reference spectrum, but the final spectra have still the same significant wave height
Hm0 = 4 m. The shapes of the spectra are shown in figures 5.1 and 5.2.

Fig. 5.1: Double-peak spectrum Fig. 5.2: Double-peak spectrum

(fp2/fp1 = 0.5) (fp2/fp1 = 1.5)
Wave heights still follow the RAYLEIGH-distribution as all other spectra do. The distributions
of the periods, however, deviate, as shown in Fig. 5.3 (absolute periods) and Fig. 5.4 (relative
periods). In both cases of double peak spectra, the distributions are broader and the higher
periods are more frequent, compared to the plain JONSWAP-spectrum.
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 8
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig. 5.3: Distributions of wave periods Fig. 5.4: Distributions of relative

in double-peak spectra compared to wave periods in double-peak spectra
plain JONSWAP-spectrum compared to plain JONSWAP-spectrum

6. Application to simulation of wave run-up of irregular waves at sloped sea dikes

6.1 Some remarks on wave run-up at sloped structures
The influence of wave height and period statistics on design will be illustrated with the exam-
ple of wave run-up at sea dikes, or more general, sloped structures.
For a certain range of wave steepness and slope angle α the wave run-up R of regular waves
can be characterized by the formula
R = 1.27 ⋅ H ⋅ T ⋅ tan α
which is based on HUNT (1959).
For irregular waves usually the parameter Ru2% is used as design value for German sea dikes.
The common design formula for Ru2% can be written in the following form:
R u 2% = a ⋅ H char ⋅ Tchar ⋅ tan α

Hchar and Tchar are characteristic wave parameters from time-series or spectral analysis.
Hchar is generally accepted to be the significant height Hs (either H1/3 or Hm0). For Tchar the
mean periods Tm or T0,2, the significant period Ts = TH1/3 or the peak period Tp have been used
in the past. Recently VAN GENT (VAN GENT, 1999) recommended the spectral period T-1,0 to
be used in the design formula for wave run-up in irregular waves for the first time.
Generally the coefficient a is determined by hydraulic model tests in irregular waves, using
standard spectra and various characteristic wave parameters. For a combination of characteris-
tic parameters H1/3 and Tp the coefficient is widely accepted to be around a = 1.87 (or
1.5 ⋅ g 2π ). Using other combinations like H1/3 and Tm or Hm0 and T-1,0 with an other coeffi-
cient is possible in principle and has been used by various authors.
The fact, that the relations between wave period parameters are not at all constant for various
types of wave spectra, highlights already that we have a principle problem with such design
formulae as long as we do not have a real problem depending “significant” combination of
wave parameters (strictly speaking, such a design formula requires that the relation between
the distribution of individual wave parameters and the distribution of wave run-ups is equal
for all types of sea states or spectra, which seems not to be realistic, when we consider the
results from the previous chapters). The scatter of the reference data, from which by curve
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 9
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

fitting and/or correlation the “best” coefficient a has to be established, is a further hint to the
basic problem.
The situation is getting worse when the design spectra are not standard spectra, but e.g. dou-
ble-peak spectra, where especially the today most preferred period parameter Tp is question-
able. But also the latest parameter T-1,0 is not without doubts.
Instead of trying various combinations of characteristic parameters it could be consequent
(looking to the physical relationship for the wave run-up in regular waves) to relate the design
wave run-up Ru2% to a (combined) statistical parameter H ⋅ T 2% . ( )
However, this is not a standard parameter in wave analysis and there are not many hydraulic
model investigations, where this parameter has been calculated.
Therefore, in this paper, the problem will be investigated on the basis of wave time-series
generated by linear superposition as described in the previous chapters.
The straight forward way would be, to attribute to each individual irregular wave event a
wave run-up, calculated from the related H and T according to the formula for regular waves
R = 1.27 ⋅ H ⋅ T ⋅ tan α
and to find Ru2% from the results of the simulation. This is what some previous authors did or
recommended (SAVILLE, 1962; BATTJES, 1971).
In case of wave run-up at sloped structures, however, the situation is more complex. The
wave run-up in irregular waves is strongly influenced by the wave run-down from the previ-
ous wave run-up event. TAUTENHAIN (1981, 1982) has done intensive investigations in hy-
draulic models and theory on this topic. He developed the only method to consider the pre-
wave influence on wave run-up. According to his results, the wave run-up R generated by an
individual wave event can be calculated from
~
( ~
R n = R n ⋅ 3 2 ⋅ Ψ − R n −1 R n )3

with
~ th
R n = wave run-up in the n wave without pre-wave influence
th
R n = wave run-up in the n wave with pre-wave influence
Ψ = coefficient to be verified by measurements (according to theory: Ψ = 1 )
This methodology is used for the theoretical calculations of the wave run-up statistics in the
following.
This method leads to an increase of the significant wave run-up Ru2% and to a reduction of the
number of wave run-up events, compared to the number of wave events, what is confirmed by
hydraulic model tests.

6.2 Influence of various spectra on wave run-up

Calculating wave run-ups first without pre-wave influence, results in the wave run-up distri-
butions for standard JONSWAP-spectra (γ = 1, 3.3 and 7) and a TMA-spectrum (γ = 3.3,
d = 10 m) shown in Fig. 6.1. It is to be seen clearly, that for JONSWAP-spectra with various
peak enhancement factors the significant wave run-up Ru2% is almost equal (Ru2% ≈ 3.9 m),
although the mean wave run-ups are quite different. The TMA-spectrum comes out with a
slightly different Ru2% (about 6% less).
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 10
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

In Fig. 6.2 the corresponding results are shown, when wave run-up is calculated with pre-
wave influence. For a number of wave events, the calculation results in a negative wave run-
up, which has to be interpreted as “no wave run-up”. In the diagrams only the positive results
are plotted, however, the frequency is still related to the number of wave events.

