1993 PhD VanDaalen

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Daalen, Edwin Frank George van

Numerical and theoretical studies of water waves and floating bodies /Edwin Frank George van Daalen. - [S.l. : s.n.]. - Ill., fig., tab.Proefschrift Enschede. - Met index, lit. opg. - Met samenvatting inhet Nederlands.ISBN 90-9005656-4Trefw.: hydrodynamica / integraalvergelijkingen / variatierekening

Numerical and Theoretical Studies of

Water Waves and Floating Bodies

Proefschrift

ter verkrijging van

de graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus

prof.dr. Th.J.A. Popma,

volgens besluit van het College van Dekanen

in het openbaar te verdedigen

op vrijdag 8 januari 1993 te 16.45 uur

door

Edwin Frank George van Daalen

geboren op 13 juni 1965 te Amsterdam

Dit proefschrift is goedgekeurd door de promotor:

prof.dr.ir. P.J. Zandbergen

(Faculteit der Toegepaste Wiskunde)

en de overige leden van de promotiecommissie:

prof.dr.ir. E.W.C. van Groesen

(Faculteit der Toegepaste Wiskunde)

prof.dr.ir. L. van Wijngaarden

(Faculteit der Technische Natuurkunde)

prof.dr.ir. A.J. Hermans

(Technische Universiteit Delft)

prof.dr.-ing. O. Mahrenholtz

(Technische Universitat Hamburg-Harburg)

to Sonja

Contents

1 Introduction 11.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The investigations: past and present . . . . . . . . . . . . . . . . . . . . . 31.3 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Suggested references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Mathematical Formulation and Numerical Algorithm 11

2 Mathematical Statement of the Problem 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Force mechanisms in fluid flow . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Impermeable fixed boundaries . . . . . . . . . . . . . . . . . . . . . 192.4.2 Impermeable moving boundaries . . . . . . . . . . . . . . . . . . . 202.4.3 Free boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.4 Open boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Wave-body formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.1 Ship wave-making and wave-resistance . . . . . . . . . . . . . . . . 232.5.2 Linearized oscillatory motion . . . . . . . . . . . . . . . . . . . . . 242.5.3 Forward speed radiation-diffraction . . . . . . . . . . . . . . . . . . 262.5.4 Nonlinear motion at zero forward speed . . . . . . . . . . . . . . . 28

2.6 Problem definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Boundary Integral Equation Formulations 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Integral equations in potential theory . . . . . . . . . . . . . . . . . . . . . 373.3 Well-posedness: existence and uniqueness . . . . . . . . . . . . . . . . . . 413.4 Choice of integral equation method . . . . . . . . . . . . . . . . . . . . . . 433.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

v

4 Algorithm for Wave-Body Simulations 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Review of methods for nonlinear ship motions . . . . . . . . . . . . . . . . 504.3 Basic algorithm for nonlinear water waves . . . . . . . . . . . . . . . . . . 524.4 Extension to nonlinear ship motions . . . . . . . . . . . . . . . . . . . . . 554.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

II Numerical Results 65

5 Impulsive Wavemaker Motion 675.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Governing equations and small time expansions . . . . . . . . . . . 705.2.2 Leading order solutions . . . . . . . . . . . . . . . . . . . . . . . . 715.2.3 Initial and small time behaviour . . . . . . . . . . . . . . . . . . . 73

5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.1 Initial behaviour on the wavemaker . . . . . . . . . . . . . . . . . . 775.3.2 Initial behaviour on the bottom . . . . . . . . . . . . . . . . . . . . 775.3.3 Initial behaviour on the free surface . . . . . . . . . . . . . . . . . 785.3.4 Free surface elevation for small time . . . . . . . . . . . . . . . . . 78

5.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Hydrodynamic Mass and Damping 956.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2 Mathematical model for two-dimensional motion . . . . . . . . . . . . . . 976.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3.1 Circular cylinder in heaving motion . . . . . . . . . . . . . . . . . 1006.3.2 Circular cylinder in swaying motion . . . . . . . . . . . . . . . . . 103

6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Cylinders in Free Motion 1137.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2 Circular heaving cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.3 Rectangular heaving cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 1177.4 Rectangular rolling cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

III Variational Principles and Hamiltonian Formulations 129

8 Lagrangian and Hamiltonian Formulations 1318.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.2 Luke’s variation principle for water waves . . . . . . . . . . . . . . . . . . 1328.3 Zakharov & Broer’s Hamiltonian formulation . . . . . . . . . . . . . . . . 1368.4 Wave-body formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.4.1 A variation principle for wave-body interactions . . . . . . . . . . . 1428.4.2 A Hamiltonian formulation for wave-body interactions . . . . . . . 146

8.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9 Symmetries and Conservation Laws 1579.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.2 Conserved densities for water waves . . . . . . . . . . . . . . . . . . . . . 1599.3 Conservation laws for the wave-body problem . . . . . . . . . . . . . . . . 164

9.3.1 Invariants for the two-dimensional case . . . . . . . . . . . . . . . . 1649.3.2 Invariants for the three-dimensional case . . . . . . . . . . . . . . . 171

9.4 The virial and conservation law no.8 . . . . . . . . . . . . . . . . . . . . . 1759.4.1 The virial connection . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.4.2 The circulation alternative . . . . . . . . . . . . . . . . . . . . . . 1799.4.3 Fluid-filled deformable bodies . . . . . . . . . . . . . . . . . . . . . 1809.4.4 The ‘broken symmetry’ argument . . . . . . . . . . . . . . . . . . . 181

9.5 Symmetry-breaking boundaries . . . . . . . . . . . . . . . . . . . . . . . . 1829.6 Numerical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

10 Radiation Boundary Conditions 19310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19310.2 Theory for continuous systems . . . . . . . . . . . . . . . . . . . . . . . . 19410.3 Application to one-dimensional wave equations . . . . . . . . . . . . . . . 20210.4 Application to the water-wave problem . . . . . . . . . . . . . . . . . . . . 218

10.4.1 The Lagrangian-Hamiltonian approach . . . . . . . . . . . . . . . . 21910.4.2 The Eulerian approach . . . . . . . . . . . . . . . . . . . . . . . . . 226

10.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

11 Conclusions and Recommendations 233

IV Appendices 235

A An Expression for φtn on the Body 237

B A Body Surface Integral Condition 243

C The Angular Body Momentum 247

Preface

This thesis was written within the framework of a research project entitled

Further development of a three-dimensional model for nonlinear surface waves,with respect to the interaction with floating objects and the influence of bound-aries.

As already indicated by the title, one of the main objectives was the extension of anexisting method — developed by Romate1 — to an algorithm for the numerical simulationof nonlinear free surface waves in hydrodynamic interaction with floating bodies; thisparticular part of the investigations is described here.

In the development of the numerical model and the computer code, many problemsof both theoretical and practical nature were encountered. From time to time the dailysearch for ‘engineering’ solutions was interrupted by periods in which these problemswere considered from a more fundamental point of view. This theoretical research — alsorecorded here — resulted in some basic changes in the numerical model. In addition, anumber of theories concerning water waves and floating bodies was developed.

The investigations were financially supported by the Netherlands Technology Foundation(STW) — a subdivision of the Dutch Foundation for Scientific Research (NWO) — undergrants TWI88.1460 and TWI80.1460.

Most of the computations were done on supercomputers; from 1989 to 1990 on aCRAY-XMP, and from 1991 on a CRAY-YMP. Computation time was granted by theDutch Foundation for Supercomputer Facilities (NCF). Technical support from the Stich-ting Academisch Rekencentrum Amsterdam (SARA) is acknowledged.

A committee of future users of the computer code accompanied the project. The commit-tee members were from the Maritime Research Institute Netherlands (MARIN), DelftHydraulics, the Royal Dutch Shell Laboratory Amsterdam (KSLA), the University ofTwente, and the Netherlands Technology Foundation.

I would like to thank the board of directors of MARIN for the unique opportunity to dothis work in such an inspiring research institute.

Thanks go to Hans P. van der Kam (Automation and Instrumentation Department)and Hans Zeller (Audio-Visual Department) for their enthusiasm and assistance in visu-alization and video-editing, and to Henk Luisman (Offshore Research Department) formaking the illustrations.

1Romate, J.E. 1989. The Numerical Simulation of Nonlinear Gravity Waves in Three Dimensionsusing a Higher Order Panel Method. Ph.D. thesis, University of Twente. Enschede, The Netherlands.

ix

Special thanks go to Ir. Rene H.M. Huijsmans (Offshore Research Department) andIr. Hoyte C. Raven (Ship Research Department) for introducing me to the field of shiphydrodynamics.

Also I want to thank Dr. Huib J. de Vriend (Delft Hydraulics) for his interest andparticipation in this project.

Very special thanks go to Ir. Jan Broeze for his good-fellowship during the past fouryears, and for our fruitful cooperation which made our investigations so successful.

Next, I would like to thank Dr. Johan E. Romate (KSLA) for his invaluable contributionsto this research, both in the past and in the present.

Furthermore, I would like to thank some members of the staff of the Faculty of AppliedMathematics of the University of Twente.

I am grateful to Dr. Douwe Dijkstra for his helpful comments in ‘numerical matters’.I feel very much indebted to professor van Groesen for his interest and active partic-

ipation in this research; our frequent discussions culminated in four joint papers, whichin turn formed a sound basis to part III of this thesis.

It is a great pleasure to express my deep gratitude to professor Zandbergen for his crit-icism, advice, and encouragement; it has been a privilege to do this research under hissupervision.

I thank my parents and my brother for their stimulation in my study and work.

Finally, I would like to express my sincere love to my wife Sonja; in the end, her enormoussupport and endless patience have helped me to complete this work.

Deventer, The Netherlands Ed van DaalenJanuary 8th, 1993

The Argonautica

A narrow escape: the Argo passes through the Symplegades.

Turning over the pages of this thesis, the interested reader inevitably will encounter sev-eral passages from the Argonautica, an ancient Greek tale written by Apollonius Rhodius.To those who are familiar with the mathematics and physics described here, each selectedfragment may come as a pleasant break while reading consecutive chapters. Perhaps evenmore important is the fact that this dissertation, despite its technical nature, still offerssomething worthwhile to read to those who are not familiar with exact sciences.

Several motives induced me to lard this thesis with a selection of passages from theArgonautica. First of all, the tale of the Argonauts has a pronounced nautical character.In this sense, the Argonautica is in close harmony with the investigations described here.The second reason is that ever since my first lesson in classical languages I have felta natural interest in Greek and Roman literature. This particular part of the literaryclassics is full of fetching parables and moralizing legends; immortal gods and mortalheroes are involved in an eternal struggle for the sympathy of the reader. Finally, inmy opinion the Argonautica has unjustly been eclipsed by Homer’s well-known Iliad andOdyssee; this is the third motive for this personal mixture of literature and exact sciences.

xi

Cited from the jacket of the Loeb Classical Library publication, with an English transla-tion by R.C. Seaton:

‘Apollonius ‘of Rhodes’ was a Greek grammarian and epic poet of Alexandriain Egypt and lived late in the third century and early in the second centuryb.c. While still young he composed his extant epic poem of four books on thestory of the Argonauts. When this work failed to win acceptance he went toRhodes where he not only did well as a rhetorician but also made a success ofhis epic in a revised form, for which the Rhodians gave him the ‘freedom’ oftheir city; hence his surname. On returning to Alexandria he recited his poemagain, with applause. In 196 b.c. Ptolemy Epiphanes made him the librarianof the Museum (the University) at Alexandria. His Argonautica is one of thebetter minor epics, remarkable for originality, powers of observation, sincerefeeling, and depiction of romantic love. His Jason and Medea are natural andinteresting, and did much to inspire Virgil (in a very different setting) in thefourth book of the Aeneid.’

The motive of the voyage of the band of Greek heroes (named Argonauts after their ship,the Argo) is the command of Pelias, king of Iolcus, to bring back the ‘golden fleece’ fromColchis. This command is based on Pelias’ desire to destroy the hero Jason, while thedivine aid given to Jason results from the intention of the goddess Hera to punish Peliasfor his neglect of the honour due to her.

The first and second books describe the history of the voyage to Colchis, whereApollonius interweaves with his narrative legends, accounts of local customs. This partof the Argonautica is filled with many exciting episodes, such as the rape of Hylas, theboxing match between Polydeuces and Amycus, the prophecies and counsels of Phineus,and the passing through the Symplegades. The third book is occupied for the greaterpart by the episode of the love of Jason and Medea2, and the accomplishment of Jason’stask with her aid. The fourth book, describing the return voyage, is invaluable for itsamazing geography, such as the supposed junction of the Rhine, Rhone, and Po rivers,the Libyan desert, and the so-called Tritonian Lake. Each book has its own specific topicand character, and the unity of the legend is that of the voyage itself.

May it please the reader!

2Medea was a daughter of the king of Colchis.

Chapter 1

Introduction

Beginning with thee, O Phoebus1, I will recount the famous deeds of menof old, who, at the behest of King Pelias, down through the mouth of Pontus2

and between the Cyanean rocks3, sped well-benched Argo in quest of the goldenfleece.

Such was the oracle that Pelias heard, that a hateful doom awaited him— to be slain at the prompting of the man whom he could see coming forthfrom the people with but one sandal. And no long time after, in accordancewith that true report, Jason crossed the stream of wintry Anaurus on foot,and saved one sandal from the mire, but the other he left in the depths heldback by the flood. And straightway he came to Pelias to share the banquetwhich the king was offering to his father Poseidon and the rest of the gods,though he paid no honour to Pelasgian Hera. Quickly the king saw him andpondered, and devised for him the toil of a troublous voyage, in order that onthe sea or among strangers he might lose his home-return.

The ship, as former bards relate, Argus wrought by the guidance of Athena.But now I will tell the lineage and the names of the heroes, and of the longsea-paths and the deeds they wrought in their wanderings; may the Muses bethe inspirers of my song!

Argonautica, Book I, Verses 1-22.

1i.e. Apollo2i.e. the Black Sea3i.e. the Symplegades

1

2 CHAPTER 1. INTRODUCTION

1.1 Historical background

Over the past three centuries, surface wave problems have interested a considerablenumber of mathematicians, beginning in the early eighteenth century with Euler and theBernoullis in Switzerland, and continuing in the late eighteenth and the early nineteenthcentury with Lagrange, Cauchy, Navier, and Poisson in France. Later the British schoolof mathematical physicists paid attention to these problems, and notable contributionswere made by Airy, Stokes, Rayleigh, Kelvin, Michell, and Lamb. In the latter part of thenineteenth century the French once more took up the subject; the work done by Boussi-nesq in this field is historic, as well as the Dutch contributions from Korteweg and deVries. Later, Poincare made excellent contributions with regard to figures of equilibriumof rotating and gravitating liquids. One of the most outstanding accomplishments fromthe purely mathematical point of view — the proof of the existence of progressing wavesof finite amplitude — was made by Levi-Civita; the extension of this proof to waves ina canal of finite depth was accomplished by Struik.4

The subject of surface gravity waves covers a wide range, whether regarded from theviewpoint of physical problems which occur, or from the point of view of the mathemat-ical ideas and methods to solve these problems. The physical problems vary from wavemotion over sloping beaches to flood waves in rivers, the motion of ships in a sea-way,free oscillations of enclosed bodies of water such as lakes and harbors, to mention justa few. The mathematical tools employed comprise the whole of the methods developedin the classical linear mathematical physics concerned with partial differential equations,as well as a good part of what has been learned about the nonlinear problems of math-ematical physics. Thus potential theory and the theories of linear and nonlinear waveequations, together with tools such as conformal mapping and complex variable methodsin general, the Laplace and Fourier transform techniques, methods employing a Green’sfunction, integral equations, etc. are used. The nonlinear problems are of both ellipticand hyperbolic type.

Nowadays, the evolution of nonlinear gravity driven water waves interacting with fixedor freely floating objects is an important field of research in ocean engineering. For largeobjects with characteristic dimensions of the order of the wave length, viscous effectsof the fluid flow can be neglected, as well as effects of compressibility and surface ten-sion. This is the so-called diffraction regime of fluid-structure interaction. Under theadditional assumption of irrotational fluid flow, a velocity potential can be introduced todescribe the flow characteristics. This potential satisfies a linear field equation, namelyLaplace’s equation, by continuity. The free surface boundary conditions render the prob-lem nonlinear.

In many cases it is sufficient to linearize the free surface conditions and solve the linearproblem. However, this applies only to waves with small amplitude compared with thewavelength and the mean water depth. For steep waves this linearization procedurecan not be justified and other techniques must be developed. In this case the mutualinteraction of waves due to the nonlinearity can be very strong; this may result, forinstance, in the overturning of waves. If a floating body is involved, nonlinear effectsmay have strong influence on the wave evolution and the body motion; in such a case alinearized approximation would provide inadequate results. In many of these problemshigher order approximations to the nonlinear equations are also inadequate. Typical

4Of course this historical survey is incomplete; for the classical publications mentioned here theinterested reader is referred to the bibliography at the end of this chapter.

1.2. THE INVESTIGATIONS: PAST AND PRESENT 3

examples where approximations to the nonlinear equations give unsatisfactory resultsare:

• wave slamming on fixed and floating structures,

• wave run-up on slopes,

• problems in which a part of the structure is submerged, causing the waves on topof it to break or nearly break.

Of course the assumptions of potential flow are not valid at all stages of the physicalprocess under consideration, but in such a case the potential solution can be very usefulas input for a more general model.

