HAWASSI-AB User Manual by © LabMath

HAWASSI-AB User Manual by © LabMath
HAWASSI-AB
User Manual
by
© LabMath-Indonesia
ver.1.150829
mail address
e-mail
home page
: LabMath-Indonesia
Lawangwangi - LMI
Jl. Dago Giri No.99
Warung Caringin, Mekarwangi
Bandung 40391, Indonesia
: [email protected]
: www.hawassi.labmath-indonesia.org
Copyright ©2015 LabMath-Indonesia
Contents
Preamble .................................................................................................................................................... 1-3
1
Introduction ........................................................................................................................................ 1-5
2
Description of HAWASSI-AB........................................................................................................... 2-6
2.1
Introduction HAWASSI-AB ...................................................................................................... 2-6
2.2
Units and computational grid ..................................................................................................... 2-7
2.3
Model features and capabilities.................................................................................................. 2-8
2.3.1
Geometry............................................................................................................................ 2-8
2.3.2
Bathymetry and run-up ...................................................................................................... 2-8
2.3.3
Embedded wave influxing of various wave types.............................................................. 2-8
2.3.4
Initial value problems......................................................................................................... 2-9
2.3.5
Model versions (evolution equations) ................................................................................ 2-9
2.3.6
Internal flow calculations [5] ........................................................................................... 2-11
2.4
3
4
Software facilities .................................................................................................................... 2-11
Installing HAWASSI-AB software .................................................................................................. 3-12
3.1
System requirements ................................................................................................................ 3-12
3.2
First step: Installing MCR ........................................................................................................ 3-12
3.3
Second step: Installing HAWASSI-AB ................................................................................... 3-12
GUI’s of HAWASSI-AB ................................................................................................................. 4-13
4.1
Main GUI ................................................................................................................................. 4-14
4.2
Post-Processing GUI ................................................................................................................ 4-16
4.3
Internal Flow GUI .................................................................................................................... 4-18
4.4
Required lay-out of user defined input files............................................................................. 4-19
4.4.1
Influx time signal ............................................................................................................. 4-19
4.4.2
Initial wave profile ........................................................................................................... 4-19
4.4.3
Bathymetry ....................................................................................................................... 4-19
4.4.4
External measurement data .............................................................................................. 4-19
4.4.5
Interior flow External measurement data ......................................................................... 4-19
4.4.6
Dispersion relation ........................................................................................................... 4-19
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5
Test cases ......................................................................................................................................... 5-20
5.1
Non-breaking waves above flat bottom ................................................................................... 5-21
5.1.1
1F001Foc: MARIN_202002, Strong focussing wave ...................................................... 5-21
5.1.2
IF002Draup: MARIN204001, Draupner Wave ............................................................... 5-21
5.1.3
IF003Irreg MARIN_224002F, Irregular wave in deep water ......................................... 5-21
5.1.4
IF004Wall AB1_Wall, Full wave reflection from wall [5].............................................. 5-22
5.1.5
1F005WallFreq, Frequency dependent wall reflection [5] .............................................. 5-22
5.1.6
1F006Bor, Undular bore .................................................................................................. 5-22
5.2
Breaking waves above flat bottom ........................................................................................... 5-23
5.2.1
1FBr001Foc MARIN203001, Strong focussing wave ................................................... 5-23
5.2.2
1FBr002Foc TUD1403Foc7, Focussing Breaking Wave [3]........................................ 5-23
5.2.3
1FBr003BiB TUD1403Bi8, Bichromatic wave breaking .............................................. 5-23
5.2.4
1FBr004Ir MARIN_223002F, Irregular wave breaking [10] ........................................ 5-24
5.2.5
1FBr005Bor Undular_breaking bore [5]........................................................................ 5-24
5.3
Non-breaking waves above non-flat bathymetry ..................................................................... 5-25
5.3.1
2B001IrSlope MARIN_103001, Irregular wave above a slope [5, 13] .......................... 5-25
5.3.2
2B002HarmBar, Harmonic over bar ............................................................................... 5-25
5.3.3
2B003IrBar, Irregular wave over bar ............................................................................... 5-25
5.4
5.4.1
2BBr001HarmBar [10] .................................................................................................... 5-26
5.4.2
2BBr002IrBar Irregular wave, Spilling breaker [5] ....................................................... 5-26
5.5
6
Breaking waves above bathymetry .......................................................................................... 5-26
Run-up of waves (breaking and non-breaking) ........................................................................ 5-27
5.5.1
3R001Harm: Harmonic Run-up (non-breaking) .............................................................. 5-27
5.5.2
3RBr001Harm: Spilling Breaker Run-up (with interior flow) [5] .................................. 5-27
References ........................................................................................................................................ 6-28
6.1
References to basic papers and applications ............................................................................ 6-28
6.2
Other references ....................................................................................................................... 6-29
1-2 | P a g e
Preamble
Waves are fascinating, important and challenging.
The importance can be substantiated from some well-known observations:
• Half of the world population lives less than 150 km from the coast
• The sea is a relatively easy medium for transport of people and goods (half of all the world crude
oil and increasingly more natural gas) and for intercontinental telecommunication through cables
• Ocean resources of food and minerals are only at the start of discovery, profits from wind parks
and harvesting of wave energy in coastal areas is expanding.
Therefore, a sustainable and safe development of the oceanic and coastal areas is of paramount importance.
Nowadays that means that for the design of harbours, breakwaters and ships, calculations are performed
with increasingly more accurate and fast simulation tools. Tools that are, packaged in software, based on
the basic physical laws that describe the properties of waves, the wave-ship interaction, the forces on
structures, etc.
HAWASSI software is aimed to contribute to extend the accuracy, capability and speed of existing
numerical methods and software using applied-mathematical modelling methods that are at the basis.
A basis with a rich history that is fascinating and challenging. Starting in the 18th century with Euler who
generalized Newton’s law for fluids, in the 19th century Airy ‘solved’ the problem to describe small
amplitude surface water waves. In that same century, many renowned scientists like Scott Russel, Stokes,
Boussinesq, Rayleigh and Korteweg & De Vries investigated the nonlinear aspects of finite amplitude
waves. As much as possible without the need to fully calculate the internal fluid motion; started with
Boussinesq in an approximative way, this was formulated accurately in the 1960-1970’s by Zakharov and
Broer by providing the Hamiltonian form of the dynamic equations.