Fig. 6.1: Run-up distributions for JONSWAP Fig. 6.2: Run-up distributions for JONSWAP
and TMA-spectra without pre-wave influence and TMA-spectra with pre-wave influence
Taking into account pre-waves, there are slightly different values of Ru2%. The value of Ru2%
is in the range of Ru2% ≈ 4.2 ÷ 4.5 m.
Using the design formula with the coefficient a = 1.87, and Hm0 and Tp as characteristic wave
parameters would result in Ru2% = 5 m. Insofar the results are possibly about 10 to 15% below
results reported from hydraulic model tests.
The reason is not yet quite clear. However, the initial coefficient (or the assumed trend) for
wave run-up in regular waves (where the results strongly depend on) is somewhat question-
able, as TAUTENHAIN has measured about 10% higher wave run-ups in his hydraulic model
tests, compared to the results published by HUNT. This would explain a part of the deviations.
On the other hand, the pre-wave method is still subject of investigations.
Whereas the influence of various types of JONSWAP- and TMA-spectra was moderate, results
from double-peak spectra show, that unrealistic results are to be expected, when the design
formula for wave run-up is applied under these conditions in the usual way (Fig. 6.3 and 6.4).

Fig. 6.3: Run-up distributions for double-peak Fig. 6.4: Run-up distributions for double-peak
spectra without pre-wave influence spectra with pre-wave influence
2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 11
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

6.3 Usefulness of characteristic wave parameters in the design formula for wave run-up
On the basis of simulated waves and wave run-ups the related coefficients a can be calculated
for various combinations of characteristic wave parameters. To give an impression on the
influence of spectral shape, the variation of the coefficient a is calculated for TMA-spectra in
various water depths from deep water (d/L0p = 0.5) to shallow water (d/L0p = 0.05) and for
double-peak spectra in deep water, with variations of the frequency of the second peak in the
range fp2/fp1 = 0.1 ÷ 2.0. Exemplarily the energy of the second peak is selected to be 50% of
the reference spectrum.
To find the related coefficient a, the design formula for wave run-up is arranged as follows:
R u 2%
R u 2% = a ⋅ H char ⋅ Tchar ⋅ tan α ⇒ a =
H char ⋅ Tchar ⋅ tan α
The variation of the coefficient a is determined for the following combinations of characteris-
tic wave parameters
Hm0 Tp
Hm0 T0,2
H1/3 T-1,0
H1/3 Tm
and for the favoured combined parameter
( H ⋅T )
2%

The results are plotted for the variations of the TMA-spectra in Fig. 6.5 and for the double-
peak spectra in Fig. 6.6.
From the course of the coefficients a the usefulness can be judged. For standard TMA-spectra
the “best” parameters are Tp (in combination with Hm0) and the combined parameter
( )
H ⋅ T 2% .

For the double-peak spectra all parameters, except the combined parameter H ⋅ T 2% are not ( )
at all close to constant. The period parameter T-1,0 (in combination with Hm0) is the relatively
best of the usual period parameters, when the whole investigated range is considered. For sec-
ondary peaks at frequencies higher than the peak of the reference spectrum, however, the
peak-period parameter Tp is more stable.

Fig. 6.5: Variation of coefficient a with water depth (TMA-spectra)

2nd Chinese - German Joint Symp. on Coastal and Ocean Eng. October 11 to 20, 2004 Nanjing, China 12
Influence of Spectral Density Distribution on Wave Parameters and Simulation in Time Domain

Fig. 6.6: Variation of coefficient a with frequency of the second peak (m02 = 0.5⋅m01)

7. Further research
The linear superposition method should be extended to non-linear superposition (e.g. by the
LAGRANGEian method, see WOLTERING and DAEMRICH, 2004) and checked by measurements
in nature and/or hydraulic model tests. The handling of “random” phase setting has to be in-
vestigated with respect to getting distributions of wave heights around the breaker zone dif-
ferent from the RAYLEIGH-distribution.

8. References
BATTJES, J. A. (1971): Run-up Distribution of Waves Breaking on Slopes. Proc. ASCE, Jour-
nal of the Waterways, Harbors and Coastal Engineering Division, Vol. 97, No.WW1, 1971
HUNT, I. A. (1959): Design of Seawalls and Breakwaters. Proc. ASCE, Journal of the Water-
ways and Harbors Division, Vol. 85, No.WW3, 1959
SAVILLE, T., Jr. (1962) : An Approximation of the Wave Run-up Frequency Distribution.
Proc. 8th International Conference on Coastal Engineering, Mexico, 1962
TAUTENHAIN, E. (1981): Der Wellenüberlauf an Seedeichen unter Berücksichtigung des Wel-
lenauflaufs – Ein Beitrag zur Bemessung. Mitt. des Franzius-Instituts f. Wasserbau und
Küsteningenieurwesen, Univers. Hannover, Heft 53, 1981
TAUTENHAIN, E., KOHLHASE, S., PARTENSCKY, H.W. (1982): Wave Run-up at Sea Dikes Un-
der Oblique Wave Approach. Proc. 18th International Conference on Coastal Engineering,
Cape Town, 1980
VAN GENT, M.R.A. (1999):Wave run-up and wave overtopping for double peaked wave en-
ergy spectra. Rep. H3351, Delft Hydraulics, Delft, The Netherlands, 1999
WOLTERING, S., DAEMRICH, K.-F. (2004): Nonlinearity in Irregular Waves from Linear LA-
th
GRANGEian Superposition. Proc. 29 International Conference on Coastal Engineering, Lis-
bon, 2004