1.2 The investigations: past and present

The main objective of these investigations is the extension of a higher order panel methodfor nonlinear gravity wave simulations to water waves in hydrodynamic interaction withfloating bodies.

The major difficulty in solving the nonlinear potential model for this particular prob-lem is the presence of a moving free surface and a freely floating body. Due to thetime dependent free surface conditions both the potential at, and the position of thefree surface are unknown variables, which have to be determined as part of the solution.Similarly, due to the equations of motion for the freely floating body, the potential onthe wetted body surface and the body position and orientation are unknown; these vari-ables have to be determined as part of the solution too. The difficulty in solving thenonlinear potential model is also indicated by the fact that this particular problem hasboth elliptic and hyperbolic properties, through the field equation and the free surfaceconditions respectively.

In the past decades numerous attempts have been made to obtain analytical solu-tions for general free surface wave problems. Substitution of perturbation series for thevariables into the governing equations and the use of expansions to sometimes very highorder provided many nonlinear approximations. However, analytical solutions of the fullynonlinear equations have not been found so far. Therefore, it seems that the developmentof numerical methods is a more promising way towards the solution of these problems.

Romate (1989) developed a very efficient boundary element method for three-dimensionalnonlinear gravity wave simulations, using a Green’s formulation for the velocity poten-tial. Higher order approximations for the singularity distributions and the geometry,combined with a robust time stepping scheme, ensured the accuracy and stability of thealgorithm. By the end of 1988, his panel method gave excellent results for linear andweakly nonlinear waves.

From 1989 on, manpower was doubled; Broeze and van Daalen continued Romate’sresearch, aiming to fit his method for highly nonlinear waves and wave interactions withfixed and freely floating objects. As a first step towards these goals, a simplified panelmethod for two-dimensional nonlinear gravity wave simulations was derived from Ro-mate’s original method. With a number of basic modifications this method was madesuitable for highly nonlinear waves and waves interacting with fixed structures. Af-ter a period of close cooperation, it was decided that Broeze focused his attention on


highly nonlinear waves in three dimensions. The extension of the panel method to two-dimensional waves in hydrodynamic interaction with freely floating bodies was entrustedto van Daalen; the results of these investigations cover the larger part of this thesis.

During the investigations many engineering problems were encountered, ranging fromgrid motion control to the development of effective radiation boundary conditions. If itwas felt that these problems called for a more fundamental approach, a reasonable amountof time was spent on gaining insight through theoretical considerations. Sometimes theseviews induced significant adaptations to the numerical algorithm; at other times, ideaswere profoundly worked out to novel theories. The findings of these investigations areembodied in the remaining part of this thesis.

1.3 Dissertation outline

This thesis comprises a number of numerical and theoretical studies of gravity drivenwater waves and the interaction with fixed structures and floating bodies. In spite of thediversity of the material, it is not merely a collection of disconnected topics, lacking unityand coherence. Considerable effort was made to supply the fundamental backgroundin hydrodynamics — and also some of the mathematics needed — and to plan thisdissertation such that a self-contained and readable whole was arrived at.

This work is split up into four main parts:

I. Mathematical Formulation and Numerical Algorithm (ch. 2–4)II. Numerical Results (ch. 5–7)

III. Variational Principles and Hamiltonian Formulations (ch. 8–10)IV. Appendices (A–D)

Each part (excepting part IV) has been written such that — to some extent — it can beread independently of the other parts. We have also striven to write separate chapters;an outline will be given next.

Parts I and II reflect the investigations with respect to the development of the math-ematical model and the numerical (boundary element) algorithm for water waves andfloating bodies.

In chapter 2 the governing equations for the nonlinear water-wave problem, includingthe interaction with fixed objects and floating bodies, are presented. The transition fromthis set of governing equations to a boundary integral equation formulation is describedin chapter 3. In chapter 4 the numerical algorithm for water waves and floating bodiesis outlined, with a short discussion of the discrete (boundary element) approximation ofthe problem.

Numerical results — obtained with our computer code TIPHYS5 — for an impulsivelystarted wavemaker are discussed in chapter 5. The numerically computed hydrodynamicmass and damping coefficients for two-dimensional cylinders in forced harmonic motionhave been compared to experimental data and analytical solutions; the findings of thisstudy are reported in chapter 6. In chapter 7 another validation study for two-dimensionalcylinders — but now in free motion — is presented.

Part III is concerned with special descriptions of the hydrodynamic problems considered5TIPHYS: a time domain panel method for nonlinear gravity waves and floating bodies; named after

Tiphys, the helmsman of the Argo.

1.4. SUGGESTED REFERENCES 5

here. Chapter 8 reviews so-called variation principles and Hamiltonian formulationsfor the classical water-wave problem; novel extensions of these theories to water wavesinteracting with floating bodies are presented. Chapter 9 is devoted to invariants andconservation laws for wave-body problems. A theory for radiation boundary conditions— for wave problems that are governed by a Lagrangian principle — that conserve acharacteristic density (for instance, the energy density) is presented in chapter 10.

Concluding remarks and recommendations for future research are given in chapter 11.

Part IV consists of four appendices containing extensive derivations (Appendices A-C)and a flow-chart of the TIPHYS-code (Appendix ??).

The bibliographical footnotes have been borrowed from the New Webster’s Dictionaryand Thesaurus of the English Language (1991 edition, Lexicon Publications), and fromC.B. Boyer’s A History of Mathematics (1985, Princeton University Press).

Finally, it is noted that the equations have not been made dimensionless, unless statedotherwise; all variables are expressed in SI-units.

1.4 Suggested references

For a general introduction to water waves and wave-body interactions, the reader is re-ferred to the many books and (review) articles on these subjects; a number of them willbe mentioned hereafter.

All of the basics of fluid dynamics used in this thesis is covered by the works of Batchelor(1967) and Milne-Thomson (1968). Valuable information regarding the field of hydro-dynamics is gathered in the historic work of Lamb (1932), of which almost a third isconcerned with surface gravity waves. A mathematically oriented treatment of waterwaves is given by Stoker (1957). Another important source of information on wave the-ories is the work of Wehausen and Laitone (1960). Wind-generated water waves arediscussed extensively by Kinsman (1965). Whitham (1974) discusses waves in a moregeneral context; his work includes a number of chapters on water waves and the use ofvariational principles. An introduction to the science of wave motions in fluids, with achapter devoted to water waves only, is given by Lighthill (1978). A very readable text onthe dynamics of ocean surface waves is due to Mei (1983). An introductory presentationof the basic theories of water waves, using direct mathematical techniques, is given byCrapper (1984).

The following reviews on various wave types should also be brought to the attentionof the reader: solitary waves, by Miles (1980); trapped waves, by Mysak (1980); waveinstabilities, by Yuen and Lake (1980); strongly nonlinear waves, by Schwartz and Fen-ton (1982); breaking waves, by Peregrine (1983) and Battjes (1988); tsunamis (i.e. hugewaves caused by large submarine earthquakes or landslides), by Voit (1987). Numericalmethods in free surface flows are reviewed by Mei (1978) and Yeung (1982). The useof boundary integral equation methods in inviscid fluid mechanics is discussed by Hess(1990). A survey of Hamiltonian formulations in fluid mechanics is due to Salmon (1988).

Mathematical treatments of the motion of bodies in waves have been given by Landwe-ber (1961) and Wehausen (1971). Newman’s (1977) textbook on the hydrodynamics of


marine objects is recommendable. Wave-structure interactions have been dealt with bySarpkaya and Isaacson (1981); more recent contributions with regard to this subject aredue to Faltinsen (1990ab). An update of the theory of floating bodies since John (1949,1950) is given by Kleinman (1982). Timman, Hermans, and Hsiao (1985) have written avery readable textbook on water waves and ship hydrodynamics.

For a survey of the different numerical methods used in the solution of free surfaceflow problems, with and without the presence of solid structures or floating bodies, theproceedings of the International Conferences on Numerical Ship Hydrodynamics, theproceedings of the Symposia on Naval Hydrodynamics, and the abstracts of the Inter-national Workshops on Water Waves and Floating Bodies should be consulted. Thesepublications include almost any method in most areas concerning fluid-structure inter-action, such as potential flows, nonlinear waves, lifting bodies, vortex flows, cavitation,ship wave-making and wave-resistance, propulsion, boundary layers, and viscous flows.

1.5 Bibliography

Abstracts of the International Workshops on Water Waves and Floating Bodies. 1986–. . .

Airy, Sir G.B. 1845. On tides and waves. Encyclopaedia Metropolitana, Series 5,5:241-396.

Batchelor, G.K. 1967. An Introduction to Fluid Dynamics. Cambridge UniversityPress.

Battjes, J.A. 1988. Surf-zone dynamics. Annual Review of Fluid Mechanics 20:257-293.

Bernoulli, D. 1738. Hydrodynamica. Argentorati.

Boussinesq, J. 1871. Theorie de l’intumescence liquide, appelee onde solitaire ou detranslation, se propageant dans un canal rectangulaire. Institut de France, Academie desSciences, Comptes Rendus 72:755-759.

Boussinesq, J. 1877. Essai sur la theorie des eaux courantes. Memoires par diverssavants, Series 2, 23:1-680.

Cauchy, A.-L. 1816. Memoire sur la theorie des ondes. Memoires de l’Academie Royaledes Sciences.

Crapper, G.D. 1984. Introduction to Water Waves. John Wiley and Sons.

Euler, L. 1755. Principes generaux du mouvement des fluides. Histoires de l’Academiede Berlin.

Faltinsen, O.M. 1990a. Wave loads on offshore structures. Annual Review of FluidMechanics 22:35-56.

Faltinsen, O.M. 1990b. Sea loads on ships and offshore structures. Cambridge Univer-sity Press.

Hess, J.L. 1990. Panel methods in computational fluid dynamics. Annual Review ofFluid Mechanics 22:255-274.

John, F. 1949. On the motion of floating bodies I. Communications on Pure and Applied

1.5. BIBLIOGRAPHY 7

Mathematics 2:13-57.

John, F. 1950. On the motion of floating bodies II. Simple harmonic motions. Commu-nications on Pure and Applied Mathematics 3:45-101.

Kelvin, Lord. 1886/1887. On stationary waves in flowing water. Philosophical Maga-zine, Series 5, 22:353-357, 445-452, 517-530, 23:52-57.

Kinsman, B. 1965. Wind Waves. Their Generation and Propagation on the OceanSurface. Prentice-Hall.

Kleinman, R.E. 1982. On the mathematical theory of the motion of floating bodies: anupdate. David Taylor National Ship Research and Development Center, Report 82/074.Bethesda, Maryland.

Korteweg, D.J., and de Vries, G. 1895. On the change of form of long waves ad-vancing in a rectangular canal and on a new type of long stationary waves. PhilosophicalMagazine, Series 5, 39:422-443.

Lagrange, J.L. 1781. Memoire sur la theorie du mouvement des fluides. Nouvellesmemoires de l’Academie de Berlin.

Lamb, Sir H. 1904. On deep water waves. Proceedings of the London MathematicalSociety, Series 2, 2:388.

Lamb, Sir H. 1913. Some cases of wave motion on deep water. Annali di Matematica,Series 3, 21:237.

Lamb, Sir H. 1932. Hydrodynamics. Cambridge University Press.

Landweber, L. 1961. Motion of immersed and floating bodies. In Handbook of FluidDynamics. McGraw-Hill.

Levi-Civita, T. 1925. Determination rigoreuse des ondes permanentes d’ampleur finie.Mathematische Annalen 93:264-314.

Lighthill, Sir M.J. 1978. Waves in Fluids. Cambridge University Press.

Mei, C.C. 1978. Numerical methods in water-wave diffraction and radiation. AnnualReview of Fluid Mechanics 10:393-416.

Mei, C.C. 1983. The Applied Dynamics of Ocean Surface Waves. John Wiley and Sons.

Michell, J.H. 1893. The highest waves in water. Philosophical Magazine, Series 5,56:430.

Miles, J.W. 1980. Solitary waves. Annual Review of Fluid Mechanics 12:11-43.

Milne-Thomson, L.M. 1968. Theoretical Hydrodynamics. MacMillan.

Mysak, L.A. 1980. Topographically trapped waves. Annual Review of Fluid Mechanics12:45-76.

Navier, C.L.M.H. 1822. Memoire sur les lois du mouvement des fluides. Memoires del’Academie des Sciences 6:389.

Newman, J.N. 1977. Marine Hydrodynamics. The MIT Press.

Peregrine, D.H. 1983. Breaking waves on beaches. Annual Review of Fluid Mechanics15:149-178.

Poincare, H. 1885. Sur l’equilibre d’une masse fluide animee d’un mouvement de ro-


tation. Acta Mathematica 7:259.

Poisson, S.D. 1816. Memoire sur la theorie des ondes. Memoires de l’Academie Royaledes Sciences.

Proceedings of the International Conferences on Numerical Ship Hydrodynamics. 1975–. . .

Proceedings of the Symposia on Naval Hydrodynamics. 1956–. . .

Rayleigh, Lord. 1876. On periodical irrotational waves at the surface of deep water.Philosophical Magazine, Series 5, 1:381-389.

Romate, J.E. 1989. The Numerical Simulation of Nonlinear Gravity Waves in ThreeDimensions using a Higher Order Panel Method. Ph.D. thesis, University of Twente.Enschede, The Netherlands.

Sarpkaya, T., and Isaacson, M. de St. Q. 1981. Mechanics of Wave Forces onOffshore Structures. Van Nostrand Reinhold.

Salmon, R. 1988. Hamiltonian fluid mechanics. Annual Review of Fluid Mechanics20:225-256.

Schwartz, L.W., and Fenton, J.D. 1982. Strongly nonlinear waves. Annual Reviewof Fluid Mechanics 14:39-60.

Stoker, J.J. 1957. Water Waves. The Mathematical Theory with Applications. Inter-science Publishers.

Stokes, Sir G.G. 1847. On the theory of oscillatory waves. Transactions of the Cam-bridge Philosophical Society 8(4):441-455.

Struik, D.J. 1926. Determination rigoreuse des ondes irrotationelles periodiques dansun canal a profondeur finie. Mathematische Annalen 95:595-634.

Timman, R., Hermans, A.J., and Hsiao, G.C. 1985. Water Waves and Ship Hy-drodynamics. An Introduction. Kluwer Academic Publishers.

Voit, S.S. 1987. Tsunamis. Annual Review of Fluid Mechanics 19:217-236.

Wehausen, J.V., and Laitone, E.V. 1960. Surface Waves. In Handbook of Physics,Volume IX, pp.445-778. Springer-Verlag.

Wehausen, J.V. 1971. The motion of floating bodies. Annual Review of Fluid Mechan-ics 3:237-268.

Whitham, G.B. 1974. Linear and Nonlinear Waves. John Wiley and Sons.

Yeung, R.W. 1982. Numerical methods in free surface flows. Annual Review of FluidMechanics 14:395-442.

Yuen, H.C., and Lake, B.M. 1980. Instabilities of waves on deep water. AnnualReview of Fluid Mechanics 12:303-334.

1.5. BIBLIOGRAPHY 9

First then let us name Orpheus whom once Calliope bare, it is said, weddedto Thracian Oeagrus, near the Pimpleian height. Men say that he by the musicof his songs charmed the stubborn rocks upon the mountains and the courseof rivers . . .

Tiphys, son of Hagnias, left the Siphaean people of the Thespians, wellskilled to foretell the rising wave on the broad sea, and well skilled to inferfrom sun and star the stormy winds and the time for sailing. Tritonian Athenaherself urged him to join the band of chiefs, and he came among them awelcome comrade. She herself too fashioned the swift ship; and with herArgus, son of Arestor, wrought it by her counsels. Wherefore it proved themost excellent of all ships that have made trial of the sea with oars . . .

Next to him came a scion of the race of divine Danaus, Nauplius. Hewas the son of Clytonaeus son of Naubolus; Naubolus was son of Lernus;Lernus we know was the son of Proetus son of Nauplius; and once Amymonedaughter of Danaus, wedded to Poseidon, bare Nauplius, who surpassed allmen in naval skill . . .

After them from Taenarus came Euphemus whom, most swift-footed ofmen, Europe, daughter of mighty Tityos, bare to Poseidon. He was wont toskim the swell of the grey sea, and wetted not his swift feet, but just dippingthe tips of his toes was borne on the watery path . . .

Yea, and two other sons of Poseidon came; one Erginus, who left thecitadel of glorious Miletus, the other proud Ancaeus, who left Parthenia, theseat of Imbrasion Hera; both boasted their skill in sea-craft and in war . . .

So many then were the helpers who assembled to join the son of Aeson6.All the chiefs the dwellers thereabout called Minyae, for the most and thebravest avowed that they were sprung from the blood of the daughters ofMinyas; thus Jason himself was the son of Alcimede who was born of Clymenethe daughter of Minyas7 . . .

Now when all things had been made ready by the thralls, all things thatfully-equipped ships are furnished withal when men’s business leads them tovoyage across the sea, then the heroes took their way through the city to theship where it lay on the strand that men call Magnesian Pagasae . . .

Argonautica, Book I, Fragments from Verses 23-238.

6i.e. Jason7Minyas was a son of Aeolus, who was a son of Zeus; hence, Jason was a descendant of the supreme

god.