HAWASSI software is based on these last findings, with methods for making the principal description into
a practical (numerical) modelling and implementation tool.
The first release of the software deals with wave propagation, but the developers are in the process to extend
the capabilities to include coupled wave-ship interactions, amongst others, in later releases.
We sincerely hope that the use of the software, just as the design of it has been, will be fascinating and
challenging for students and academicians as well as for practitioners; from both groups we hope to receive
comments and suggestions for further improvements and extensions in a way that can be profitable for both
sides.
Let nature tell its secrets
Listen to the physics in its mathematical language
Restrain from idealization
Only then models will serve us in abundance
1-3 | P a g e
© Copyright of HAWASSI software is with LabMath-Indonesia, an independent research institute
under the Foundation Yayasan AB in Bandung, Indonesia.
The software has been developed over the past years in collaboration with the University of Twente,
Netherlands, with additional financial support of Netherlands Technology Foundation STW and Royal
Netherlands Academy of Arts and Sciences KNAW.
By downloading and using the software you agree that Yayasan AB is not liable for any loss or damage
arising out of the use of the Software. Although much care is taken to arrive at trustful results of
simulations with HAWASSI, Yayasan AB cannot be held responsible for any result of simulations
obtained with the software, or consequential actions or calculations that are based on the results, e.g.
because of possible bugs, wrong use of the software, or other causes.
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1 Introduction
This document is the Manual of HAWASSI-AB software that serves as a guide for using and running the
software.
HAWASSI-AB simulates phase resolved waves in 1 Horizontal Direction (1HD, long crested waves), as
are generated in wave tanks to simulate on scale coastal and oceanic waves above flat and varying
bathymetry, and with (partially) reflecting walls and damping zones.
Section 2 describes briefly the mathematical background and the capabilities of the code, such as the various
dispersive and nonlinear properties, together with the features of the software; it is advised to read this
Section before continuing to the rest of the manual 1.
Section 3 provides a step-by-step installation procedure of the software.
A condensed description to handle the software, regarding GUIs and input/output parameters, is given in
Section 4.
Section 5 describes briefly the 18 TestCases that show capabilities of the code and its use. (More test-cases
will become available on www.hawassi.labmath-indonesia.org)
DEMO-version with restricted functionality
The Demo-version of HAWASSI-AB has restricted functionality and facilities:
 Only exact dispersion
 Only linear and 2nd order nonlinearity
 Only non-breaking waves
 Partially reflective wall with reflection coefficient the same for all frequencies
 No internal flow calculations
 Comparison of demo-simulations with AB-simulations using full functionality
(instead of comparing with measurement data)
Full functionality and facilities under licence
• Licence for University Thesis Projects
• Research Licence for extending capabilities and/or functionalities
• Licence for companies / commercial use, tailor made on demand; all proceeds
will be used at Foundation Yayasan AB for improving/extending the software
Visit www.hawassi.labmath-indonesia.org for further information
or send email to [email protected]
1
Users with limited experience in mathematical-physical wave modelling may consult the service booklet [1]
Water Wave Modelling & Simulation, with Introduction to HAWASSI-software, YAB LabMath
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2 Description of HAWASSI-AB
2.1 Introduction HAWASSI-AB
This section provides background information of HAWASSI-AB about the basic scientific ideas.
HAWASSI – AB is a software package for the simulation of realistic waves in wave tanks (1HD), i.e. longcrested waves as can appear in oceanic and coastal areas, with the option of reflections from walls with
various reflection properties.
The acronym HAWASSI stands for
Hamiltonian Wave-Ship-Structure Interaction.
HAWASSI –AB is a spatial-spectral implementation of the Analytic Boussinesq Model (AB).
Presently the code is for simulation of wave-structure interactions; coupled wave-ship interaction is
foreseen in future releases.
Underlying Modelling Methods
HAWASSI-AB is based on the following principles
• The free surface dynamics for inviscid, incompressible fluid in irrotational motion is governed by
a set of Hamilton equations for the surface elevation η and the potential φ at the surface.
•
With H (φ ,η ) the Hamiltonian, the sum of potential and kinetic energy, the Hamilton equations
are given by (Zakharov 1968, Broer 1974)
•
 ∂ tη =δφ H (φ ,η )

−δη H (φ ,η )
∂ tφ =
Here 𝜕𝜕𝑡𝑡 denotes the time derivative and δφ the variational derivative with respect to φ and
similarly for 𝜂𝜂.
By approximating the kinetic energy functional K (φ ,η ) explicitly as an expression in η and φ
the simulation of the interior flow can be avoided, the Boussinesq character of the code.
•
The way of approximating K (φ ,η ) is based on Dirichlet’s principle for the boundary-value
problem in the fluid domain. By restricting the set of competing functions in the minimization, an
approximation of K (φ ,η ) is obtained. The variational derivative
δφ K (φ ,η ) =∂ N Φ
•
•
is the corresponding consistent approximation of the Dirichlet-to-Neumann operator.
The (approximate) Hamilton system conserves the (approximate) positive definite total energy
exactly, avoiding sources of instability.
The time dynamics is explicit, no CFL-conditions are required. Time stepping is done with matlab
odesolver code, with automatic variable time step.
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In AB (with exact dispersion) the interior flow is approximated by using a nonlinear extension of the
potential as given by the Airy theory of small amplitude waves, and Taylor expansion of the kinetic energy
leading to Hamiltonian consistent approximations.
Numerical Implementation
Fourier Integral Operators (FIO’s) multiply the Fourier Transform of a function by the symbol of the
operator. These are generalizations of (Pseudo-) Differential Operators since for FIO the symbol will
depend both on the wave number and on the spatial variable; the spatial dependence is for nonlinear
extensions and varying bottom. FIO’s are used in a spatial-spectral implementation; these are approximated
by interpolation techniques to enable efficient Fast FT-methods [13]. Localization methods (a difficult point
in Fourier-type implementations) have been successfully implemented to deal with walls, run-up, breaking
waves, etc. [5]
2.2 Units and computational grid
HAWASSI-AB expects all quantities to be expressed in S.I units: m, kg, s (meter, kilogram, second). As a
consequence, the wave height and water depth are in m, wave period in s, etc.
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2.3 Model features and capabilities
HAWASSI-AB accounts for the following physical circumstances and phenomena of waves in 1 HD, i.e.
long -crested waves.
2.3.1 Geometry
Simulation interval and Grid
A uniform grid is defined by specifying an x-interval  xleft , xright  and a grid with grid size
dx =
( xright − xleft ) / ( N − 1) where N = 2 p with p an integer.
Wave numbers k in Fourier space are defined in accordance with the spatial grid:
k=
2π ( xright − xleft )
 − ( N / 2 ) + 1: N / 2  × dk , with dk =
Fourier Boundary
For the use of Fast Fourier Transformations, all quantities (except bathymetry) are tapered to vanish near
the end points; this takes place is the so-called Fourier Boundary, the length of which can be specified. The
Fourier Boundary should be such that reflection of outgoing waves has to be prevented; hence the Fourier
Boundary also acts as a damping zone.
Walls [5]
The position of a wall inside the simulation interval can be specified. Depending on the reflection properties,
the following choices can be made:
 A uniform, partially reflecting wall by specifying the reflection coefficient in [0,1] for all wave
lengths (frequencies)
 A frequency depending (non-uniform) partially reflecting wall by providing a reflection function
with reflection coefficients depending on frequency in the input panel, using only the frequency f
as variable in matlab-formula style.
2.3.2 Bathymetry and run-up
The bathymetry can be user-specified. The software provides parameterized bathymetries, for flat bottom
(depth), and linear sloping (parts of the) bottom, including run-up.
Bottom friction
Bottom friction can be applied at a specified part of the bottom; in the bottom friction formula
Rf = −
(
Cf
D +η
uu
)
typical values for the friction coefficient are C f ∈ 10−3 ,10−2 depending on Reynolds number and bottom.
2.3.3
Embedded wave influxing of various wave types
Wave influxing [9]
In AB the wave influxing is done through a source in the continuity equation.
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The influx position and a time signal of the desired elevation at the influx position have to be specified.
A choice can be made between
uni-directional influxing: for waves propagating in one direction,
in the direction of the positive (uni+) or negative (uni-) x-axis,
or bi-directional influxing for waves running symmetrically in both directions.
The spatial extent over which the influxing takes place can be chosen:
With a point-influx the ‘generation area’ is restricted to the immediate neighbourhood of the influx
point (using Dirac delta-functions). Then the desired time signal is modified into a (much) higher
modified time signal to guarantee the correct waves being influxed.
A smoother influx (better suited for high, steep waves) is area- (or area-short) influxing; then the
waves are generated over a broader interval, better suited for high, steep waves.
Nonlinear adjustment zone [9]
In order to prevent spurious modulations in the influxed wave when using a nonlinear wave model, a
nonlinear adjustment zone of length to be specified has to be applied; in the adjustment zone a coefficient
in front of the nonlinear terms in the Hamiltonian grows from 0 at the influx point to 1 at the end of the
zone. Typically, especially for harmonic waves, the required length will be at least 2 times the peakwavelength (and substantially more for steeper waves on shallower water).
Wave types
Any type of waves can be influxed from a user-specified time signal.
The software makes it possible to specify parameters for harmonic waves and for irregular waves with
Jonswap (JS) spectrum; for irregular waves, the phases are chosen randomly. The parameterized influx
signal will be stored for possible re-use for comparison of different evolution models.
Any influx signal will start and end by default with a smooth ramp function, the length of which can be
specified as a number of periods.
2.3.4 Initial value problems
Instead of wave influxing, data for an initial value problem (initial elevation profile and initial potential)
can be user-specified or chosen from predefined parameterized cases: a ‘Gaussian’ as a single hump, and
‘Nwave’ for an N-wave shaped wave, all with zero initial velocity.
The Gaussian is given by specifying the three parameters in
(
η ( x,=
0 ) A exp − ( x − xc ) σ 2
2
)
and the N-wave is the derivative with adjusted amplitude.
2.3.5 Model versions (evolution equations)
All combinations are possible of choices for dispersion (also user-specified), nonlinearity, and breaking,
with various choices for each item as described below.
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2.3.5.1 Dispersion
The main property of HAWASSI-AB is that it can handle exact dispersion of small-amplitude (so-called
linear) waves of any wave length, owe to the Fourier character and implementation of AB.
As a consequence, the continuity equation is exactly satisfied above flat bottom, and in a very good
approximation above varying bottom.
Besides that, mainly for educational/academic purposes, to enable simulations for models with
approximate dispersion, other predefined or user-defined dispersive models can be chosen:
 Shallow Water (SW) dispersion
 KdV (Korteweg-de Vries) dispersion [KdV 1895]
 BBM (Benjamin-Bona-Mahony) dispersion [BBM 1972]
 User-specified dispersion to be specified in input panels.
For the given or user-specified dispersion, all nonlinear terms are calculated consistently, above flat as well
as above varying bottom.
A brief explanation is given, referring to the table for the explicit expressions. The exact dispersion relation
ω = Ωex ( k )
is the correct expression for small amplitude waves. For kd → 0 , i.e. ‘rather’ long waves or
‘rather’ shallow water, this relation can be Taylor expanded, leading to

SW dispersion (first order Taylor)
ω = Ω SW ( k ) , which gives a translation of waves of any wave
length with the limiting speed c0 =

KdV – dispersion (3th order Taylor)
gd
ω = Ω KdV ( k ) ; note that short waves with kd > 1/ 6
will
travel in the opposite direction.
Remarks: Since influxing uses properties of the group-velocity, Uni-directional influxing in the
KdV model will show the short waves running in the ‘wrong’ direction, corresponding to the
dispersion relation. Bi-directional influxing will include these wrongly-directed waves, which is not
corresponding to the original KdV dispersion relation for uni-directional waves.

BBM-dispersion
ω = Ω BBM ( k ) , same as KdV in 3th order, but uni-directional
Note: To avoid problems with too poor dispersion, KdV and BBM uses exact dispersion for influxing.