Part I

Mathematical Formulation andNumerical Algorithm

11

Chapter 2

Mathematical Statement of theProblem

. . .And they heaped their garments, one upon the other, on a smooth stone,which the sea did not strike with its waves, but the stormy surge had cleansedit long before. First of all, by the command of Argus, they strongly girded theship with a rope well twisted within, stretching it tight on each side, in orderthat the planks might be well compacted by the bolts and might withstand theopposing force of the surge. And they quickly dug a trench as wide as thespace the ship covered, and at the prow as far into the sea as it would runwhen drawn down by their hands. And they ever dug deeper in front of thestem, and in the furrow laid polished rollers; and inclined the ship down uponthe first rollers, that so she might glide and be borne on by them. And above,on both sides, reversing the oars, they fastened them round the thole-pins, soas to project a cubit’s space. And the heroes themselves stood on both sides atthe oars in a row, and pushed forward with chest and hand at once. And thenTiphys leapt on board to urge the youths to push at the right moment; andcalling on them he shouted loudly; and they at once, leaning with all theirstrength, with one push started the ship from her place, and strained withtheir feet, forcing her onward; and Pelian Argo followed swiftly; and they oneach side shouted as they rushed on. And then the rollers groaned under thesturdy keel as they were chafed, and round them rose up a dark smoke owingto the weight, and she glided into the sea; but the heroes stood there and keptdragging her back as she sped onward. And round the thole-pins they fittedthe oars, and in the ship they placed the mast and the well-made sails and thestores.


13

14 CHAPTER 2. MATHEMATICAL STATEMENT OF THE PROBLEM

2.1 Introduction

In this chapter the complete set of governing equations for the motion of an ideal fluidwith a free surface is presented. These equations include the field equation for the interiorfluid flow and the conditions on all physical and artificial boundaries. The additionalequations describing the hydrodynamic interaction with floating bodies, either partiallyor totally submerged, are also presented. These equations comprise the appropriateboundary conditions on the body surface and the equations of motion for the body.

In order to give full account for assumptions that would have been made tacitly other-wise, the present derivation of the governing equations is rather extensive. In section 2.2we discuss assumptions with respect to various types of forces which are active in fluidflow. Then, the set of equations describing potential flow is derived in section 2.3. Insection 2.4 the conditions for several types of boundaries are presented. The governingequations for a number of formulations describing three-dimensional wave-body interac-tions are discussed in section 2.5. Finally, formal definitions of the particular water-waveand wave-body problems treated in this thesis are given in section 2.6.

2.2 Force mechanisms in fluid flow

The motion of a fluid, like the motion of rigid bodies, is governed by opposing actions ofdifferent forces. In fluid dynamics, these forces are distributed continuously throughouta volume filled with infinitesimal fluid particles. One may distinguish force mechanismsassociated with the fluid inertia, the fluid weight, the fluid viscosity, and other (sec-ondary) effects such as surface tension.1 In order to analyze the three principal forcemechanisms — inertial, gravitational, and viscous — it is useful to estimate the orders ofmagnitude. Suppose that the problem under consideration is characterized by a physicallength L, a velocity U , a fluid density ρ, a gravitational acceleration g, and a dynamicfluid viscosity µ. An order estimation of the force magnitudes is given in the table below.

Type of Force Symbol Order of MagnitudeInertial Fi ρU2L2

Gravitational Fg ρgL3

Viscous Fv µUL

Table 2.1: Order estimation of force magnitudes.

1In section 8.2 we show that surface tension effects are negligible for all waves but the shortest or‘ripple’ waves.

2.3. POTENTIAL FLOW 15

These order estimates merely indicate how changes in any of the physical parameters L,U , ρ, g, or µ affect the balance of the various force mechanisms. For instance, doublingthe length scale L corresponds to multiplicative factors of 22, 23, and 21 for the inertial,gravitational, and viscous forces respectively. The above considerations are useful notonly in predicting full-scale phenomena from tests with a scale model, but they alsoindicate which effects can be neglected in the mathematical model of the problem underconsideration.

In order to gain insight in the relative magnitudes of the various forces, three nondi-mensional parameters are introduced to describe the fluid flow:

Fi

Fg=

ρU2L2

ρgL3= U2/gL , (2.1)

Fi

Fv=

ρU2L2

µUL= ρUL/µ , (2.2)

Fg

Fv=

ρgL3

µUL= ρgL2/µU . (2.3)

From any two of these ratios the third can be calculated. Hence, any pair of these ratiosdetermines the balance of forces in the fluid motion. Usually, the first two are used todefine the characteristic numbers

Fr =U

(gL)1/2(Froude number) , (2.4)

Re =ρUL

µ=

UL

ν(Reynolds number) , (2.5)

where ν = µ/ρ is the kinematic fluid viscosity; common values of ν are 10−6 m2/s forwater and 1.5 × 10−5 m2/s for air. For typical values of the characteristic velocity andthe length scale, say U = 1 m/s and L = 10 m, this implies that the Reynolds numberRe will be large, and hence that viscous forces will be small compared to inertial forces.Therefore, effects of viscosity can be neglected for the bulk of the fluid. However, inspecial regions — such as the boundary layer2 very close to a body — viscosity shouldbe included.

2.3 Potential flow

Assuming the laws of classical mechanics and thermodynamics to apply, the motion ofa fluid can be described by a set of partial differential equations expressing conservationof mass, conservation of momentum, and conservation of energy per unit volume of thefluid. If necessary, this set of equations is complemented with an equation of state.

2For a detailed discussion of ship boundary layers we refer to the review article written by Landweberand Patel (1979).


Let ~e1, ~e2, ~e3 denote a Cartesian3 coordinate system fixed in space, with correspondingspatial coordinates (x, y, z), and let ~v = (u, v, w)T denote the fluid velocity field. Follow-ing the motion of an infinitesimal control volume δV , the equation of mass conservation— or continuity equation — reads

D

Dt[ρδV ] =

Dρ

DtδV + ρ

DδV

Dt= 0 , (2.6)

where D/Dt denotes the material derivative (i.e. the substantial derivative or the totaltime derivative) and ρ = ρ (x, y, z; t) is the local time dependent fluid density.

The control volume is subject to changes due to strain only; hence, in terms of thevelocity field the material derivative of the control volume is given by

DδV

Dt=

(∂u

∂x+

∂v

∂y+

∂w

∂z

)δV = (∇ · ~v) δV , (2.7)

where ∇ represents the three-dimensional gradient-operator:

∇ =(

∂

∂x,

∂

∂y,

∂

∂z

)T

. (2.8)

With (2.7) the equation of mass conservation (2.6) can be simplified to

Dρ

Dt+ ρ (∇ · ~v) = 0 . (2.9)

Then, assuming the fluid to be absolutely incompressible and homogeneous, we haveρ = constant throughout the fluid domain, and the continuity equation reduces to

∇ · ~v = 0 or div~v = 0 , (2.10)

expressing that the velocity field is free of strain (or divergence-free).

Next, for the control volume δV , the equation of momentum conservation reads

D

Dt[ρ~v ] = ~Fi + ~Fg + ~Fv , (2.11)

where we have dropped the incompressibility assumption. This equation expresses thechange of momentum due to forces acting on the control volume. These forces are, ingeneral, inertial (pressure) forces (~Fi), gravity forces (~Fg), and frictional forces due toviscosity (~Fv). Substitution of explicit expressions for these different forces yields thewell-known Navier-Stokes equations:

Dρ

Dtu + ρ

Du

Dt= −∂p

∂x+ µ

(∇2u +

∂∇ · ~v∂x

), (2.12)

Dρ

Dtv + ρ

Dv

Dt= −∂p

∂y+ µ

(∇2v +

∂∇ · ~v∂y

), (2.13)

Dρ

Dtw + ρ

Dw

Dt= −∂p

∂z+ µ

(∇2w +

∂∇ · ~v∂z

)− ρg , (2.14)

3Descartes, Rene (1596-1650), French philosopher, physicist and mathematician. He founded thescience of analytical geometry (‘la Geometrie’, 1637) and discovered the laws of geometric optics. In‘Discours de la Methode’ (1637) he divests himself of all previously held beliefs, to rebuild on his ownbasis of certitude, i.e. the fact of his self-conscious existence: ‘dubito ergo cogito: cogito ergo sum’ (Idoubt, therefore I think: I think, therefore I am).

2.3. POTENTIAL FLOW 17

where p denotes the pressure, µ is the uniform dynamic fluid viscosity, g is the gravita-tional acceleration (acting downwards along the z-axis), and ∇2 is the Laplace-operator:4

∇2 = ∇ · ∇ =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2

). (2.15)

Under the assumption of incompressibility, the above set of equations reduces to theNavier-Stokes equations for incompressible fluid flow:

Du

Dt=

∂u

∂t+ ~v · ∇u = −1

ρ

∂p

∂x+ ν∇2u , (2.16)

Dv

Dt=

∂v

∂t+ ~v · ∇v = −1

ρ

∂p

∂y+ ν∇2v , (2.17)

Dw

Dt=

∂w

∂t+ ~v · ∇w = −1

ρ

∂p

∂z+ ν∇2w − g , (2.18)

where ν = µ/ρ denotes the kinematic viscosity of the fluid. If ν = 0, these equationssimplify to the Euler5 equations for incompressible and inviscid fluid flow:

D~v

Dt=

∂~v

∂t+ (~v · ∇)~v = −1

ρ∇p− g~e3 . (2.19)

In general, a fluid flow problem is characterized by a pressure p, a density ρ, a tempera-ture T , a viscosity µ, and a velocity ~v = (u, v, w). For an incompressible, homogeneous,and inviscid fluid with uniform temperature, ρ and T are constant and µ equals zero,and only four equations are needed to solve the flow problem. Summarizing, it can bestated that the motion of an ideal and isothermal fluid is governed by the continuityequation (2.10) and the three Euler equations (2.19).

4Laplace, Pierre Simon, Marquis de (1749-1827), French physicist and astronomer. With Lavoisierhe made the first determination of the coefficient of expansion of a metal rod, and initiated the study ofthermochemistry.

5Euler, Leonhard (1707-1783), Swiss mathematician. He was responsible for the revision of nearlyall the branches of pure mathematics then known and the foundation of new methods of analysis. Themathematical symbols π, e, and i were introduced by Euler; his celebrated equality eiπ +1 = 0 combinesthe five most important numbers in mathematics in a single equation.


Next, consider the fluid vorticity

~ω = ∇× ~v or (ωx, ωy, ωz) =(

∂w

∂y− ∂v

∂z,∂u

∂z− ∂w

∂x,∂v

∂x− ∂u

∂y

). (2.20)

Assuming that the fluid flow is irrotational at some time t = t0, then it follows from thecurl of equation (2.19) that the flow will remain irrotational at all times:

~ω = ~0 for t ≥ t0 . (2.21)

This conclusion is important because an irrotational vector field can be represented asthe gradient of a scalar field.6 Thus we are allowed to introduce a potential φ for thevelocity field:

~v = ∇φ . (2.22)

With (2.22) the continuity equation (2.10) becomes

∇2φ = 0 (2.23)

i.e. Laplace’s equation for the velocity potential φ.With (2.22) the Euler equations (2.19) can be integrated to a general form of Bernoulli’s7

equation:

∂φ

∂t+

12

(∇φ · ∇φ) + gz = −p− p0

ρ+ B (t) (2.24)

where p0 is a constant reference pressure and B (t) is an arbitrary function of time.When, as in the cases which will be considered, the boundary conditions are kinematical,the solution process consists in finding a harmonic potential φ satisfying (2.24) and theprescribed boundary conditions. The pressure p is then indeterminate to the extent ofan additive function of t. It becomes determinate when the value of p at some point ofthe fluid is given for all values of t. Since the term B (t) has no influence on resultantpressures, it is frequently omitted.8

Thus it has been shown that under the assumptions of an incompressible, homogeneous,and inviscid (i.e. ideal) fluid and in the absence of vorticity (i.e. an irrotational flow), themotion of the fluid is governed by Laplace’s equation for the velocity potential and thenonlinear Bernoulli equation. If the potential φ is known throughout the fluid domain,the velocity field ~v and the pressure field p are easily obtained from (2.22) and (2.24)respectively. From these quantities the potential and kinetic energies can be determined,due to the absence of friction.

6This is a direct result of Helmholtz’ theorem in vector analysis, which states that any continuousand finite vector field can be expressed as the sum of the gradient of a scalar function and the curl ofa zero-divergence vector. The divergence-free vector vanishes if the original vector field is irrotational.The proof of this theorem can be found in Morse and Feshbach (1953).

7Bernoulli, Swiss family of Dutch extraction, several members of which made distinguished contri-butions to physics and mathematics. Jacques (1654-1705) worked on analytical geometry, his brotherJean (1667-1748) discovered the exponential calculus and a method of integrating rational functions,and Jean’s son Daniel (1700-1782) developed the kinetic theory of gases.

8Suppose, for instance, that a solid body moves through a fluid completely enclosed by fixed bound-aries, and that it is possible — say, by means of a piston — to apply an arbitrary pressure at some pointof the boundary. Whatever variations are made in the magnitude of the force applied to the piston, themotion of both the fluid and the solid will be absolutely unaffected, since at all points the pressure willinstantaneously rise or fall by equal amounts. Physically, the origin of this paradox is that the fluid istreated as absolutely incompressible. In actual fluids changes of pressure propagate with very great, butnot infinite, velocity.

2.4. BOUNDARY CONDITIONS 19

2.4 Boundary conditions

To complete the set of governing equations, initial conditions are needed, as well asboundary conditions for the elliptic field equation (2.23). The initial conditions dependon the specific problem under consideration, and are incorporated in the final problemspecifications. Obviously, the initial conditions must be compatible with the governingequations, in order to obtain a well-posed problem.

Figure 2.1: Fluid domain Ω and bounding surfaces.

The boundary conditions depend on the type of boundary under consideration. In gen-eral, we have a certain amount of fluid occupying a simply connected9 transient domainΩ (t), see Figure 2.1. The problem of gravity driven water waves introduces a free sur-face F , which is one part of the domain boundary ∂Ω(t). Other parts are, for instance,the bottom B and the hull of a ship S. These and other types of boundaries and thecorresponding conditions are discussed hereafter.

2.4.1 Impermeable fixed boundaries

In most practical configurations, at least one fixed boundary is present; the bottom —denoted by B in Figure 2.1 — which is not necessarily even. If such a fixed boundary isimpermeable, then fluid particles can not penetrate it; hence, the normal component ofthe fluid velocity must vanish there:

vn = ~v · ~n = 0 . (2.25)

9A region such that any simple closed curve can be shrunk to a point without leaving the region iscalled simply (or singly) connected.


Here ~n is the unit normal vector along ∂Ω ⊃ B, pointing outwards.Under the assumptions of potential flow, this so-called zero-flux condition can be

reformulated to

∂φ

∂n= ∇φ · ~n = 0 (2.26)

stating that the normal derivative of the velocity potential vanishes.

2.4.2 Impermeable moving boundaries

The motion of fluid particles in the neighbourhood of impermeable moving boundaries— such as a wavemaker or the hull of a ship — is influenced by the boundary velocity.Effects of viscosity can be taken into account by introducing a boundary layer, whereinstress forces influence the fluid motion. However, assuming the fluid to be inviscid, themotion of a fluid particle on an impermeable moving boundary with velocity ~V satisfies

vn = ~v · ~n = ~V · ~n , (2.27)

expressing that the normal velocity of the fluid particle and the normal boundary velocityare equal.

In case of potential flow, this condition reads

∂φ

∂n= ∇φ · ~n = ~V · ~n (2.28)

Of course condition (2.26) for impermeable fixed boundaries is a special case of condi-tion (2.28).

2.4.3 Free boundaries

The free surface — denoted by F in Figure 2.1 — differs from other physical boundariessince it is a free boundary indeed; not only the potential φ must be determined there,but also the position of the free surface itself. Therefore, it is obvious that two boundaryconditions are needed for a proper mathematical description of the free surface.

In the absence of surface tension, the pressure at the free surface must equal the atmo-spheric pressure p0; the first condition is then obtained from Bernoulli’s equation (2.24)by substitution of p = p0 = 0 and by the choice B (t) = 0:

−p

ρ=

∂φ

∂t+

12

(∇φ · ∇φ) + gz = 0 , (2.29)

where z = 0 corresponds to the mean water level. Condition (2.29) is known as thedynamic free surface condition, since it has been deduced from a momentum equation,i.e. a balance of forces.

In Lagrangian notation (2.29) reads

Dφ

Dt=

12

(∇φ · ∇φ)− gz (2.30)

The second condition concerns the free surface velocity; fluid particles at the free sur-face have the property that they remain part of the free surface until they encounter a

2.4. BOUNDARY CONDITIONS 21

boundary of another type. Following such a free surface particle, this implies that itstransient position ~x = ~xF is determined by the Lagrangian expression

D~xF

Dt= ~v = ∇φ (2.31)

which is known as the kinematic free surface condition, since it is concerned with thevelocity field only.

In many applications the free surface elevation can be assumed to be a single-valuedfunction of the horizontal coordinates and the time. In these cases the free surface isdescribed in terms of its elevation η by

F (x, y, z; t) ≡ z − η (x, y; t) = 0 . (2.32)

The requirement that the material derivative of z − η vanishes on the free surface thengives

∂η

∂t+

∂η

∂xu +

∂η

∂yv = w , (2.33)

or, in terms of the elevation η and the potential φ:

∂η

∂t+

∂η

∂x

∂φ

∂x+

∂η

∂y

∂φ

∂y=

∂φ

∂z. (2.34)

Condition (2.31) is usually preferred to (2.34), since the latter condition excludes phe-nomena as overturning waves.