In the table the explicit formulas and plots for the various cases are given.
Model
Dispersion relation
Exact
ω=
Ωex ( k ) =
sign ( k )
dispersion
Shallow
Ω SW ( k ) =
ω=
c0 k
Water
KdV
(
Plots Disp. relation & Phase velocity
gk tanh ( kd )
with c0 =
gd
ω=
Ω KdV ( k ) =
c0 k 1 − 16 ( kd )
(
2
)
)
BBM
ω=
Ω BBM ( k ) =+
c0 k 1 16 ( kd )
Userspecified
Provide dispersion relation and group velocity
in input panels, using only wave number k and
depth d as variables, in matlab-formula style.
2
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2.3.5.2 Nonlinearity
The order of nonlinearity of the Hamiltonian System (HS) is specified by the number in the present
version of HAWSSI-AB
HSm for orders m=1 (linear), 2, 3 and 4
To avoid aliasing in the Fourier implementation, the wave numbers have to be restricted to be at most a
fraction 1/(2m), with m the so-called cutfrac in the input for the specified code. (Note: This cut only applies
to the terms in the nonlinear equation; the spatial grid remains as determined by p, i.e. 2p gridpoints.)
2.3.5.3 Breaking [10]
Breaking of waves is modelled with an eddy viscosity method; to initiate the breaking process, the value
of a kinematic initiation breaking criterion has to be specified, the quotient of fluid velocity at the crest
and the velocity of the crest: U/C, usually in the interval [0.6, 1]
2.3.6 Internal flow calculations [5]
As an option it is possible to calculate (in a post-processing step, but indicated in the preparation-step)
 the horizontal and vertical velocities and accelerations of the interior fluid motion,
 all components of the total pressure,
at a user defined grid in horizontal and vertical direction in a specified time interval.
2.4 Software facilities
Facilities of the software include (to be described in Section 4)
• GUI for input of wave characteristics and model parameters, with efficient project management
• GUI for post-processing of the output of the wave simulation and comparison with data
• GUI for internal flow calculations
• Wave Calculator
• Time partitioned simulation to reduce (computer) hardware requirements
• Pre-processing step with warnings/suggestions for improved settings
• 18 TestCases (see Chapter 5) with examples of various kind, several of which include measurement
data to compare with simulations.
• Comparison with experimental data that have been reported in various publications (see the
references in Section 6 and the examples of TestCases in Section 5).
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3 Installing HAWASSI-AB software
`
The HAWASSI-AB installer will install the HAWASSI-AB code, including documentation.
HAWASSI-AB software is programmed under the MATLAB environment. The compiled MATLAB
applications can be run on PC’s that do not have MATLAB installed using the MATLAB Compiler
Runtime (MCR). The MCR will install MATLAB Runtime Libraries on the computer. The installation
consists therefore of two main steps: the installation of MCR and the installation of HAWASSI-AB.
3.1 System requirements
HAWASSI-AB (v.1.1) can run on Windows operating system with 64bit architecture. The minimum
memory (RAM) needed is 2GB (for some test cases and extensive applications 4GB RAM or more).
3.2 First step: Installing MCR
The HAWASSI-AB package (v.1.1) requires MCR installer for version MATLAB R2014b for Windows
operating system 64bit. The MCR installer can be downloaded directly from the MATLAB website:
http://www.mathworks.com/products/compiler/mcr/; after downloading install the MCR by double
clicking the installer and following the instruction in the installation wizard.
3.3 Second step: Installing HAWASSI-AB
After installing MCR, the installation of HAWASSI-AB can be started by double clicking the AB-installer
‘setup_HAWASSI_AB_v1.1.exe’ and following the instructions in the installation wizard.
During the installation process a copyright and non-liability agreement should be accepted to be able to
proceed.
After the installation is finished, start HAWASSI-AB from the shortcut on the Desktop. In the Main-GUI
that appears, under ‘Help’ go to ‘Activation’ and load ‘licence.lic’. Closing the software and starting again,
the licence will have been activated and the software can run for the licence-period. If a new version is
downloaded and installed, the same licence.lic file will be valid for the new version until expiration time.
In the Help- Menu of Main-GUI, the ‘Documentation’ will show this manual.
Test Cases can be found in ‘User \ My Documents \HAWASSI-AB1 \Testcases’.
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4 GUI’s of HAWASSI-AB
For ease of operation, HAWASSI-AB software includes three GUI’s, Graphical User Interfaces, as inputoutput managers.
Main GUI for providing
input for the simulation
Post-Processing GUI
specifying output of
simulation
for
the
Internal Flow GUI for
calculating
interior
flow
properties
The GUI’s will be described briefly in the next 3 sections. The meaning of most required input fields needs
no or little explanation; the choices that can be made will be described. The function of, and required input
format for input panels is indicated when the cursor is moved over it; when an optional panel is checked,
additional input fields may appear that have to be filled out.
User-input is accepted for various purposes to replace pre-programmed choices;
this is the case for
• influx signal
• initial wave profile
• Bathymetry
• dispersion function
• measurement data for comparison with simulations
The required lay-out of such files or formula’s is described in Section 4.4.
There is a simple Wave-Calculator that expects as input the period, frequency
or wave-length of a harmonic wave and the depth, and will then calculate all
other wave-relevant quantities; by also specifying the amplitude, the
calculated steepness is added.
4-13 | P a g e
4.1 Main GUI
Choosing the wave model characteristics, wave parameters, the domain, input signal and initial profile are
all managed in the Main GUI. An overview of the GUI with its main functionalities and input requirements
is shown in the Figure below; some of the ingredients are described thereafter.
4-14 | P a g e
Opening File will show
 Open Project: to go to a ‘project’ that has been created before (including the provided test cases) from
which data can be loaded to be inserted in the GUI.
 Save Project: Saves all data entered in the GUI (after pre-processing this info will be stored)
 Clear: clears the GUI from inserted input
 Quit HAWASSI
Opening Modules will give possibility to activate the Post-Processing GUI, the Internal Flow GUI and
the Wave Calculator.
Help contains info about the loaded version in About, this manual in Documentation, and Activation is
used for loading the licence (first use) or renewing the licence.
The Working Directory can be chosen and specified; the software will create a new folder named ‘Output’
if the working directory does not contain this folder yet; if the folder already exists, it will keep and use it.
By specifying a ‘Project Name’, the software will create a subfolder with that name under ‘Output’.
ALL output of a simulation will be stored in this subfolder, together with selected output of post-processing.
A User Note gives the possibility to provide details or a short description of the specific simulation; this
text will be copied to the log-file.
Input panels are separated to provide various details of the simulation to be executed:
 Model
 Initial condition
 Wave Input
 Xspace
 Bathymetry
 Options: there are two major options:
o Internal Flow: details can be given of the times at which in a post-processing step interior
flow properties have to be calculated (see the Interior Flow GUI)
o ODE partition: to reduce hardware requirements, it is possible to split one simulation time
interval in various consecutive time intervals; the data of each subinterval will be stored at
the end of that time interval, so that memory requirements and size of data are restricted.
The option to collect the data in one file after finishing the simulation can be checked.
Clicking the Pre-Proc (pre-processing) button will prepare the input before the actual evolution simulation
starts. A pop-up figure will summarise graphically the input, including the geometric lay-out and the quality
of dispersion used in the computation compared to exact dispersion. Warnings /suggestions may be given
to optimise the results of the simulation; the input can be changed, after which a new Pre-Proc step is
required.
The log-file is available after Pre-Proc, and contains info about the waves to be simulated; note that
calculated data may slightly differ from input values because of the statics used to calculate wave length,
period etc. The log-file will be updated after finishing the calculation with info about the computation time.
Successive simulations under the same ‘Project Name’ will be added successively in the log-file, but files
of computations will be overwritten (as warned on the GUI).
At the bottom of the screen there will be warnings/suggestions when specifying input.
After ‘RUN’, during the simulation a time-indicator estimates the remaining time.
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4.2 Post-Processing GUI
After a simulation is finished the Post-Processing (PP) GUI will automatically pop-up loaded with the
simulation data. The GUI can also be called directly from the Main GUI, and selected data can be loaded.
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The working directory will be as selected in Main-GUI after automatic pop-up when a simulation is
finished. Else the directory can be selected.
There are several panels in the PP-GUI:
 Simulation data is automatically loaded with results after finishing a simulation case; data of
previous projects (including Test Cases) can be selected to be loaded using ‘Other’ .
 Plotting Profile and Buoy to plot (multiple) wave profile(s)/time-signal(s) at specified time(s)/
position(s), with various options to include spectra, bottom, MTA (maximal temporal crests and
troughs), and quantitative information (see below)
 Animation, with options to make a gif-movie, on specified x- and t- interval
 Validation, to compare simulation results with experimental data or other simulations
For comparing at a certain position a time signal as simulated with a signal from a measurement or
a previous simulation, there is the option Time shifting to shift the simulated signal: automatically
optimized for best correlation or with a user-specified number of time steps. Quantitative
information is provided also (see below)
 Setting, options for any of the above graphical/animation output methods. (Be aware that
coarsening may change the quality, such as time traces or profiles, spectra, etc, and will influence
the validation results.)
Quantitative information
To analyse properties of simulated results, and/or to compare simulation results with experimental data or
other simulations, two formats are available:
 Graphical output: for time signals at specified positions calculated (amplitude or energy) spectra,
spatial values of the Energy, of MTA and MWL (height of Mean Water Level), Hs, Skewness,
Asymmetry and Kurtosis and a position vs. time plot of breaking events.
 Quantitative information for time signals at a specified position:
o the correlation of the signals (for validation)
o the value (or quotient) of the Variance, Skewness, Asymmetry and Kurtosis of simulated
and measured signal
The definitions of these quantities is as follows, for signals with zero-mean and with < > denoting
time averaging.
The correlation of the simulated signal s(t) and the measured signal m(t) is
corr ( s, m ) = s.m
s 2 . m2
and for a signal s (with H the Hilbert transform)
=
H s 4 var
=
, var
=
s 2 , Sk
s3
3/2
s2 =
, As
( Hs )3
3/2
s 2=
, Ku
s4
s2
2
4-17 | P a g e
4.3 Internal Flow GUI
In order to calculate (a posteriori) internal flow properties, it is needed that before the simulation is started
this has been indicated in the Main-GUI, since some data during the calculation will be stored to be used
for the interior flow calculations. This storage will slow down the simulation, and therefore the times of
interest can be indicated. Quantities that can be computed are
 the horizontal and vertical velocities and accelerations of the interior fluid motion,
 each of the components of the total pressure.
Be aware that the amount of data can become very large, depending on the chosen discretization settings.
4-18 | P a g e
4.4 Required lay-out of user defined input files
User input of data-files for various purposes need to be prepared with an extension (.mat, .dat,
.txt, etc) with a specified format as described below.
4.4.1 Influx time signal
A 2-column matrix (time, elevation)
first column the (equidistant) time ([s]),
second column the corresponding elevation ([m]).
4.4.2 Initial wave profile
A 3-column matrix (space point x, elevation)
first column the (equidistant) x-value ([m]) (covering the whole interval; if only partially, the data
will be taken to have value 0),
second column the prescribed elevation ([m]),
third column the prescribed tangential velocity (space derivative of the potential, [m/s])
4.4.3 Bathymetry
A 2-column matrix (space point x, bathymetry)
first column the (equidistant) x-value ([m]) (covering the whole interval),
second column the corresponding bathymetry ([m]).
4.4.4 External measurement data
Time signals at m measurement positions. Matrix with (m+1) columns:
First row, columns 2 to m+1: specify measurement position (0, position x_1, …, position x_m)
Next rows: time and elevation at the measurement positions (time, elevation_1, …, elevation_m)
4.4.5
Interior flow External measurement data
Make a separate file for each horizontal measurement point with a name depending on the
quantity that has been measured, for instance data_U_X1.mat , for a measurement of horizontal
(U) or vertical (V) velocity at position X1. Each data file has the following format.