For waves with small amplitude A compared to the wavelength λ and the mean waterdepth h, the free surface conditions (2.29) and (2.34) can be linearized about the meanwater level:

φt = −gηηt = φz

at z = 0 , (2.35)

where a suffix denotes partial differentiation, i.e. φt = ∂φ/∂t etc. The second approxi-mate boundary condition states that the vertical velocity of a free surface particle coin-cides with the vertical velocity of the free surface itself, thus ignoring the small horizontaldeviations.

In case of a horizontal bottom, the zero-flux condition (2.26) gives

φz = 0 at z = −h . (2.36)

A two-dimensional periodic solution satisfying Laplace’s equation (2.23), the linearizedfree surface conditions (2.35), and the bottom condition (2.36) is given by the velocitypotential

φ (x, z; t) = Aω

k

cosh k (z + h)sinh kh

cos (kx− ωt) , (2.37)

and the corresponding free surface elevation

η (x; t) = −A sin (kx− ωt) , (2.38)

where ω is the wave frequency and k = 2π/λ is the wave number.


Substitution of (2.37-2.38) into (2.35) yields the first order dispersion relation forStokes waves:

ω2 = gk tanh kh . (2.39)

For finite-amplitude waves on deep water we have kh → ∞ and consequently ω2 = gk.Hence, for deep water waves the phase velocity cf and the group velocity cg are given by

cf ≡ ω

k= (g/k)1/2

, cg ≡ dω

dk=

12

(g/k)1/2 =12cf . (2.40)

For finite-amplitude waves on shallow water we have kh → 0, and as a result ω2 = ghk2.It follows that for shallow water waves cf and cg are equal and independent of the wavenumber k:

cf = cg = (gh)1/2. (2.41)

2.4.4 Open boundaries

In general, the fluid domain Ω extends to infinity in the horizontal directions. In thiscase a so-called radiation condition at infinity is required to obtain a uniquely solvableproblem. This condition — based on conservation of energy — states that the wavesshould behave at infinity like progressing waves moving away from the source of thedisturbance. In problems concerning electromagnetic wave propagation this condition isknown as the Sommerfeld radiation condition, see Sommerfeld (1964) or Stoker (1957).

With regard to the water-wave problem, this radiation condition states that thesolution corresponds to outgoing waves only. However, for computational reasons (limitedcpu-time and computer memory) it will be necessary to truncate the fluid domain atsome distance from the area of interest. The artificial (i.e. non-physical) boundariesthus introduced are part of the bounding surface ∂Ω. To obtain a well-posed problem,appropriate conditions are needed on these boundaries.

There are two main criteria for a radiation condition on an artificial boundary. First of all,the condition should yield a well-posed problem. This boils down to the requirement thatthe solution to the problem exists, is unique, and depends continuously upon the initialand boundary conditions, i.e. well-posed in the sense of Hadamard (1923). Secondly,the radiation condition should simulate the behaviour of the excluded domain as well aspossible; the solution should approximate the solution that one would have obtained ifthe boundaries would have been chosen at infinity. With regard to the free surface waveproblem, this comes down to the requirement that radiated surface waves approaching anartificial boundary should be fully transmitted or ‘absorbed’. In this sense, the radiationcondition should provide an ‘open’ boundary.

Romate (1992) has reviewed methods for the numerical simulation of open boundariesfor linear and nonlinear water waves. On the basis of his literature survey, he decidedto use Higdon’s (1987) first- and second-order partial differential equations as absorbingboundary conditions for the linearized model; the question of well-posedness was ad-dressed too. The numerical implementation in his panel method for linear free surfacewave simulations and the stability of these first- and second-order absorbing boundaryconditions have been discussed — in a subsequent paper — by Broeze and Romate (1992).

More recently, van Daalen, Broeze, and van Groesen (1992) introduced an analyticaltechnique for the development of radiation boundary conditions for general wave systems.

2.5. WAVE-BODY FORMULATIONS 23

At the outset of a variational principle for the wave problem under consideration, addi-tional boundary conditions are derived such that a characteristic density (for instance,the energy density) is conserved. In chapter 10 this theory is presented, and applicationsto a number of wave problems — including the three-dimensional nonlinear water-waveproblem — are discussed.

2.5 Wave-body formulations

The presence of a floating body, either partly or totally submerged, implies that extraequations are needed for a correct mathematical description of the fluid-body interaction.Usually, a second coordinate system ~e1

′, ~e2′, ~e3

′ is introduced, which moves with thebody, see Figure 2.2. The origin of this coordinate system is located at some fixed pointinside the body — for example, the centre of mass G — and the unit vectors are chosenalong the body principal axes of inertia. The wetted (submerged) part of the bodysurface is denoted by S.

Figure 2.2: Free surface with a floating body.

Generally, different assumptions on the motion of the body and the free surface lead todifferent sets of governing equations. To illustrate this dependence, four frequently usedwave-body formulations are discussed next.

2.5.1 Ship wave-making and wave-resistance

Ever since Kelvin (1886) analyzed the potential waves generated by a pressure pointmoving with constant velocity in otherwise calm water and Michell (1898) derived thepotential wave-resistance of a thin ship, much interest has been paid to theoretical (i.e.analytical) ship wave-resistance calculations. With the development of fast computersthis problem has also been addressed numerically.

Consider a body moving at a constant speed in otherwise calm water. For thisparticular system a stationary problem can be formulated, using a moving frame of


reference fixed to the body. Assuming that the uniform inflow velocity is given by

~V = (U, 0, 0)T, (2.42)

the following steady-state formulation is obtained:

∇2φ = 0 in Ω , (2.43)12

(∇φ · ∇φ) + gz =12U2 on F , (2.44)

∂φ

∂n= 0 on F and S , (2.45)

∇φ = ~V for r =(x2 + y2

)1/2 →∞ . (2.46)

Condition (2.44) is the dynamic free surface condition, and is easily obtained by applica-tion of Bernoulli’s theorem to a free surface streamline extending to infinity in horizontaldirection. The kinematic condition (2.45) expresses that both the free surface flow andthe flow around the body are stationary.

Linearization of the free surface conditions about the mean water level z = 0 andabout the uniform flow ∇φ = ~V results in the so-called Neumann-Kelvin problem. Thisproblem is usually solved with integral equation techniques; in the ‘Havelock-source’ (or‘Kelvin-source’) approach the Green’s function satisfies all boundary conditions exceptthe hull boundary condition, while in the ‘Rankine-source’ approach the Green’s functionsatisfies none of the boundary conditions.

Another solution procedure is the linearization about the flow at zero Froude number.Successful computations for this so-called slow ship wave-making and wave-resistanceproblem have been initiated by Gadd (1976) and Dawson (1977), followed by Sclavounosand Nakos (1988), Raven (1988), and many others. In recent literature a shift towardssolution methods for the nonlinear problem is observed, see Ni (1987), Jensen, Bertram,and Soding (1989), Campana, Lalli, and Bulgarelli (1989), and Raven (1992).

2.5.2 Linearized oscillatory motion

If the body motion is time-harmonic and non-translatory, the amplitude of the motioncan be used to linearize the governing equations. This has the important advantage thatthe free surface grid is fixed in time, thus simplifying a numerical solution procedureconsiderably. Introducing the body frequency ω by

φ (x, y, z; t) = Re[φ (x, y, z) e−iωt

], (2.47)

Vn (x, y, z; t) = Re[Vn (x, y, z) e−iωt

], (2.48)

where the body velocity Vn is imposed, the governing equations can be Fourier10 trans-formed in time to obtain a much simpler frequency domain problem:

10Fourier, Jean Baptiste Joseph, Baron de (1768-1830), French physicist, mathematician and politician,important for his theorem that any periodic function may be resolved into sine and cosine terms.


∇2φ = 0 in Ω , (2.49)

ω2

gφ =

∂φ

∂zon F , (2.50)

∂φ

∂n= Vn on S , (2.51)

where F and S denote the mean positions of the free surface and the wetted bodysurface respectively. Condition (2.50) is obtained by substitution of (2.47-2.48) into the(combined) linearized free surface conditions (2.35).

In order to render this problem uniquely solvable, Sommerfeld’s radiation conditionat infinity must be imposed:

limr→∞

(kr)1/2

(∂φ

∂r− ikφ

)= 0 , r =

(x2 + y2

)1/2, (2.52)

where k is the wave number. Mei (1978) reviews two approaches towards the solution ofthis so-called harmonic radiation problem: hybrid (coupled finite element and boundaryelement) methods and integral equation methods.

The use of integral equation methods in the study of wave fields goes back to Lamb(1932), who examined the scattering of linear waves by a surface piercing body. Thepotential of the wave field can be described in terms of a source distribution on the meanwetted part of the body surface, where the Green’s function satisfies Laplace’s equation,the linearized free surface conditions, and the radiation condition at infinity. Green’sfunctions of this type are rather complicated and have been developed in the fifties byJohn (1950) and Stoker (1957); for a more recent discussion, see Noblesse (1983). Thecorresponding integral equation methods have been employed by, for instance, Brown,Eatock Taylor, and Patel (1983), and Nestegard and Sclavounos (1984) in the frequencydomain, and by Adachi and Ohmatsu (1980), Beck and Liapis (1987), and Newman(1985) in the time domain.

Unfortunately, this method fails to give a unique solution at the so-called irregularfrequencies — which are the eigenfrequencies of the interior Dirichlet11 problem of thebody — even though the solution of the original boundary value problem is unique.Much effort has been spent to solve this problem; Sclavounos and Lee (1985) give a shortsurvey of methods for the removal of irregular frequencies. For further results on this(non-physical) phenomenon, see Ursell (1981), Martin (1981), Hulme (1983), and Forbes(1984). The question of well-posedness for this specific problem was addressed by John(1950); more general existence and uniqueness proofs are given by Lenoir and Martin(1981), and Simon and Ursell (1984).Another way to treat this problem is to employ a simpler Green’s function, which onlysatisfies the boundary condition on the bottom. In this case singularity distributionsare needed on all other boundaries. Angell, Hsiao, and Kleinman (1986) show that— for a restricted class of three-dimensional body geometries — this formulation hasno irregular frequencies, provided that the original boundary value problem is uniquelysolvable. Liu (1991) proved — in a different setting — that the same integral equationfor the two-dimensional problem does not suffer from irregular frequencies either.

11Dirichlet, Peter Gustav Lejeune (1805-1859), German mathematician who contributed to the theoryof numbers and to the establishment of Fourier’s theorem.


2.5.3 Forward speed radiation-diffraction

Another important problem in offshore technology is the slow drift motion of a float-ing marine structure, such as a moored ship or a moored oil platform. The motion isgenerated by resonance between the moored structure and slowly oscillating waves, andmay result in very large horizontal displacements. The viscous damping and the wave(radiation) damping are often small; in many sea states the so-called wave drift damping— defined as the increase in wave drift forces due to a small forward velocity of a bodymoving in waves — may be the dominant damping effect.

The concept of wave drift damping has been introduced by Wichers and Sluijs (1979).It has been discussed further by Wichers and Huijsmans (1984), and Wichers (1988). Anextensive study — among many other contributions — of wave drift forces was publishedby Pinkster (1980).

Consider a body moving horizontally with constant forward speed U in response toincoming regular waves with small amplitude-wavelength ratio A/λ. Let the referenceframe be fixed to the body, with the undisturbed free surface in the (x, y)-plane, thex-axis in the direction of forward motion, and the z-axis vertically upwards. In thisframe the body performs small oscillations due to the incoming waves and is embeddedin a uniform current with speed −U along the x-axis. Assuming the fluid to be idealand of infinite extent in the lower half-space, and the flow to be free of vorticity, andneglecting effects of viscosity and surface tension, then there exists a velocity potentialφ that satisfies Laplace’s equation:

∇2φ = 0 . (2.53)

This potential can be split up as follows:

φ (~x; t) = φs (~x) + φd (~x; t) + φr (~x; t) . (2.54)

The first part φs represents the steady flow and is independent of time. The unsteadyparts φd and φr are time-harmonic with frequency of encounter σ, determined by theangle of incidence β and the orbital frequency ω of the incoming waves:

σ = ω − Uk cosβ , (2.55)

where k = ω2/g is the deep water approximation of the zero-speed wave number. The caseβ = 0 corresponds to following waves, while β = π corresponds to head (i.e. opposing)waves.

The steady potential φs can be put as

φs (~x) = Uχs (~x) = U [χ (~x)− x] , (2.56)

where the term −Ux accounts for the uniform current, and Uχ (~x) is the steady distur-bance due to the presence of the body.

The total radiation potential φr represents the effects of the body oscillations; it canbe written as

φr (~x; t) = Re

iσeiσt

6∑

j=1

ξjφj (~x)

, (2.57)

where ξj is the amplitude in the j-mode of motion (corresponding to surge, sway, heave,roll, pitch, and yaw) and φj is the corresponding radiation potential for unit amplitudeof motion.


The total diffraction potential φd can be expressed as

φd (~x; t) = Re[Aeiσt φ0 (~x) + φ7 (~x)] , (2.58)

where φ7 is the scattering potential, and φ0 is the incoming wave potential:

φ0 (~x) =ig

ωekze−ik(x cos β+y sin β) . (2.59)

The steady potential (2.56) fulfills

∇φs · ~n = 0 ⇒ ∂χ

∂n= n1 on S , (2.60)

corresponding to a zero flux through the wetted body surface S; the normal vector~n = (n1, n2, n3)

T points out of the fluid domain.The boundary conditions for the unknown potentials φj are — see, for instance,

Newman (1978) —

∂φj

∂n= nj +

U

iσmj , j = 1, 2, . . . , 6 , (2.61)

∂φ7

∂n= −∂φ0

∂n, (2.62)

where (n4, n5, n6)T ≡ ~x× ~n. The so-called m-terms mj are defined as

(m1,m2, m3)T = −~n · ∇ (∇χs) , (2.63)

(m4,m5, m6)T = −~n · ∇ (~x×∇χs) . (2.64)

Thus, the normal derivative of each radiation potential consists of two parts. The first,the n-term, represents the oscillatory normal velocity of the body, while the second, them-term, represents the change in the local steady field due to the motion of the body.

Let L be the characteristic dimension of the body. If the Froude number Fr =U/ (gL)1/2 is small, the free surface condition for the steady potential can be approxi-mated to first order in Fr by

∂φs

∂z= 0 at z = 0 . (2.65)

The steady-state problem defined by (2.53), (2.60), and (2.65) can now easily be solvedby means of a source distribution method.

The radiation potentials φj , with j = 1, 2, . . . , 6, and the diffraction potential φd =φ0 + φ7 will then have to satisfy — to first order in Fr — the combined kinematic anddynamic free surface conditions:

−σ2φj + 2iσ∇φs · ∇φj + iσφj∇2φs + g∂φj

∂z= 0 at z = 0 , (2.66)

where ∇ ≡ (∂/∂x, ∂/∂y)T denotes the horizontal gradient operator.When φs is known, this is a linear boundary condition with spatially dependent

coefficients. Far away from the body, we have φs = −Ux, and (2.66) simplifies to

−σ2φj − 2iσU∂φj

∂x+ g

∂φj

∂z= 0 at z = 0 , (2.67)


which contains constant coefficients only.A radiation condition at infinity — stating that all potentials φj must correspond

to outgoing waves — renders this forward speed radiation-diffraction problem uniquelysolvable.

Recently, Nossen, Grue, and Palm (1991) presented a boundary integral equation methodto compute the first order unsteady forces and wave drift forces for arbitrary three-dimensional bodies at small forward speed. Following the approach introduced by Hui-jsmans and Hermans (1985), the forward velocity U is put in nondimensional form asτ ≡ Uσ/g. Typical values for the wave period and the body velocity in offshore problemsare T = 10 s and U = 1 m/s respectively, giving τ ≈ 0.06. The velocity potential andthe Green’s function — with the latter satisfying the radiation condition at infinity —are then expressed in power series of τ , retaining linear terms only. The solution can beexpressed as an integral over the wetted body surface and the free surface.

Another way to solve the above radiation-diffraction problem has been indicatedby Zhao and Faltinsen (1989). They use a hybrid method, where close to the body aboundary element method with Rankine-sources is applied. This region is matched toan outer regime where a multipole expansion is used. Finally, we mention the approachof Wu and Eatock Taylor (1990), wherein the velocity potential is developed in a seriesof τ over the whole free surface. Their examples, however, are all two-dimensional.

A discussion of the state of affairs with regard to the solution of the radiation-diffraction problem — also in coherence with the steady-state ship wave-making andwave-resistance problem — has recently been given by Newman (1991).

2.5.4 Nonlinear motion at zero forward speed

Finally, we present the governing equations for an unrestrained body at zero forwardspeed, i.e. a freely floating body. Let the position and the orientation of the bodybe specified by the three-component vectors ~xG = (x1, x2, x3)

T and ~θG = (θ1, θ2, θ3)T

respectively, as shown in Figure 2.2.The motion of the fluid near the wetted body surface S is determined by the same

mechanisms which are active in the vicinity of an impermeable moving boundary. So,under the assumptions of an ideal fluid and an irrotational flow, the proper boundarycondition is given by (2.28), repeated here for convenience:

∂φ

∂n= ∇φ · ~n = ~V · ~n , (2.68)

where ~V is the velocity of a point on S and the unit normal vector ~n points out of thebody. In terms of the body position and orientation, this condition reads

∂φ

∂n= ∇φ · ~n = ~xG · ~n + (~r × ~n) · ~θG (2.69)

where a dot denotes differentiation with respect to time and ~r = ~xS − ~xG represents theposition of a point ~xS on S with respect to the centre of gravity G, see Figure 2.2. In caseof a floating body under forced motion, ~xG and ~θG are prescribed, and the fluid-bodyinteraction is completely determined by (2.69).