Time signals at m measurement positions in the vertical direction: matrix (2+T_length, (m+1))
(1, 1) = [horizontal position x], (1, 2) = [water depth at position x];
(2, 2: m+1) = [vertical positions of measurement];
(3: T(end),1:m+1) = [time, u_1: u_m]
4.4.6 Dispersion relation
The software can handle different dispersion relations. Default is the exact dispersion, but other dispersion
relations for Shallow Water, KdV and BBM dispersion are predefined and can be dealt with for simulations
of nonlinear waves over bathymetry.
A user defined dispersion relation can be given through input-panels in matlab-formula style. Needed are the
 dispersion relation ω = Ωuser ( k , d ) (which should be defined as an odd function) and
 the corresponding group velocity Vuser = d Ωuser dk .
4-19 | P a g e
5 Test cases
HAWASSI-AB provides 18 Test-cases which are identified with a code of which the first letter has the
following meaning:
F: for various cases of non-breaking waves above Flat bottom
B: for various cases of non-breaking waves above non-flat bathymetry
Br: for various cases of breaking waves above flat or varying bottom
R: for run-up on a coast
The basic properties of the test cases are listed with references to relevant publications in the next sections.
Acknowledgements:
We are very grateful to be allowed to use measurement data of
 MARIN (Maritime Research Institute Netherlands), Dr. T. Bunnik
 TUD (Technical University of Delft), Prof.dr. R.H.M. Huijsmans
 Authors of publications:
o Beji & Battjes
 S. Beji, J. Battjes, Numerical simulation of nonlinear wave propagation
over a bar, Coastal Engineering 23 (1994) 1 – 16.
 S. Beji, J. Battjes, Experimental investigation of wave propagation over a
bar, Coastal Engineering 19 (1993) 151 – 162.
o Ting& Kirby
 F. C. Ting, J. T. Kirby, Observation of undertow and turbulence in a
laboratory surf zone, Coastal Engineering 24 (1994) 51 – 80.
o Wei e.a.
 G. Wei, J. T. Kirby, S. T. Grilli, R. Subramanya, A fully nonlinear
Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady
waves, Journal of Fluid Mechanics 294 (1995) 71–92.
Only by testing with realistic data the software can be validated and improved.
5-20 | P a g e
5.1 Non-breaking waves above flat bottom
5.1.1
1F001Foc: MARIN_202002, Strong focussing wave
Dynamic Model
Dispersion Model
Bathymetry
: HS2F
: OmExact
: Flat; Depth: 1[m]
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
5.1.2
0.013[m]
1.304[s]
4.817[rad/s]
2.404
2.613[m]
2.003[m/s]
0.015
2.6133
2.404
Intermediate depth
Focussing wave: using dispersion to
generate high waves in wave tanks.
Simulation of MARIN – measurement.
IF002Draup: MARIN204001, Draupner Wave
Dynamic Model
Dispersion Model
Bathymetry
: HS2F
: OmExact
: Flat; Depth: 1[m]
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepnest (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
0.076[m]
2.061[s]
3.048[rad/s]
1.156
5.437[m]
2.638[m/s]
0.044
5.437
1.156
Intermediate depth
Environmental Freak Wave, measured at
the Draupner platform in the NorthSea
1995 (18.5 crest height on 75m depth);
here for simulation in MARIN wave tank.
Simulation of MARIN – measurement
5.1.3 IF003Irreg MARIN_224002F, Irregular wave
in deep water
Dynamic Model
: HS4F
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 510[m]
Significant wave Height (Hs) : 9.829[m]
Peak period (Tp)
: 13.887[s]
Peak frequency (nu)
: 0.452[rad/s]
Peak wave-number (kp)
: 0.021
Peak wave-length
: 301.056[m]
Peak phase speed
: 21.679[m/s]
Steepness (kp*(Hs./2))
: 0.103
Relative wave-length(lambda/h): 0.59031
(kp*h)
: 10.644
Category
: Deep water
Observe nonbreaking freak wave on deep
water
Simulation of MARIN – measurement
5-21 | P a g e
5.1.4
IF004Wall AB1_Wall, Full wave reflection from wall [5]
Dynamic Model
: HS1FWall
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 5[m]
Signal type
: Harmonic
Significant wave Height (Hs) : 0.1[m]
Peak period (Tp)
: 2[s]
Peak frequency (nu)
: 3.141[rad/s]
Peak wave-number (kp)
: 1.006
Peak wave-length
: 6.247[m]
Peak phase speed
: 3.123[m/s]
Steepness (kp*(Hs./2))
: 0.05
Relative wave-length(lambda/h): 1.2494
(kp*h)
: 5.029
Category
: Deep water
5.1.5
1F005WallFreq, Frequency dependent wall reflection [5]
Dynamic Model
: HS2FWall
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 25[m]
Signal type
: User-defined
Significant wave Height (Hs) : 0.1[m]
Peak period (Tp)
: 12.362[s]
Peak frequency (nu)
: 0.508[rad/s]
Peak wave-number (kp)
: 0.036
Peak wave-length
: 172.264[m]
Peak phase speed
: 13.935[m/s]
Steepness (kp*(Hs./2))
: 0.002
Relative wave-length(lambda/h): 6.8906
(kp*h)
: 0.912
Category
: Intermediate depth
5.1.6
Full wave reflection of a linear harmonic
wave from a wall
Reflection of an irregular wave with
frequency dependent reflection at a wall:
from full reflection of long waves to half
for short waves, shown in the spectrum
plot (red line)
1F006Bor, Undular bore
Dynamic Model
: HS2F
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 1[m]
INITIAL VALUE PROBLEM
Undular bore, non-breaking. Settings and
results as in Wei e.a.
See also Testcase 1FBr005Bor (section
5.2.5) for the breaking bore
5-22 | P a g e
5.2 Breaking waves above flat bottom
5.2.1 1FBr001Foc MARIN203001, Strong focussing wave
Dynamic Model
: HS4brF
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 1[m]
Signal type
: User-defined
Significant wave Height (Hs) : 0.038[m]
Peak period (Tp)
: 1.446[s]
Peak frequency (nu)
: 4.346[rad/s]
Peak wave-number (kp)
: 1.998
Peak wave-length
: 3.145[m]
Peak phase speed
: 2.176[m/s]
Steepnest (kp*(Hs./2))
: 0.038
Relative wave-length(lambda/h): 3.1454
(kp*h)
: 1.998
Category
: Intermediate depth
5.2.2
Simulation of MARIN – measurement.
1FBr002Foc TUD1403Foc7, Focussing
Breaking Wave [3]
Dynamic Model
: HS3brF
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 2.132[m]
Significant wave Height (Hs) : 0.078[m]
Peak period (Tp)
: 1.99[s]
Peak frequency (nu)
: 3.157[rad/s]
Peak wave-number (kp)
: 1.041
Peak wave-length
: 6.038[m]
Peak phase speed
: 3.034[m/s]
Steepness (kp*(Hs./2))
: 0.04
Relative wave-length(lambda/h): 2.832
(kp*h)
: 2.219
Category
: Intermediate depth
5.2.3
Strong focussing, breaking wave
Strong focussing breaking wave
Simulation of TUD – measurement.
1FBr003BiB TUD1403Bi8, Bichromatic wave breaking
Dynamic Model
: HS3brF
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 2.132[m]
Significant wave Height (Hs) : 0.356[m]
Peak period (Tp)
: 1.634[s]
Peak frequency (nu)
: 3.845[rad/s]
Peak wave-number (kp)
: 1.511
Peak wave-length
: 4.157[m]
Peak phase speed
: 2.544[m/s]
Steepness (kp*(Hs./2))
: 0.269
Relative wave-length(lambda/h): 1.9498
(kp*h)
: 3.222
Category
: Deep water
Bichromatic breaking wave
5-23 | P a g e
5.2.4 1FBr004Ir MARIN_223002F, Irregular wave breaking [10]
Dynamic Model
Dispersion Model
Bathymetry
: HS4brF
: OmExact
: Flat; Depth: 510[m]
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
5.2.5
10.23[m]
12.656[s]
0.496[rad/s]
0.025
250.057[m]
19.758[m/s]
0.129
0.49031
12.815
Deep water
1FBr005Bor Undular_breaking bore [5]
Irregular wave breaking
Simulation of MARIN – measurement.
Dynamic Model
: HS2brF
Dispersion Model
: OmExact
Bathymetry
: Flat; Depth: 1[m]
INITIAL VALUE PROBLEM
Breaking undular bore, at two different
times. Settings and results as Wei e.a., for
higher amplitudes for which no validation
data are available.
See also Testcase 1F006Bor (section
5.1.6) for the non-breaking bore
5-24 | P a g e
5.3 Non-breaking waves above non-flat bathymetry
5.3.1
2B001IrSlope MARIN_103001, Irregular wave above a slope [5, 13]
Dynamic Model
: HS2B
Dispersion Model
: OmExact
Bathymetry
: Slope;
Max Depth: 0.6[m] Min Depth : 0.3[m]
Slope: 0.05[m] Foot slope position: -30[m]
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
5.3.2
0.062[m]
1.701[s]
3.694[rad/s]
1.769
3.551[m]
2.088[m/s]
0.055
5.9186
1.062
Intermediate depth
2B002HarmBar, Harmonic over bar
Dynamic Model
Dispersion Model
Bathymetry
Non-breaking irregular wave over slope.
Observe Freak Wave above deep part at
x=-71, t=282.8
Simulation of MARIN – measurement.
: HS2U
: OmExact
: Under-water bar;
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
0.027[m]
2.026[s]
3.101[rad/s]
1.675
3.75[m]
1.851[m/s]
0.023
9.3752
0.67
Intermediate depth
5.3.3 2B003IrBar, Irregular wave over bar
Harmonic waves over under water bar.
Strong mode generation: compare spectra
at x=8 (red) and x=16 (black) below
Simulation of Beji & Battjes Experiment
BB94SL:Jonswap;Low Freq: Non breaking
Dynamic Model
Dispersion Model
: HS2U
: OmExact
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Influx position
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
0.021[m]
1.666[s]
3.772[rad/s]
2.109
2.979[m]
1.788[m/s]
0.022
5.7157[m];
7.447
0.844
Intermediate depth
Simulation of Beji & Battjes Experiment
5-25 | P a g e
5.4 Breaking waves above bathymetry
5.4.1
2BBr001HarmBar [10]
Dynamic Model
Dispersion Model
Bathymetry
: HS2brU
: OmExact
: Under-water bar
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepnest (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
5.4.2
0.058[m]
2.537[s]
2.476[rad/s]
1.305
4.816[m]
1.898[m/s]
0.038
12.0391
0.522
Intermediate depth
Breaking harmonic waves over bar
Simulation of Beji & Battjes experiment
2BBr002IrBar Irregular wave, Spilling breaker [5]
Dynamic Model
Dispersion Model
Bathymetry
: HS2brU
: OmExact
: Under-water bar
Significant wave Height (Hs) :
Peak period (Tp)
:
Peak frequency (nu)
:
Peak wave-number (kp)
:
Peak wave-length
:
Peak phase speed
:
Steepness (kp*(Hs./2))
:
Relative wave-length(lambda/h):
(kp*h)
:
Category
:
0.035[m]
1.898[s]
3.31[rad/s]
1.806
3.479[m]
1.832[m/s]
0.032
8.6966
0.722
Intermediate depth
Spilling breaker over bar
Simulation of Beji & Battjes experiment
5-26 | P a g e
5.5 Run-up of waves (breaking and non-breaking)
5.5.1 3R001Harm: Harmonic Run-up (non-breaking)
Dynamic Model
: HS2BR
Dispersion Model
: OmExact
Bathymetry
: Slope (Shore);
Max Depth: 0.5[m]
Signal type
: Harmonic
Significant wave Height (Hs) : 0.006[m]
Peak period (Tp)
: 10[s]
Peak frequency (nu)
: 0.62[rad/s]
Peak wave-number (kp)
: 0.281
Peak wave-length
: 22.384[m]
Peak phase speed
: 2.207[m/s]
Steepness (kp*(Hs./2))
: 0.001
Relative wave-length(lambda/h): 44.7688
(kp*h)
: 0.14
Category
: Shallow water
Harmonic nonbreaking wave run-up on
1:25 coast
5.5.2 3RBr001Harm: Spilling Breaker Run-up (with interior flow) [5]
Dynamic Model
: HS2brUR
Dispersion Model
: OmExact
Bathymetry
: Slope (Shore);
Max Depth: 0.4[m]
Signal type
: Harmonic
Significant wave Height (Hs) : 0.125[m]
Peak period (Tp)
: 2[s]
Peak frequency (nu)
: 3.141[rad/s]
Peak wave-number (kp)
: 1.7
Peak wave-length
: 3.695[m]
Peak phase speed
: 1.847[m/s]
Steepness (kp*(Hs./2))
: 0.106
Relative wave-length(lambda/h): 9.2387
(kp*h)
: 0.68
Category
: Intermediate depth
Harmonic wave run-up on 1:35 coast,
spilling breaker
Simulation, including interior flow
properties, of Ting & Kirby experiment
5-27 | P a g e
6 References
6.1 References to basic papers and applications
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
E. van Groesen & Andonowati, Hamiltonian Boussinesq formulation of Wave-Ship interactions, Part 1: Evolution
equations, submitted
R. Kurnia, T. van den Munckhof, C.P. Poot, P. Naaijen, R.H.M. Huijsmans & E. van Groesen, Simulations for
design and reconstruction of breaking waves in a wavetank, OMAE 2015 Vol2,41633, 7 pages
R. Kurnia & E. van Groesen, Spatial-spectral Hamiltonian Boussinesq wave simulations, Advances in
Computational and Experimental Marine Hydrodynamics, VOL. 2 Conference Proceedings, 2015, pp. 19-24,
ISBN: 978-93-80689-22-7R.