However, if the body is floating freely, then ~xG (t) and ~θG (t) are not known a priori,and these variables have to be determined as part of the solution. In this case, the set of

2.6. PROBLEM DEFINITIONS 29

governing equations is supplemented with the equations of motion for a rigid body with,say, mass M and moment of inertia ~I about its principal axes. The gravitational forceacting on the body is given by

~Fg = −Mg~e3 . (2.70)

The inertial forces and moments acting on the body are directly obtained from hydro-dynamical considerations; integration of the pressure exerted by the fluid over the bodysurface gives

~Fi =∫

S

∫p~n dS , ~Li =

∫

S

∫p (~r × ~n) dS , (2.71)

where the pressure p equals the Bernoulli pressure from (2.24), where we have chosenp0 = 0 and B (t) = 0:

p = −ρ

(∂φ

∂t+

12

(∇φ · ∇φ) + gz

). (2.72)

Thus, in the absence of viscosity, the equations of motion for the freely floating bodyread

M~xG =∫

S

∫p~n dS −Mg~e3 , ~I ⊗ ~θG =

∫

S

∫p (~r × ~n) dS (2.73)

where the symbol ⊗ is used to define a component-wise product of two vectors:

~a⊗~b ≡ (a1b1, a2b2, a3b3)T

. (2.74)

The Neumann boundary condition (2.69) and the hydrodynamic equations of motion (2.73)are the additional governing equations for a body floating freely at zero forward speedin or below the free surface of an ideal and vorticity-free fluid.

2.6 Problem definitions

The potential problem that will be dealt with in this thesis is described as

Nonlinear gravity driven water waves travelling over a bottom, and the hy-drodynamic interaction with solid fixed structures and rigid bodies floating inor below the free surface at zero forward speed.

The evolution of this particular type of dynamical systems is governed by the followingset of (partial) differential equations:

1. Laplace’s equation for the velocity potential throughout the fluid domain:

∇2φ = 0 . (2.75)

2. Nonlinear dynamic and kinematic conditions on the free surface:

Dφ

Dt=

12

(∇φ · ∇φ)− gz ,D~xF

Dt= ∇φ . (2.76)


3. Zero-flux condition on impermeable fixed boundaries:

∂φ

∂n= 0 . (2.77)

4. Contact condition on impermeable moving boundaries:

∂φ

∂n= ~V · ~n . (2.78)

5. Hydrodynamic equations of motion for freely floating bodies:

M~xG =∫

S

∫p~n dS −Mg~e3 , ~I ⊗ ~θG =

∫

S

∫p (~r × ~n) dS . (2.79)

6. Bernoulli’s equation throughout the fluid domain:

−p

ρ=

∂φ

∂t+

12

(∇φ · ∇φ) + gz . (2.80)

In general, supplementary initial conditions and the imposition of a Sommerfeld radiationcondition at infinity render these problems uniquely solvable.

From now on, the problem of free surface waves travelling on water of finite depth— governed by (1-3) — is referred to as a (classical) water-wave problem. Likewise, asystem of free surface waves in hydrodynamic interaction with floating bodies at zeroforward speed — governed by (1-6) — is labelled a wave-body system, and any problemmatching this definition is called a wave-body problem hereafter.

The above set of equations constitutes an elliptic spatial problem with time dependentboundary conditions. In most numerical procedures these water-wave and wave-bodyproblems are treated as such; the elliptic boundary value problem is solved at a certaintime level, and then the boundary conditions — and, if a floating body is involved, theequations of motion for the body — are integrated in time to obtain a new boundaryvalue problem, and so forth. In chapter 4 we shall outline a method, proceeding alongthese lines, for the numerical simulation of the physical systems described above. Thispanel method is based on a Green’s formulation for the velocity potential; the transitionfrom an elliptic boundary value problem to a boundary integral equation is treated inthe next chapter.

2.7 Bibliography

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Beck, R.F., and Liapis, S. 1987. Transient motions of floating bodies at zero forwardspeed. Journal of Ship Research 31(3):164-176.

Broeze, J., and Romate, J.E. 1992. Absorbing boundary conditions for free surface

2.7. BIBLIOGRAPHY 31

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Newman, J.N. 1991. The quest for a three-dimensional theory of ship-wave interactions.Philosophical Transactions of the Royal Society of London, Series A, 334:213-227.

Ni, S.-Y. 1987. A Method for Calculating Non-linear Free Surface Potential Flows usingHigher Order Panels. Ph.D. thesis, Chalmes University. Gothenborg, Sweden.

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Pinkster, J.A. 1980. Low Frequency Second Order Wave Exciting Forces on FloatingStructures. Ph.D. thesis, Delft University of Technology. Delft, The Netherlands. Also:MARIN Publication No.650. Wageningen, The Netherlands.

Raven, H.C. 1988. Variations on a theme by Dawson. Proceedings of the SeventeenthSymposium on Naval Hydrodynamics, The Hague, The Netherlands.

Raven, H.C. 1992. A practical nonlinear method for calculating ship wavemaking andwave resistance. Proceedings of the Nineteenth Symposium on Naval Hydrodynamics,Seoul, Korea.

Romate, J.E. 1992. Absorbing boundary conditions for free surface waves. Journal ofComputational Physics 99(1):135-145.

Sclavounos, P.D., and Lee, Ch.-H. 1985. Topics on boundary element solutions ofwave radiation-diffraction problems. Proceedings of the Fourth International Conferenceon Numerical Ship Hydrodynamics, Washington, DC.

Sclavounos, P.D., and Nakos, D.E. 1988. Stability analysis of panel methods forfree-surface flows with forward speed. Proceedings of the Seventeenth Symposium onNaval Hydrodynamics, The Hague, The Netherlands.


Simon, M.J., and Ursell, F. 1984. Uniqueness in linearized two-dimensional water-wave problems. Journal of Fluid Mechanics 148:137-154.

Sommerfeld, A. 1964. Partial Differential Equations in Physics. Volume VI of Lec-tures on Theoretical Physics. Academic Press.

Stoker, J.J. 1957. Water Waves. The Mathematical Theory with Applications. Inter-science Publishers.

Ursell, F. 1981. Irregular frequencies and the motion of floating bodies. Journal ofFluid Mechanics 105:143-156.

Wichers, J.E.W. 1988. A Simulation Model for a Single Point Moored Tanker. Ph.D.thesis, Delft University of Technology. Delft, The Netherlands. Also: MARIN Publica-tion No.797. Wageningen, The Netherlands.

Wichers, J.E.W., and Huijsmans, R.H.M. 1984. On the low frequency hydrody-namic damping forces acting on offshore moored vessels. Proceedings of the OffshoreTechnology Conference, Houston, Texas, OTC 4831.

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Now when gleaming dawn with bright eyes beheld the lofty peaks of Pelion,and the calm headlands were being drenched as the sea was ruffled by thewinds, then Tiphys awoke from sleep; and at once he roused his comrades togo on board and make ready the oars. And a strange cry did the harbour ofPagasae utter, yea and Pelian Argo herself, urging them to set forth. Forin her a beam divine had been laid which Athena had brought from an oakof Dodona and fitted in the middle of the stem. And the heroes went to thebenches one after the other, as they had previously assigned for each to rowin his place, and took their seats in due order near their fighting gear. In themiddle sat Ancaeus and mighty Heracles, and near him he laid his club, andbeneath his tread the ship’s keel sank deep. And now the hawsers were beingslipped and they poured wine on the sea. But Jason with tears held his eyesaway from his fatherland. And just as youths set up a dance in honour ofPhoebus either in Pytho or haply in Ortygia, or by the waters of Ismenus,and to the sound of the lyre round his altar all together in time beat the earthwith swiftly-moving feet; so they to the sound of Orpheus’ lyre smote withtheir oars the rushing sea-water, and the surge broke over the blades; and onthis side and on that the dark brine seethed with foam, boiling terribly throughthe might of the sturdy heroes. And their arms shone in the sun like flameas the ship sped on; and ever their wake gleamed white far behind, like a pathseen over a green plain. On that day all the gods looked down from heavenupon the ship and the might of the heroes, half-divine, the bravest of men thensailing the sea; and on the topmost heights the nymphs of Pelion wondered asthey beheld the work of Itonian Athena, and the heroes themselves wieldingthe oars. And there came down from the mountain-top to the sea Chiron,son of Philyra, and where the white surf broke he dipped his feet, and, oftenwaving with his broad hand, cried out to them at their departure, “Good speedand a sorrowless home-return!”


Chapter 3

Boundary Integral EquationFormulations

Now when they had left the curving shore of the harbour through the cun-ning and counsel of prudent Tiphys son of Hagnias, who skilfully handled thewell-polished helm that he might guide them steadfastly, then at length they setup the tall mast in the mast-box, and secured it with forestays, drawing themtaut on each side, and from it they let down the sail when they had hauled it tothe top-mast. And a breeze came down piping shrilly; and upon the deck theyfastened the ropes separately round the well-polished pins, and ran quietly pastthe long Tisaean headland. And for them the son of Oeagrus1 touched his lyreand sang in rhythmical song of Artemis, saviour of ships, child of a glorioussire, who hath in her keeping those peaks by the sea, and the land of Iolcos;and the fishes came darting through the deep sea, great mixed with small, andfollowed gambolling along the watery paths. And as when in the track of theshepherd, their master, countless sheep follow to the fold that have fed to thefull of grass, and he goes before gaily piping a shepherd’s strain on his shrillreed; so these fishes followed; and a chasing breeze ever bore the ship onward.


1i.e. Orpheus

35

36 CHAPTER 3. BOUNDARY INTEGRAL EQUATION FORMULATIONS

3.1 Introduction

In this chapter the (numerical) solution of the time independent field equation (2.23) isconsidered. Solutions of Laplace’s equation

∇2φ (~x) = 0 , ~x in Ω (3.1)

are referred to as harmonic functions.The velocity potential φ satisfying (3.1) in the entire fluid domain Ω is subject to

conditions on the bounding surface ∂Ω. These conditions can be written in the generalform

Γ(

φ,∂φ

∂n, ~x

)= 0 , ~x on ∂Ω (3.2)

In the following it is assumed that the boundary value problem (3.1-3.2) is well-posed.

In the past, numerous methods — both analytical and numerical — have been developedto solve the above problem. Since the field equation (3.1) is linear and elliptic, one mightthink that this problem can easily be solved with standard methods. However, difficultiesmay arise from the supplementary boundary conditions (3.2); for instance, geometricalsingularities (sharp edges, corners, etc.) hamper the solution of this boundary valueproblem.

As far as numerical techniques are concerned, finite difference methods, finite vol-ume methods, finite element methods, and integral equation methods are used mostfrequently. The first three types of methods are so-called field discretization methods;in these techniques the field equation — here Laplace’s equation (3.1) — is discretizeddirectly. In integral equation methods the first step is to transform the field equationinto a (set of) boundary integral equation(s) by using a fundamental identity — hereGreen’s second identity — for the problem under consideration. The next step in inte-gral equation methods is the discretization of these boundary integral equations.

So far, none of the above methods has been proven to be superior to all other meth-ods. Each method has its specific advantages and disadvantages; for instance, integralequation methods lower the problem dimension by one, thus reducing data storage andallowing a more efficient use of computer memory. On the other hand, the (derived)integral equations are more difficult to solve than the (original) field equation; in gen-eral, a full system matrix is obtained, while field discretization methods usually yieldband matrices. As a consequence, relatively expensive matrix solvers have to be usedin integral equation techniques. Furthermore, integral equation methods are sensitiveto the discretization, and later extensions to the solution of problems that are not gov-erned by the field equation concerned, are difficult or even impossible. However, forproblems with moving boundaries, like free surface wave problems, integral equationmethods have a clear advantage compared to the field discretization methods; the mo-tion of (two-dimensional) surface grids is much easier to compute than the motion of(three-dimensional) volume grids. A discussion of the advantages and disadvantages offield discretization methods and integral equation methods — including an analysis ofthe computational effort involved — has been given by Romate and Zandbergen (1989).

The use of integral equation methods goes back to the fifties, when these techniqueswere introduced in various fields, such as aerodynamics and elastostatics. The soundtheoretical basis of potential theory and the fast growing knowledge of integral equations

3.2. INTEGRAL EQUATIONS IN POTENTIAL THEORY 37

since the pioneering work of Fredholm2 made integral equation methods good competitorsof other numerical techniques.

For the above and other reasons Romate (1989) decided to use a boundary integralequation method to solve the nonlinear free surface wave problem. Elaborating on hispanel method, a new simulation procedure for water waves interacting with fixed andfloating bodies was developed; this numerical algorithm is outlined and discussed in thenext chapter.

The aim of this chapter is to present a compact introduction to integral equationsin potential theory. In section 3.2 a number of integral equations is derived using basicconcepts of potential theory. The well-posedness of the boundary value problem (abbre-viated: BVP) (3.1-3.2) is considered in section 3.3. The choice of the integral equationmethod to solve this boundary value problem is accounted for in section 3.4. For a moredetailed introduction to integral equation methods for potential problems we refer toRomate (1989).

3.2 Integral equations in potential theory

In this section the BVP (3.1-3.2) will be reformulated in terms of boundary integralequations. Those elements of potential theory that are essential for the panel methoddescribed in chapter 4 will be discussed briefly, assuming that the reader is familiarwith the basic concepts. For an extensive mathematical treatment of potential theorythe reader is referred to Kellogg (1953), and Courant and Hilbert (1962). An excellentintroduction to integral equation methods in potential (scattering) theory is given byColton and Kress (1983).

The well-posedness of the elliptic BVP (3.1-3.2) implies that solutions φ (~x) and allspatial derivatives of these solutions are finite and continuous throughout Ω, except possi-bly at some points on ∂Ω. Therefore, it can be expected that the properties of a harmonicsolution φ depend strongly on the shape of ∂Ω and the boundary conditions (3.2). Indeed,these boundary data determine the solution φ, as will be shown later on.The three-dimensional singly connected3 domain Ω in Figure 3.1 is bounded by thepiecewise smooth surface ∂Ω. For two arbitrary scalar fields ϕ and ψ which are C2-continuous4 throughout Ω, Green’s second identity states:

∫∫

Ω

∫[ϕ∆ψ − ψ∆ϕ] dΩ =

∫

∂Ω

∫ [ψ

∂ϕ

∂nξ− ϕ

∂ψ

∂nξ

]dSξ , (3.3)

where ∆ ≡ ∇2 is Laplace’s operator and ~nξ is the unit normal vector in ~ξ on ∂Ω, pointinginto Ω.

Substituting the unknown potential φ (~x) for ϕ, with φ satisfying (3.1), and replacingψ by the Green’s function

G(~ξ; ~x

)=−14πr

, r = ‖~x− ~ξ ‖ (3.4)

2Fredholm, Ivar (1866-1927), Swedish mathematician who made important contributions to the theoryof integral equations.

3See section 2.5 for a definition.4This means that ϕ and ψ are continuously differentiable and have continuous spatial derivatives up

to second order.


Figure 3.1: Definition of Ω, ∂Ω, and ~nξ.

with ~x fixed and ~ξ on ∂Ω, we obtain the following identity:∫∫

Ω

∫φ (~x)∆G

(~ξ; ~x

)dΩ =

∫

∂Ω

∫G

(~ξ; ~x

) ∂φ

∂nξ

(~ξ)

dSξ

−∫

∂Ω

∫φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)dSξ . (3.5)

If ~x is in the interior of Ω, then the Green’s function G(~ξ; ~x

)has a singularity in ~ξ = ~x.

Introducing a neighbourhood Ωε (~x) of ~x with boundary Sε (~x), a small sphere with centreat ~x and radius ε, identity (3.5) applied to Ω \ Ωε (~x) — with boundary ∂Ω ∪ Sε (~x) —reduces in the limit ε → 0 to

φ (~x) =∫

∂Ω

∫ [∂φ

∂nξ

(~ξ)

G(~ξ; ~x

)− φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ . (3.6)

If ~x is on the boundary ∂Ω, then identity (3.5) reduces in a similar way to

ϑ (~x)4π

φ (~x) =∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ)

G(~ξ; ~x

)− φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ , (3.7)

where ϑ (~x) denotes the interior space angle of ∂Ω in ~x; if ~x is on a smooth part of ∂Ω,then ϑ (~x) = 2π. The symbol

∫∫− denotes the finite part of the integral∫∫

in the sense ofHadamard (1923); see also Courant and Hilbert (1962).

Since all surface integrals in (3.5) are regular, the ‘normal derivative operator’ ∂/∂n =~n ·∇ can be brought under the integral signs; then, by the above token a second integralequation is obtained:

ϑ (~x)4π

∂φ

∂n(~x) =

∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ) ∂G

∂nx

(~ξ; ~x

)− φ

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)]dSξ , (3.8)

where the normal derivative operator is applied to G both in the field point ~x and in thesource point ~ξ.