Kurnia & E. van Groesen, Localization for spatial-spectral implementations of 1D Analytic Boussinesq equations,
Wave Motion to be published
R. Kurnia & E. van Groesen, Design of Wave Breaking Experiments and A-Posteriori Simulations, Memorandum
2042, (January 2015), http://www.math.utwente.nl/publications ISSN 1874−4850
R. Kurnia & E. van Groesen, Localization in Spatial-Spectral Methods for Water Wave Applications, Proceedings
ICOSAHOM 2014, Submitted
W. Kristina. O. Bokhove & E. van Groesen, Effective coastal boundary conditions for tsunami wave run-up
over sloping bathymetry, Nonlin.Processes GeoPhys, 21(2014) 987-1005
Lie S Liam, D. Adyia & E. van Groesen, Embedded wave generation for dispersive surface wave models, Ocean
Engineering 80 (2014) 73-83
R. Kurnia & E. van Groesen, High Order Hamiltonian Water Wave Models with Wave-Breaking Mechanism,
Coastal Engineering 93 (2014) 55–70
Lie She Liam, Mathematical modelling of generation and forward propagation of dispersive waves PhD-Thesis
UTwente, 15 May 2013
A.L. Latifah & E. van Groesen, Coherence and Predictability of Extreme Events in Irregular Waves, Nonlin.
Processes Geophys, 19 (2012)199-213
E. van Groesen & I. van der Kroon, Fully dispersive dynamic models for surface water waves above varying
bottom, Part 2: Hybrid spatial-spectral implementations, Wave Motion 49 (2012) 198-211
E. van Groesen & Andonowati, Fully dispersive dynamic models for surface water waves above varying bottom,
Part 1: Model equations, Wave Motion 48 (2011) 657-666
E. van Groesen & Andonowati, Time-accurate AB-simulations of irregular coastal waves above bathymetry,
Proceedings of the Sixth International Conference on Asian and Pacific Coasts (APAC 2011) December 14 – 16,
2011, Hong Kong, China, World Scientific ISBN: 978-981-4366-47-2, pp.1854-1864
E. van Groesen, T. Bunnik & Andonowati, Surface wave modelling and simulation for wave tanks and coastal
areas, International Conference on Developments in Marine CFD, 18 - 19 November 2011, Chennai, India,
RINA, ISBN: 978-1-905040-92-6, p. 59-63
L. She Liam & E. van Groesen, Variational derivation of KP-type equations, Physics Letters A, 374(2010) 411415
E. van Groesen, Andonowati, L. She Liam & I. Lakhturov, Accurate modelling of uni-directional surface waves,
Journal of Computational and Applied Mathematics 234 (2010) 1747-1756
E. van Groesen, L. She Liam, I. Lakhturov & Andonowati, Deep water Periodic waves as Hamiltonian Relative
Equilibria, Proceedings of Waves 2007, N. Biggs e.a. (eds), Reading UK, 23-27 July 2007, pp. 482-484;
E. van Groesen & Andonowati, Variational derivation of KdV-type of models for surface water waves, Physics
Letters A 366 (2007)195-201
6-28 | P a g e
6.2 Other references
Beji & Battjes
 S. Beji &J. Battjes, Numerical simulation of nonlinear wave propagation over a bar,
Coastal Engineering 23 (1994) 1 – 16.
 S. Beji & J. Battjes, Experimental investigation of wave propagation over a bar, Coastal
Engineering 19 (1993) 151 – 162.
BBM
 T.B. Benjamin, J.L. Bona & J.J. Mahony, On model equations for long waves in
nonlinear dispersive systems, Phil. Trans. Roy. Soc. London A272 (1972)47
Broer
 L.J.F. Broer, Approximate equationsn for long water waves, Appl. Sc. Res., 31(1975)377395
KdV
 D.J. Korteweg & G. de Vries, On the change of form of long waves advancing in a
rectangular canal and anewtype of long stationary waves, Phil. Mag. 39(1895)422
Ting& Kirby
 F. C. Ting & J. T. Kirby, Observation of undertow and turbulence in a laboratory surf zone,
Coastal Engineering 24 (1994) 51 – 80.
Wei e.a.
 G. Wei, J. T. Kirby, S. T. Grilli & R. Subramanya, A fully nonlinear Boussinesq model for
surface waves. Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics 294
(1995) 71–92.
YAB LabMath,
 YAB LabMath , Water Wave Modelling & Simulation, with Introduction to HAWASSI-software
Zakharov
 V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep
fluid, J. of Mechanics and Technical Physics 2(1968)190-194
6-29 | P a g e
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