3.2. INTEGRAL EQUATIONS IN POTENTIAL THEORY 39

The surface ∂Ω also bounds the exterior domain Ω′ = IR3 \ (Ω ∪ ∂Ω) which extends toinfinity and is singly connected. Let SR (~x) denote a sphere with centre at ~x and radiusR sufficiently large to enclose Ω, see Figure 3.1. Then, application of Green’s secondidentity (3.3) to the part of Ω′ bounded by ∂Ω and SR (~x) leads to

ϑ′ (~x)4π

φ′ (~x) =∫

∂Ω

∫−

[φ′

(~ξ) ∂G

∂nξ

(~ξ; ~x

)− ∂φ′

∂nξ

(~ξ)

G(~ξ; ~x

)]dSξ

+∫

SR(~x)

∫ [φ′

(~ξ) ∂G

∂r

(~ξ; ~x

)− ∂φ′

∂r

(~ξ)

G(~ξ; ~x

)]dSξ , (3.9)

ϑ′ (~x)4π

∂φ′

∂n(~x) =

∫

∂Ω

∫−

[φ′

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)− ∂φ′

∂nξ

(~ξ) ∂G

∂nx

(~ξ; ~x

)]dSξ

+∫

SR(~x)

∫ [φ′

(~ξ) ∂2G

∂nx∂r

(~ξ; ~x

)− ∂φ′

∂r

(~ξ) ∂G

∂nx

(~ξ; ~x

)]dSξ ,

(3.10)

where φ′ satisfies Laplace’s equation in Ω′, and ϑ′ (~x) denotes the exterior space angleof ∂Ω in ~x; obviously, ϑ (~x) + ϑ′ (~x) = 4π for all ~x on ∂Ω. For increasing radius R, thecontributions of the spherical boundary SR (~x) vanish, assuming that φ′ (R) = O (

R−1)

for R →∞.

From now on it is assumed that ~x is on a regular (smooth) part of ∂Ω, and henceϑ (~x) = ϑ′ (~x) = 2π. Addition of (3.7) and (3.9), and (3.8) and (3.10) results in

12

(φ + φ′) (~x) =∫

∂Ω

∫−

(∂φ

∂nξ− ∂φ′

∂nξ

) (~ξ)

G(~ξ; ~x

)dSξ

−∫

∂Ω

∫− (φ− φ′)

(~ξ) ∂G

∂nξ

(~ξ; ~x

)dSξ , (3.11)

12

(∂φ

∂n+

∂φ′

∂n

)(~x) =

∫

∂Ω

∫−

(∂φ

∂nξ− ∂φ′

∂nξ

) (~ξ) ∂G

∂nx

(~ξ; ~x

)dSξ

−∫

∂Ω

∫− (φ− φ′)

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)dSξ . (3.12)

Then, if we put(

∂φ

∂nξ− ∂φ′

∂nξ

) (~ξ)

= σ(~ξ)

, ~ξ on ∂Ω (3.13)

− (φ− φ′)(~ξ)

= µ(~ξ)

, ~ξ on ∂Ω (3.14)

the identities (3.11-3.12) can be rewritten in terms of the source distribution σ and thenormal dipole distribution µ on the surface ∂Ω:

φ (~x) +12µ (~x) =

∫

∂Ω

∫−

[σ

(~ξ)

G(~ξ; ~x

)+ µ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ (3.15)


∂φ

∂n(~x)− 1

2σ (~x) =

∫

∂Ω

∫−

[σ

(~ξ) ∂G

∂nx

(~ξ; ~x

)+ µ

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)]dSξ

(3.16)

In order to have a unique solution φ′ of Laplace’s equation, that is

∇2φ′ (~x) = 0 , ~x in Ω′ , (3.17)

boundary conditions on ∂Ω are needed; written in a general form, we have

Γ′(

φ′,∂φ′

∂n, ~x

)= 0 , ~x on ∂Ω . (3.18)

Since φ′ is fictitious5, any condition that renders the exterior boundary value prob-lem (3.17-3.18) well-posed is adequate. From (3.13-3.14) and (3.15-3.16) it is clear thatdifferent choices for the boundary conditions (3.18) lead to different boundary integralequations:

1. Choose φ′ = φ on ∂Ω ⇒ µ = 0

i.e. a source-only distribution (~x on ∂Ω):

φ (~x) =∫

∂Ω

∫−σ

(~ξ)

G(~ξ; ~x

)dSξ , (3.19)

∂φ

∂n(~x) =

12σ (~x) +

∫

∂Ω

∫−σ

(~ξ) ∂G

∂nx

(~ξ; ~x

)dSξ . (3.20)

2. Choose∂φ′

∂n=

∂φ

∂non ∂Ω ⇒ σ = 0

i.e. a dipole-only distribution (~x on ∂Ω):

φ (~x) = −12µ (~x) +

∫

∂Ω

∫−µ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)dSξ , (3.21)

∂φ

∂n(~x) =

∫

∂Ω

∫−µ

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)dSξ . (3.22)

3. Choose[φ′ = 0 on ∂Ω ⇒ ∂φ′

∂n= 0 on ∂Ω

]⇒ σ =

∂φ

∂n, µ = −φ

i.e. a Green’s or mixed source-dipole distribution (~x on ∂Ω):

12φ (~x) =

∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ)

G(~ξ; ~x

)− φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ , (3.23)

12

∂φ

∂n(~x) =

∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ) ∂G

∂nx

(~ξ; ~x

)− φ

(~ξ) ∂2G

∂nx∂nξ

(~ξ; ~x

)]dSξ .

(3.24)5This means that the potential φ′ does not represent a real physical phenomenon.

3.3. WELL-POSEDNESS: EXISTENCE AND UNIQUENESS 41

The final choice of the boundary integral equation formulation will be made in section 3.4.First, we discuss the well-posedness of the BVP (3.1-3.2) and present some results on theuniqueness of integral equations (3.7-3.8) for various boundary conditions and singularitydistributions.

3.3 Well-posedness: existence and uniqueness

For the well-posedness of the BVP (3.1-3.2) the definition due to Hadamard reads

Definition 1 : Well-posedness (Hadamard, 1923)The boundary value problem (3.1-3.2) is well-posed if for all ~x in Ω a unique solution φ,depending continuously on the boundary data, exists.

The question of existence and uniqueness of a solution φ to (3.1-3.2) has been treatedin many books and papers; for completeness, some important results are presented here.For rigorous proofs of these results the reader is referred to Kellogg (1953), Jaswon andSymm (1977), and Courant and Hilbert (1962).

Suppose that the simply connected domain Ω is bounded by a piecewise smooth surface

∂Ω =M⋃

i=1

Si , (3.25)

with all subsurfaces Si twice continuously differentiable. In addition, it is assumed that∂Ω satisfies the so-called external cone condition:

Condition 1 : External cone conditionFor each point ~x on ∂Ω there exists a circular cone lying outside Ω with angle ϑ > 0 andvertex at ~x.

Furthermore, it is assumed that the boundary conditions

Γi

(φ,

∂φ

∂n, ~x

)= 0 , ~x on Si , i = 1, 2, . . . , M , (3.26)

are all continuous.

A necessary condition for the existence of a solution to (3.1-3.2) is obtained by applyingGauss’ theorem6 to the vector velocity field ∇φ:

∫∫

Ω

∫∇2φdΩ =

∫

∂Ω

∫∇φ · ~n dS . (3.27)

By continuity we have ∇2φ = 0 throughout Ω, giving∫

∂Ω

∫∂φ

∂ndS = 0 . (3.28)

6Gauss, Johann Karl Friedrich (1777-1855), German mathematician. In addition to his work in puremathematics, he made major contributions to theoretical astronomy, geodesy, terrestrial magnetism andelectricity, and other branches of physics.


Condition (3.28) states that the total flux of fluid through the surface ∂Ω must equalzero.

With the above definitions and conditions, the following theorems on the existence anduniqueness of a solution to the BVP (3.1-3.2) can be formulated and proven:

Theorem 1 : Existence for the boundary value problem (3.1-3.2)Let Ω ⊂ IR3 be a simply connected domain, and let ∂Ω, Si, and Γi satisfy the abovementioned conditions. Further, let at each point on ∂Ω either φ, or ∂φ/∂n, or a linearcombination of φ and ∂φ/∂n be specified. Then there is at least one scalar function φsatisfying the boundary value problem (3.1-3.2), only if (3.28) holds.

Theorem 2 : Uniqueness for the boundary value problem (3.1-3.2)Let Ω ⊂ IR3 be a simply connected domain, and let ∂Ω, Si, and Γi satisfy the abovementioned conditions. Further, let at each point on ∂Ω either φ, or ∂φ/∂n, or a linearcombination of φ and ∂φ/∂n be specified. Then there is at most one scalar function φsatisfying the boundary value problem (3.1-3.2); this solution is determinate but for aconstant if the value of φ is nowhere specified on ∂Ω.

From the foregoing it appears that the solution to the boundary value problem (3.1-3.2)can be constructed by means of integral equations (3.15-3.16). However, the use of thisset of integral equations again raises the question of the existence of a unique solutionφ. From the derivation of the integral identities (3.7-3.8) — with ϑ (~x) = 2π for all ~x on∂Ω — it is clear that

φ satisfies (3.2) ⇒ [ φ satisfies (3.1) ⇐⇒ φ satisfies (3.15-3.16) ] ,

which solves the question of existence and uniqueness, assuming that φ already satisfiesthe given boundary conditions.

Next, a summary of the uniqueness results with regard to a solution φ for variousboundary conditions is given. A boundary condition is known as

• a Dirichlet condition if φ is prescribed;

• a Neumann condition if ∂φ/∂n is prescribed.

The results on uniqueness for (3.7) and (3.8) with ϑ (~x) = 2π are listed in table 3.1,where ‘non-unique’ means that the solution φ is determined but for a constant.

With these results, the uniqueness of the different singularity distributions can bedetermined for various boundary conditions. Therefore, let us return to the fictitiousproblem introduced in the previous section, where ∇2φ′ = 0 in the exterior domain Ω′.From (3.13-3.14) it is evident that the boundary conditions on ∂Ω for φ′ and ∂φ′/∂ndetermine the type of surface distribution for the real problem of ∇2φ = 0 in Ω withcertain boundary conditions. For the uniqueness of this distribution it is necessary thatunique solutions φ and φ′ of the given boundary value problems can be found withidentities (3.7-3.8).

3.4. CHOICE OF INTEGRAL EQUATION METHOD 43

boundary condition φ ∂φ/∂nintegral equation (3.7)

internal Dirichlet specified uniqueexternal Dirichlet specified uniqueinternal Neumann non-unique specifiedexternal Neumann unique specified

integral equation (3.8)internal Dirichlet specified uniqueexternal Dirichlet specified uniqueinternal Neumann non-unique specifiedexternal Neumann non-unique specified

Table 3.1: Uniqueness for solutions of (3.7-3.8) with various boundary conditions.

boundary condition source-only dipole-only Green’sintegral equation (3.7)

internal Dirichlet σ unique µ unique ∂φ/∂n uniqueexternal Dirichlet σ unique µ non-unique ∂φ/∂n uniqueinternal Neumann – – φ non-uniqueexternal Neumann – – φ unique

integral equation (3.8)internal Dirichlet – – ∂φ/∂n uniqueexternal Dirichlet – – ∂φ/∂n uniqueinternal Neumann σ unique µ non-unique φ non-uniqueexternal Neumann σ unique µ non-unique φ non-unique

Table 3.2: Uniqueness for various surface distributions.

The results on uniqueness for a source-only, a dipole-only, and a Green’s (mixed source-dipole) distribution on ∂Ω are listed in table 3.2. These results7 will be used in the nextsection to justify the choice of the boundary integral equation — and the correspondingsurface distributions — to solve the BVP (3.1-3.2).

3.4 Choice of integral equation method

In general, analytical solutions to the boundary value problem (3.1-3.2) can not be found.However, the integral equations derived in section 3.2 can be used to construct approxi-mate solutions; for that purpose, integral equations are formulated for a finite number of

7The discussions so far have referred to simply connected domains only. The degree of connectivityis determined by the minimum number of different barriers needed to make each subregion simplyconnected; if n − 1 such barriers are needed, the region is said to be n-ply connected. In a doublyconnected region the solution φ is a multi-valued function of position, and therefore non-unique. Theinsertion of a barrier which creates two simply connected regions renders φ single-valued, and the theorydescribed earlier for simply connected domains can be applied. Along the barrier the jump in φ — i.e.the circulation when crossing the barrier — has to be specified. See also Batchelor (1967).


(say, N) points on the bounding surface ∂Ω, so that N equations are obtained in termsof boundary data and unknowns. In the following we shall indicate a way to transformthis set of integral equations into a linear system, where the number of equations equalsthe number of unknowns.

Figure 3.2: Boundary ∂Ω consisting of two smooth subsurfaces S1 and S2.

Without loss of generality it is assumed that the boundary ∂Ω consists of two smoothsubsurfaces, see Figure 3.2. Suppose that a Dirichlet boundary condition is imposedon S1, while S2 is a Neumann boundary.

In order to solve the boundary value problem numerically, ∂Ω is discretized with N1

panels (surface elements) on S1 and N2 panels on S2. For each panel Taylor seriesexpansions are used to approximate the surface shape and the variables defined on theboundary; the Taylor series of the surface distributions σ and µ are expressed in termsof N = N1 + N2 singularity strengths at the N panel midpoints. The integral equationsare then replaced by a set of N linear equations involving N source strengths and/or Ndipole strengths. A point collocation method is chosen to impose the boundary conditionson ∂Ω; the panel midpoints serve as collocation points, where the proper boundaryconditions are applied. Thus, N data are substituted, leaving only N unknowns to besolved from a system of N linear equations:

A~y = ~b . (3.29)

The choice of integral equation method determines the properties of this linear system.The actual order of approximation of the surface shape and the order of expansion inthe Taylor series for the singularity distributions determine the accuracy of the method.Higher order approximations will reduce ‘leakage’ and hence the error in the solution.

By preference, the choice of the surface distributions and the integral equations shouldlead to Fredholm integral equations of the second kind. Then, the resulting systemmatrix A has large coefficients on the diagonal (due to the principal parts of the singularintegrals), and iterative methods can be used to solve (3.29). Integral equations of thesecond kind are numerically stable, which is one of the main reasons for their widespreaduse in the past. A major disadvantage of this type of equations is that special propertiesof Laplace’s equation, such as symmetry, in general are not preserved, see Hsiao (1987).

3.4. CHOICE OF INTEGRAL EQUATION METHOD 45

Fredholm integral equations of the first kind do preserve the symmetry and coercivenessproperties, which can be exploited by special matrix solvers such as conjugate gradient(CG-) methods. However, equations of the first kind give an ill-conditioned matrix A(to be more precise: without large entries on the main diagonal), which may give rise tonumerical instabilities — see, for instance, Hsiao (1987).

Another important criterium in the choice of the surface distributions and the integralequations is the computational effort to attain a certain level of accuracy. As Hunt (1978)states, the choice of a mixed source-dipole distribution reduces leakage considerably andleads to better results. For this reason a Green’s formulation is favourable. Anothermajor plus-point of a mixed formulation is that the fictitious potential φ′ equals zerothroughout Ω′. Hence, it can not introduce singularities in the solution on the boundingsurface ∂Ω.

Lastly, we mention an advantage which should not be underestimated; with a Green’sformulation the chosen singularities σ = ∂φ/∂n and µ = −φ are of direct physical inter-est. To illuminate this, consider the free surface boundary conditions (2.76), involving φand its gradient ∇φ, and suppose that the free surface potential is known. Then ∂φ/∂nis calculated with a panel method using a Green’s formulation, and the tangential deriva-tives ∂φ/∂s1 and ∂φ/∂s2 (where s1 and s2 are tangential coordinates) can be computedfrom φ, thus giving the necessary ingredients to compute the free surface velocity ∇φ.

For the above (and other less important) reasons Romate (1989) decided to use a Green’sformulation for his higher order panel method. Thus we are left with integral equa-tions (3.23-3.24) to solve the boundary value problem (3.1-3.2). From these two identi-ties, the latter is more expensive for a given level of accuracy, due to the evaluation ofthe second order normal derivative of the Green’s function G. Therefore, only the firstintegral equation will be used to construct a numerical solution procedure:

12φ (~x) =

∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ)

G(~ξ; ~x

)− φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ (3.30)


With φ known in all collocation points on ∂Ω, the tangential velocities are calculatedusing finite differences. Once ∂φ/∂n is also known in all collocation points, the velocityfield on the boundary can be determined. Finally, with identity (3.30) the potential φcan be computed throughout the fluid domain Ω, and from these data the quantities ofphysical interest — such as the velocity and pressure fields — can be obtained.

3.5 Bibliography

Batchelor, G.K. 1967. An Introduction to Fluid Dynamics. Cambridge UniversityPress.

Colton, D., and Kress, R. 1983. Integral Equation Methods in Scattering Theory.John Wiley and Sons.

Courant, R., and Hilbert, D. 1962. Methods of Mathematical Physics II. Inter-science Publishers.

Hadamard, J. 1923. Lectures on Cauchy’s Problem. Oxford University Press.

Hsiao, G.C. 1987. On the stability of boundary element methods for integral equationsof the first kind. In Boundary Element Methods IX. Volume 1: Mathematical and com-putational aspects. Springer-Verlag.

Hunt, B. 1978. The panel method for subsonic aerodynamic flows: a survey of mathe-matical formulations and numerical models, and an outline of the new British Aerospacescheme. VKI Lecture series 1978-4 on Computational Fluid Dynamics.

Jaswon, M.A., and Symm, G.T. 1977. Integral Equation Methods in Potential The-ory and Elastostatics. Academic Press.

Kellogg, O.D. 1953. Foundations of Potential Theory. Dover Publications.


Romate, J.E., and Zandbergen, P.J. 1989. Boundary integral equation formulationsfor free-surface flow problems in two and three dimensions. Computational Mechanics4:276-282.


Thereupon a spirit of contention stirred each chieftain, who should be thelast to leave his oar. For all around the windless air smoothed the swirlingwaves and lulled the sea to rest. And they, trusting in the calm, mightilydrove the ship forward; and as she sped through the salt sea, not even thestorm-footed steeds of Poseidon would have overtaken her. Nevertheless whenthe sea was stirred by violent blasts which were just rising from the riversabout evening, forspent with toil, they ceased. But Heracles by the might ofhis arms pulled the weary rowers along all together, and made the strong-knittimbers of the ship to quiver. But when, eager to reach the Mysian mainland,they passed along in sight of the mouth of Rhyndacus and the great cairnof Aegaeon, a little way from Phrygia, then Heracles, as he ploughed up thefurrows of the roughened surge, broke his oar in the middle. And one half heheld in both his hands as he fell sideways, the other the sea swept away withits receding wave. And he sat up in silence glaring round; for his hands wereunaccustomed to lie idle . . .

But the son of Zeus8 having duly enjoined on his comrades to prepare thefeast took his way into a wood, that he might first fashion for himself an oarto fit his hand. Wandering about he found a pine not burdened with manybranches, nor too full of leaves, but like to the shaft of a tall poplar; so greatwas it both in length and thickness to look at. And quickly he laid on theground his arrow-holding quiver together with his bow, and took off his lion’sskin. And he loosened the pine from the ground with his bronze-tipped cluband grasped the trunk with both hands at the bottom, relying on his strength;and he pressed it against his broad shoulder with legs wide apart; and clingingclose he raised it from the ground deep-routed though it was, together withclods of earth. And as when unexpectedly, just at the time of the stormysetting of baleful Orion, a swift gust of wind strikes down from above, andwrenches a ship’s mast from its stays, wedges and all; so did Heracles lift thepine. And at the same time he took up his bow and arrows, his lion skin andclub, and started on his return . . .

Argonautica, Book I, Verses 1153-1171 and 1187-1206.

8i.e. Heracles


Chapter 4

Algorithm for Wave-BodySimulations

Meantime Hylas with pitcher of bronze in hand had gone apart from thethrong, seeking the sacred flow of a fountain, that he might be quick in drawingwater for the evening meal and actively make all things ready in due orderagainst his lord’s1 return . . .And quickly Hylas came to the spring which thepeople who dwell thereabouts call Pegae. And the dances of the nymphs werejust now being held there . . .But one, a water-nymph was just rising fromthe fair-flowing spring; and the boy she perceived close at hand with the rosyflush of his beauty and sweet grace . . .But as soon as he dipped the pitcherin the stream, leaning to one side, and the brimming water rang loud as itpoured against the sounding bronze, straightway she laid her left arm aboveupon his neck yearning to kiss his tender mouth; and with her right hand shedrew down his elbow, and plunged him into the midst of the eddy . . .

Alone of his comrades the hero Polyphemus, son of Eilatus, as he wentforward on the path, heard the boy’s cry, for he expected the return of mightyHeracles . . .And straightway he told the wretched calamity while his heartlaboured with his panting breath. “My poor friend, I shall be the first to bringthee tidings of bitter woe. Hylas has gone to the well and has not returnedsafe, but robbers have attacked and are carrying him off, or beasts are tearinghim to pieces; I heard his cry.” Thus he spake; and when Heracles heard hiswords, sweat in abundance poured down from his temples and the black bloodboiled beneath his heart. And in wrath he hurled the pine to the ground andhurried along the path whither his feet bore on his impetuous soul . . .

Argonautica, Book I, Fragments from Verses 1207-1264.

1i.e. Heracles

49

50 CHAPTER 4. ALGORITHM FOR WAVE-BODY SIMULATIONS

4.1 Introduction

In this chapter we present a boundary element method for numerical time domain sim-ulations of wave-body interactions in two and three dimensions.

Section 4.2 is a review of numerical techniques for the solution of the nonlinear wave-body problem defined in section 2.6. For a recent survey of numerical methods for linearand nonlinear water wave simulations we refer to the reviews given by Romate (1989).

The basic algorithm for nonlinear water wave simulations is outlined in section 4.3.The major features of the discrete approximation of the problem are briefly touched inthis section; the reader is referred to the work of Romate (1988, 1989, 1990) for a detaileddiscussion of the geometry approximation, the calculation of the influence coefficients,and the applied time stepping scheme.

In section 4.4 this basic algorithm is made fit for the simulation of nonlinear waterwaves interacting with floating bodies at zero forward speed. The essential feature of thisextended algorithm is that the hydrodynamic equilibrium of the fluid and the body ispreserved at each time level. A stable time stepping mechanism for a freely floating bodycalls for the accurate computation of the hydrodynamic forces from pressure integrationsover the wetted body surface S. However, the Bernoulli pressure involves the unknownpartial time derivative of the velocity potential φ. The fact that φt, besides φ, satisfiesLaplace’s equation, makes it possible to derive an extra boundary integral equation,connecting φt and its normal derivative φtn; integration is over the fluid domain boundary,including S. But, if the body is floating freely, both φt and φtn are unknown along S; toprovide uniqueness, the hydrodynamic equations of motion for the body are transformedinto a boundary integral equation, connecting φt and φtn on S only. The discretizedversions of the extra integral equation and the transformed equations of motion togetheryield a system of linear equations, such that the number of equations and the number ofunknowns are equal. The solution of the resulting matrix equation provides the φt-termin the Bernoulli pressure on S.

4.2 Review of methods for nonlinear ship motions

It has already been mentioned in section 2.6 that a time domain solution procedure forthe nonlinear wave-body problem involves the solution of Laplace’s equation at each timelevel, and the marching in time by integration of the nonlinear free surface conditions andthe equations of motion for an unrestrained body. With very few exceptions only finitedifference methods and integral equation methods have been used to solve this problem.2

In two dimensions, integral equation methods have proven to be very suitable for theaccurate and efficient solution of the nonlinear water-wave problem. At times, rathersuccessful extensions to linear and nonlinear wave-body problems have been reported.

The first successful method for the transient water-wave problem (in the absence ofa floating body) was developed by Longuet-Higgins and Cokelet (1976). They used aLagrangian description of the free surface, which — under the assumption of periodicityin horizontal (propagation) direction — is transformed into a closed contour aroundthe origin in the complex plane. Applying Green’s theorem to the transformed Laplace

2Other solution techniques — applied to problems involving water waves only — are, for instance,spectral methods; see Fenton and Rienecker (1982), and Dommermuth and Yue (1987b). An excellentwork on the use of spectral methods in fluid dynamics is due to Canuto, Hussaini, Quarteroni, and Zang(1988); see also Hussaini and Zang (1987).

4.2. REVIEW OF METHODS FOR NONLINEAR SHIP MOTIONS 51

equation, they arrive at Fredholm integral equations of the first kind. Despite the veryaccurate high order approximation, smoothing techniques are needed to suppress thedevelopment of non-physical ‘wiggle’ instabilities.

Another successful method was initiated by Vinje and Brevig (1981a). This method— also for two-dimensional problems — is based on Cauchy’s integral theorem3 in thecomplex plane. With z = x + iy, where y is taken vertically upwards, the complexpotential is put as β (z; t) = φ (z; t) + iψ (z; t), where φ is the (real) potential and theimaginary part ψ is the stream function. Again, horizontal periodicity is assumed, butnow Laplace’s equation is solved directly in the z-plane, allowing the extension to non-periodic problems. Expressing the governing equations in terms of complex variables,the following two identities are used:

α0ψ (z0) + Re

∫

C

− β (z)z − z0

dz

= 0 , (4.1)

when φ is given in z0, and

α0φ (z0)− Im

∫

C

− β (z)z − z0

dz

= 0 , (4.2)

when ψ is given in z0. In (4.1-4.2) the interior angle at z0 is denoted by α0, which equals πif z0 is on a smooth part of the fluid contour C. In this way Fredholm integral equationsof the second kind are obtained in all cases. The resulting time domain algorithm isvery stable4, and smoothing is needed only for very large and steep waves. The sametechnique has also been applied to the nonlinear wave-body problem in two dimensions;in their ship motion computations, Vinje and Brevig (1981bc) encountered numericalinstabilities originating from the point where the body intersects the free surface. Anothermajor drawback of their method is that extension to three-dimensional problems is notpossible.For a more complete literature survey on integral equation methods for transient non-linear water-wave problems without floating bodies we refer to Romate’s doctoral thesis(1989). The number of papers describing numerical results on nonlinear wave-body prob-lems is still comparatively small. Using extensions of the integral equation techniquesdescribed above, sometimes promising results have been obtained in two and three di-mensions; a number of publications has been collected in table 4.1.

3Cauchy, Augustin-Louis (1789-1857), French mathematician, who made important contributions tothe theory of complex functions.

4Lin, Newman, and Yue (1984) use this method in their study of the singular behaviour of the freesurface near the intersection with an impulsively started vertical wavemaker; they suggest a specialtreatment of the intersection point. This problem is discussed in detail in chapter 5.


author(s) / year method dim. description time-stepFaltinsen (1977) BEM-R 2D Lagrangian RK-4Haussling/Coleman (1977/1979) FDM 2D Lagrangian mod.EulerChan/Chan (1980) FDM 3D Eulerian implicitVinje/Brevig (1981bc) BEM-C 2D Lagrangian HammingIsaacson (1982) BEM-R 3D Eulerian AB-2Vinje/Maogang/Brevig (1982) BEM-C 2D Lagrangian HammingGreenhow/Lin (1985) BEM-C 2D Lagrangian HammingTelste (1985) FDM 2D Lagrangian mod.EulerYim (1985) BEM-C 2D Lagrangian HammingGreenhow (1987) BEM-C 2D Lagrangian HammingDommermuth/Yue (1987a) BEM-R 2D mixed E-L RK-4Wang/Spaulding (1988) FDM 2D Lagrangian mod.EulerYeung/Wu (1989) FDM 2D Lagrangian FTSCointe et al (1990) BEM-R 3D mixed E-L RK-4Kang/Gong (1990) BEM-R 3D mixed E-L RK-4Tanizawa/Sawada (1990) BEM-R 2D mixed E-L AB-1Ehlers (1991) BEM-C 2D Lagrangian ABM-4Saubestre (1991) BEM-R 2D mixed E-L RK-4Yeung/Ananthakrishnan (1992) FDM 2D mixed E-L FTSBEM = boundary element method

-R = in real domain (Green’s second identity)-C = in complex domain (Cauchy’s integral)

FDM = finite difference methodRK = Runge-KuttaFTS = fractional time-steppingAB = Adams-BashforthABM = Adams-Bashforth-MoultonAn appended number denotes the order of the time stepping method.

Table 4.1: Numerical methods for nonlinear ship motion simulations.

4.3 Basic algorithm for nonlinear water waves

In this section we shall briefly discuss the basic features of our numerical algorithmfor nonlinear gravity wave simulations. For the record, it is stressed that the presentmethod for two-dimensional simulations was derived from Romate’s (1989) higher orderpanel method for three-dimensional nonlinear gravity waves; once more, we refer to hisdoctoral thesis for a detailed description of the original algorithm. Recently, Zandbergen,Broeze, and van Daalen (1992) reported on a number of improvements on his method, andpresented very promising results on various types of free surface waves, such as solitarywaves, highly nonlinear waves, and plunging breakers. An account of the progress madein three-dimensional wave simulations will be given by Broeze (1993).

It has been demonstrated in chapter 2 that — under the assumptions of an irrotationalflow of an ideal fluid — the complete set of governing equations for nonlinear free surface

4.3. BASIC ALGORITHM FOR NONLINEAR WATER WAVES 53

flow under the action of gravity is given by:

∇2φ = 0 in the fluid domain , (4.3)Dφ

Dt=

12

(∇φ · ∇φ)− gz on the free surface , (4.4)

D~xF

Dt= ∇φ on the free surface , (4.5)

∂φ

∂n= 0 on the bottom , (4.6)

i.e. Laplace’s equation throughout the fluid, the dynamic and kinematic conditions onthe free surface, and the zero-flux condition on the bottom.

The above set of equations constitutes an elliptic spatial problem, characterizedby (4.3), with time dependent boundary conditions (4.4-4.5). In most numerical pro-cedures, the water-wave problem is treated as such; the elliptic boundary value problemis solved at a certain time level, and then the boundary conditions are updated to thenew time level, and so forth. This so-called step-by-step method is very popular for itssimplicity; it will be used here to develop a numerical algorithm for water wave simula-tions.

In the first place we have to solve the spatial problem, characterized by Laplace’sequation (4.3) for the velocity potential. With Green’s second identity this field equationis transformed into a Fredholm equation of the second kind, see chapter 3:

12φ (~x) =

∫

∂Ω

∫−

[∂φ

∂nξ

(~ξ)

G(~ξ; ~x

)− φ

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ (4.7)

where ~x is the boundary point considered, and integration is over the fluid domainboundary ∂Ω. This boundary is divided into a number of smooth subsurfaces, and eachof them is projected onto a grid in the computational domain with cubic splines. Fromthese spline data, each subsurface is approximated by curved panels. For each panel onecollocation point is determined from its central counterpart in the computational domain.The boundary integral equation (abbreviated: BIE) is discretized with a linear sourcedistribution and a quadratic dipole distribution. In compact notation, the discretizedBIE, applied in collocation point i, reads

12φi =

N∑

j=1

[Cij

s φjn + Cij

d φj]

(4.8)

where Cijs and Cij

d are the source and dipole coefficients respectively, and summationis over all N collocation points. Substitution of φ for Dirichlet boundaries and φn forNeumann boundaries yields a system of linear equations, which is solved by Gaussianelimination or with a conjugate gradients squared (CGS-) method. The structure of thismatrix equation is shown in Figure 4.1, where for convenience all Neumann boundarieshave been gathered (the same holds for all Dirichlet boundaries). The right-hand sidevector ~β contains contributions from the (substituted) known variables. The solutionof this matrix equation then provides φn for Dirichlet boundaries and φ for Neumannboundaries.Next we have to solve the time dependent part of the problem, especially for the evolutionof the free surface. The new positions of the collocation points on moving boundaries


Cijs

Cijd −

12δij

Dirichlet Neumann

φjn

φj

= βi

Figure 4.1: Structure of matrix equation.

are determined by integrating the kinematic boundary conditions in time. The sameprocedure is followed for φ (or φn) in all collocation points.

Suppose, for instance, that at a certain time level tn the following set of ‘free surfacevariables’ is known:

• the free surface shape, i.e. the position of all free surface collocation points,

• the free surface potential, i.e. the value of φ in these points, and

• the free surface normal velocity, i.e. the normal derivative of φ in these points.

The first step is the determination of the tangential velocities using a finite differencescheme. Next, the free surface velocity ∇φ is computed from the normal and tangentialvelocities. Then, the positions of all collocation points and the corresponding free surfacepotential at the next time level tn+1 = tn + ∆t are obtained by time integration of thefree surface conditions (4.4-4.5):

φ (tn+1) = φ (tn) + ∆tDφ

Dt(tn) +O (

∆t2)

= φ (tn) + ∆t

12

(∇φ · ∇φ)− gz

(tn) +O (

∆t2)

, (4.9)

~xF (tn+1) = ~xF (tn) + ∆tD~xF

Dt(tn) +O (

∆t2)

= ~xF (tn) + ∆t∇φ (tn) +O (∆t2

). (4.10)

This time integration is carried out with a fourth order Runge-Kutta method, whichimplies that three intermediate time levels are used for each time step ∆t. Once thenew positions of the collocation points are known, the free surface shape is updated interms of new spline data. Finally, the solution of integral equation (4.7) provides thenew values of the normal velocity — i.e. ∂φ/∂n — in all updated free surface collocationpoints.

Summarizing, the step-by-step method for nonlinear gravity wave simulations consists ofa repetition of the following sequence of operations:

1. calculation of the source and dipole coefficients at time tn

2. substitution of φ for Dirichlet boundaries and φn for Neumann boundaries at timetn

4.4. EXTENSION TO NONLINEAR SHIP MOTIONS 55

3. solution of the BIE at time tn, giving φn for Dirichlet boundaries and φ for Neumannboundaries

4. calculation of the tangential velocities at time tn from φ using finite differences

5. calculation of the boundary velocity ∇φ at time tn from the tangential and normalvelocities

6. integration of the time dependent boundary conditions, giving the collocation pointpositions and φ for Dirichlet boundaries and φn for Neumann boundaries at timetn+1

7. spline approximation of the geometry at time tn+1

In the next section we demonstrate how this basic scheme can be extended to nonlinearinteractions of water waves with floating bodies in either forced or free motion.

4.4 Extension to nonlinear ship motions

In section 2.5 it was stated that the presence of a floating body — either partly ortotally submerged — implies that extra equations are needed for a correct mathematicaldescription of the fluid-body interaction. The problem of nonlinear ship motions atzero forward speed is usually described with two orthonormal coordinate systems. Onesystem is fixed in space and set up by the unit vectors ~e1, ~e2, ~e3, where ~e3 is verticaland pointing upwards. The origin of this system is initially located at the centre of massof the body, denoted by G. The other system is attached to the body; its origin is locatedat G and it is set up by the unit vectors ~e1

′, ~e2′, ~e3

′ along the body principal axes ofinertia. The position and orientation of the body are specified by the three-componentvectors ~xG (corresponding to surge, sway, and heave motions) and ~θG (corresponding toroll, pitch, and yaw motions) respectively. Figure 4.2 illustrates these definitions.

Figure 4.2: Body motion and geometry definitions.


The motion of the fluid at the wetted (submerged) part of the body surface, denoted byS, is determined by the Neumann boundary condition

∂φ

∂n= ∇φ · ~n = ~xG · ~n + (~r × ~n) · ~θG , (4.11)

where ~n is the unit normal vector on S, and ~r = ~xS − ~xG, with ~xS on S. In case of afloating body under forced motion, ~xG and ~θG are known, and boundary condition (4.11)completes the set of governing equations. The body position and orientation can beintegrated in time as follows:

~xG (tn+1) = ~xG (tn) + ∆t ~xG (tn) +O (∆t2

), (4.12)

~θG (tn+1) = ~θG (tn) + ∆t ~θG (tn) +O (∆t2

). (4.13)

However, if the body is floating freely, then the translational and rotational velocities ofthe body are not known a priori and have to be determined as part of the solution. Inthis case, the mathematical model is completed with (4.11) and the equations of motionfor a rigid body with, say, mass M and moment of inertia ~I about its principal axes:

M~xG =∫

S

∫p~n dS −Mg~e3 , ~I ⊗ ~θG =

∫

S

∫p (~r × ~n) dS , (4.14)

where ⊗ defines a component-wise product of two vectors; see section 2.5 for a definition.The pressure along S is obtained from the nonlinear Bernoulli equation:

p = −ρ

φt +

12

(∇φ · ∇φ) + gz

. (4.15)

Since∇φ can be computed from the solution of Laplace’s equation for φ, the calculation ofthe body accelerations reduces to the computation of φt along the wetted body surface S.An obvious approach towards the calculation of φt is to use a finite (backward) differencemethod. Due to the motion of the body, φt then has to be determined from the materialderivative of φ:

∂φ

∂t=

Dφ

Dt−∇φ · ∇φ . (4.16)

This principle can be used in combination with a predictor-corrector method, or othercomplex numerical schemes. With this choice however, it is inevitable that one makesuse of data from previous time levels, which may give rise to numerical instabilities —see, for instance, Isaacson (1982).

A less obvious way to calculate φt along S, is to use the fact that besides φ, its partialtime derivative φt satisfies Laplace’s equation as well. This enables us to derive an extraboundary integral equation for φt, similar to (4.7):

12

∂φ

∂t(~x) =

∫

∂Ω

∫−

[∂2φ

∂nξ∂t

(~ξ)

G(~ξ; ~x

)− ∂φ

∂t

(~ξ) ∂G

∂nξ

(~ξ; ~x

)]dSξ (4.17)

The discretized version of this second BIE, applied in collocation point i, reads:

12φi

t =N∑

j=1

[Cij

s φjtn + Cij

d φjt

](4.18)


Analogous to the solution of the first BIE for φ, substitution of φt for Dirichlet boundariesand φtn for Neumann boundaries into (4.18) yields a system of N linear equations.However, since the body is floating freely, both φt and φtn are unknown along S. Hence,a complementary set of equations is needed to ensure that the number of equations equalsthe number of unknowns.

Up to now, the equations of motion (4.14) have not been used; in Appendix B we showthat these equations can be transformed into a boundary integral equation involving φt

and φtn on S only:

φtn (~x) +∫

S

∫K

(~x, ~ξ

)φt

(~ξ)

dSξ = γ (~x) (4.19)

After discretization, a set of linear equations connecting φt and φtn on S is obtained:

φitn +

NS∑

j=1

Cijk φj

t = γi (4.20)

where collocation point i is on S and summation is over all NS collocation points onS. We shall refer to this system of NS equations as the discretized equations of motionhenceforth.

Cijs

Cijd −

12δij

Cijs

0 Cijk

δij

Dirichlet Neumann Extra BIE

Floating Body

φjtn

φjt

φjtn

=

γi

βi

Figure 4.3: Structure of extra matrix equation.

The discretized extra BIE (4.18) and the discretized equations of motion (4.20) togetherform a system of N +NS linear equations involving φt and φtn on all boundaries, includ-ing the free surface. Substitution of φt for Dirichlet boundaries and φtn for Neumannboundaries (excepting S) yields a linear system, where the number of equations equalsthe number of unknowns.

The structure of this matrix equation is shown in Figure 4.3. Note that this matrixsystem has NS extra equations compared to the standard matrix equation — see Fig-ures 4.1 and 4.3 — due to the double number (i.e. 2NS) of unknowns on S.

The extra influence coefficients Cijk have some favourable properties; first of all, they


depend upon the geometry of the body only. More important however, is that the coef-ficient expressions are regular and symmetric, making their calculation straightforward(see Appendix B). If the collocation points on S have fixed positions, the coefficients Cij

k

have to be computed only once; in general, a redistribution of these collocation pointswill be necessary from time to time, thus requiring an update of these coefficients.

The extra components γi in the right-hand side vector are computed from knownbody variables, such as the potential and its spatial derivatives up to second order onS (see Appendix B). Since the calculation of the source coefficients Cij

s and the dipolecoefficients Cij

d — being the most expensive part of the computations — has alreadybeen carried out for the first boundary integral equation (4.7), we arrive at the conclu-sion that the extra computational effort is relatively small; it is mainly caused by thesolution of the extra matrix equation. Thus a direct, accurate, and efficient method forthe calculation of φt on S is developed.

Once φt is known on S, the hydrodynamic forces can be computed directly from pres-sure integrations. The resulting accelerations can then be used to determine the bodyvelocities at the next time level:

~xG (tn+1) = ~xG (tn) + ∆t ~xG (tn) +O (∆t2

), (4.21)

~θG (tn+1) = ~θG (tn) + ∆t ~θG (tn) +O (∆t2

). (4.22)

Naturally, the aforementioned Runge-Kutta time stepping scheme is applied here. Fi-nally, the Neumann ‘contact’ boundary condition (4.11) at time tn+1 is determined fromthese updated body velocities.

Summarizing, the proposed step-by-step method for nonlinear wave-body simulationsconsists of a repetition of the following sequence of operations:

1. calculation of the source and dipole coefficients at time tn

2. substitution of φ for Dirichlet boundaries and φn for Neumann boundaries at timetn

3. solution of the first BIE at time tn, giving φn for Dirichlet boundaries and φ for allNeumann boundaries, i.e. including the body surface S

4. calculation of the tangential velocities at time tn from φ

5. calculation of the boundary velocity ∇φ at time tn from the tangential and normalvelocities

6. calculation of the extra influence coefficients at time tn

7. calculation of the second order tangential derivatives of φ and the first order tan-gential derivatives of φn on S at time tn

8. substitution of φt for Dirichlet boundaries and φtn for Neumann boundaries, ex-cepting S, at time tn

9. solution of the second BIE at time tn, giving φtn for Dirichlet boundaries and φt

for all Neumann boundaries, i.e. including S


10. calculation of the body accelerations from the hydrodynamic equations of motionand pressure integrations over S

11. integration of the time dependent boundary conditions and the body position andorientation, giving the collocation point positions and φ for Dirichlet boundariesand φn for all Neumann boundaries at time tn+1

12. integration of the body velocities, giving the Neumann condition on S at time tn+1

13. spline approximation of the geometry at time tn+1

The approach outlined above resembles the somewhat earlier work of Cointe (1989), whodeveloped a panel method for two-dimensional linear wave-body interactions. However,his computations for an extincting circular cylinder show poor agreement with the ana-lytical linear predictions. See also chapter 7, where our results concerning freely floatingbodies are presented.

Recently — in the course of writing this thesis and after completion of the calculationsdescribed herein — it has come to the author’s attention that Tanizawa and Sawada(1990) have successfully used the above technique in their boundary element methodfor two-dimensional nonlinear wave-body interactions. Unfortunately, their work is lessaccessible since it was written in Japanese.

We close this chapter with a few remarks on the numerical algorithm:

• The essential feature of the algorithm proposed here is that — apart from numericalspatial discretization errors — the dynamic equilibrium of the fluid and the bodyis preserved at all times. By using an extra BIE to compute φt on the wettedbody surface, unnecessary time discretization errors are avoided, thus preservingthe stability and accuracy of the algorithm.

• Another important feature of this algorithm is that it is not limited to two dimen-sions; the derivations in Appendices A and B contain no two-dimensional elementsor assumptions.

• The above approach permits more than one freely floating body; a second bodymerely introduces an extra set of equations of motion which can be transformed intoa boundary integral equation over the wetted body surface. After discretization,the two systems of linear equations (one for body I, and one for body II) are addedto the discretized extra BIE for φt.

• Note that in case of a body under forced motion, the hydrodynamic forces acting onthe body can be computed likewise; φtn on S can be computed from the prescribedbody velocities and accelerations, and then φt is solved from the extra BIE. In thiscase, the discretized equations of motion should not be added to the extra BIE. The(minor) advantage of this approach is that the hydrodynamic forces and momentscan be computed during the computations, and postprocessing is not necessaryto compute φt. Kang and Gong (1990) employ this technique for this particularpurpose.

• Restoring forces and moments acting on the body may also be incorporated in theequations of motion (4.14), thus allowing for the numerical simulation of waterwaves interacting with moored objects.


4.5 Bibliography

Broeze, J. 1993. Analysis of an Effective Method for 3-D Flow Computations with aNonlinear Free Surface. Ph.D. thesis, University of Twente. Enschede, The Netherlands.

Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A. 1988. SpectralMethods in Fluid Dynamics. Springer-Verlag.

Chan, R.K.-C., and Chan, F.W.-K. 1980. Numerical solution of transient andsteady free-surface flows about a ship of general hull shape. Proceedings of the Thir-teenth Symposium on Naval Hydrodynamics, Tokyo, Japan.

Cointe, R. 1989. Quelques Aspects de la Simulation Numerique d’un Canal a Houle.Ph.D. thesis, Ecole des Ponts et Chaussees. Paris, France. (in French)

Cointe, R., Geyer, P., King, B., Molin, B., and Tramoni, M. 1990. Nonlinearand linear motions of a rectangular barge in a perfect fluid. Proceedings of the Eigh-teenth Symposium on Naval Hydrodynamics, Ann Arbor, Michigan.

Dommermuth, D.G., and Yue, D.K.P. 1987a. Numerical simulations of nonlinearaxisymmetric flows with a free surface. Journal of Fluid Mechanics 178:195-219.

Dommermuth, D.G., and Yue, D.K.P. 1987b. A high-order spectral method for thestudy of nonlinear gravity waves. Journal of Fluid Mechanics 184:267-288.

Ehlers, J. 1991. Numerische Simulation in Schwerewellen schwimmender Korper miteinem Randelement-Zeitschrittverfahren. Master’s thesis, Technische Universitat Hamburg-Harburg, Arbeitsbereich Meerestechnik II - Strukturmechanik. Hamburg, Germany. (inGerman)

Faltinsen, O.M. 1977. Numerical solutions of transient nonlinear free surface motionoutside or inside moving bodies. Proceedings of the Second International Conference onNumerical Ship Hydrodynamics, Berkeley, California.

Fenton, J.D., and Rienecker, M.M. 1982. A Fourier method for solving nonlinearwater-wave problems; application to solitary-wave interactions. Journal of Fluid Mechan-ics 118:411-443.

Greenhow, M. 1987. Wedge entry into initially calm water. Applied Ocean Research9(4):214-223.

Greenhow, M., and Lin, W.-M. 1985. Numerical simulation of nonlinear free sur-face flows generated by wedge entry and wavemaker motion. Proceedings of the FourthInternational Conference on Numerical Ship Hydrodynamics, Washington, DC.

Haussling, H.J., and Coleman, R.M. 1977. Finite-difference computations usingboundary-fitted coordinates for free-surface potential flows generated by submerged bodies.Proceedings of the Second International Conference on Numerical Ship Hydrodynamics,Berkeley, California.

Haussling, H.J., and Coleman, R.M. 1979. Nonlinear water waves generated by anaccelerated circular cylinder. Journal of Fluid Mechanics 92(4):757-781.

Hussaini, M.Y., and Zang, T.A. 1987. Spectral methods in fluid dynamics. AnnualReview of Fluid Mechanics 19:339-367.

Isaacson, M. de St.Q. 1982. Nonlinear-wave effects on fixed and floating bodies. Jour-nal of Fluid Mechanics 120:267-281.


Kang, C.-G., and Gong, I.-Y. 1990. A numerical solution method for three-dimensional nonlinear free surface problems. Proceedings of the Eighteenth Symposiumon Naval Hydrodynamics, Ann Arbor, Michigan.

Lin, W.-M., Newman, J.N., and Yue, D.K.P. 1984. Nonlinear forced motionsof floating bodies. Proceedings of the Fifteenth Symposium on Naval Hydrodynamics,Hamburg, Germany.

Longuet-Higgins, M.S., and Cokelet, E.D. 1976. The deformation of steep sur-face waves on water. I. A numerical method of computation. Proceedings of the RoyalSociety of London, Series A, 350:1-26.

Romate, J.E. 1988. Local error analysis in 3-D panel methods. Journal of EngineeringMathematics 22:123-142.


Romate, J.E. 1990. Local error analysis of three-dimensional panel methods in termsof curvilinear surface coordinates. SIAM Journal on Numerical Analysis 27(2):529-542.

Saubestre, V. 1991. Numerical simulation of transient nonlinear free-surface flowswith body interaction. OMAE - Volume I-A, Offshore Technology ASME.

Tanizawa, K., and Sawada, H. 1990. A numerical method for nonlinear simulationof 2-D body motions in waves by means of B.E.M. Journal of the Society of Naval Ar-chitects of Japan 168:221-226.

Telste, J.G. 1985. Calculation of fluid motion resulting from large-amplitude forcedheave motion of a two-dimensional cylinder in a free surface. Proceedings of the FourthInternational Conference on Numerical Ship Hydrodynamics, Washington, DC.

Vinje, T., and Brevig, P. 1981a. Breaking waves on finite water depths. A numericalstudy. Norwegian Hydrodynamic Laboratories, Report R-111.81, Trondheim, Norway.

Vinje, T., and Brevig, P. 1981b. Nonlinear, two-dimensional ship motions. Norwe-gian Hydrodynamic Laboratories, Report R-112.81, Trondheim, Norway.

Vinje, T., and Brevig, P. 1981c. Nonlinear ship motions. Proceedings of the ThirdInternational Conference on Numerical Ship Hydrodynamics, Paris, France.

Vinje, T., Maogang, X., and Brevig, P. 1982. A numerical approach to nonlinearship motion. Proceedings of the Fourteenth Symposium on Naval Hydrodynamics, AnnArbor, Michigan.

Wang, X.M., and Spaulding, M.L. 1988. A two-dimensional potential flow modelof the wave field generated by a semisubmerged body in heaving motion. Journal of ShipResearch 32(2):83-91.

Yeung, R.W., and Ananthakrishnan, P. 1992. Oscillation of a floating body in aviscous fluid. Journal of Engineering Mathematics 26:211-230.

Yeung, R.W., and Wu, C.-F. 1989. Nonlinear wave-body motion in a closed domain.Computers and Fluids 17(2):351-370.

Yim, B. 1985. Numerical solution for two-dimensional wedge slamming with a nonlinearfree-surface condition. Proceedings of the Fourth International Conference on NumericalShip Hydrodynamics, Washington, DC.


Zandbergen, P.J., Broeze, J., and van Daalen, E.F.G. 1992. A panel methodfor the simulation of nonlinear gravity waves and ship motions. In Advances in BoundaryElement Techniques. Springer-Verlag.


But straightway the morning star rose above the topmost peaks and thebreeze swept down; and quickly did Tiphys urge them to go aboard and availthemselves of the wind. And they embarked eagerly forthwith; and they drewup the ship’s anchors and hauled the ropes astern. And the sails were belliedout by the wind, and far from the coast were they joyfully borne past thePosideian headland. But at the hour when gladsome dawn shines from heaven,rising from the east, and the paths stand out clearly, and the dewy plainsshine with a bright gleam, then at length they were aware that unwittinglythey had abandoned those men.5 And a fierce quarrel fell upon them, andviolent tumult, for that they had sailed and left behind the bravest of theircomrades . . .

But to them appeared Glaucus from the depths of the sea, the wise inter-preter of divine Nereus, and raising aloft his shaggy head and chest from hiswaist below, with sturdy hand he seized the ship’s keel, and then cried to theeager crew:

“Why against the counsel of mighty Zeus do ye purpose to lead bold Her-acles to the city of Aeetes? At Argos it is his fate to labour for insolentEurystheus and to accomplish full twelve toils and dwell with the immortals,if so be that he bring to fulfilment a few more yet; wherefore let there be novain regret for him. Likewise it is destined for Polyphemus to found a gloriouscity at the mouth of Cius among the Mysians and to fill up the measure ofhis fate in the vast land of the Chalybes. But a goddess-nymph through lovehas made Hylas her husband, on whose account those two wandered and wereleft behind.”

He spake, and with a plunge wrapped him about with the restless wave;and round him the dark water foamed in seething eddies and dashed againstthe hollow ship as it moved through the sea . . .

Argonautica, Book I, Verses 1273-1286 and 1310-1328.

5i.e. Hylas, Heracles, and Polyphemus


Part II

Numerical Results

65

Chapter 5

Impulsive Wavemaker Motion

Here were the oxstalls and farm of Amycus, the haughty king of the Be-brycians, whom once a nymph, Bithynian Melie, united to Poseidon Geneth-lius, bare — the most arrogant of men; for even for strangers he laid down aninsulting ordinance, that none should depart till they had made trial of himin boxing; and he had slain many of the neighbours. And at that time too hewent down to the ship and in his insolence scorned to ask them the occasionof their voyage, and who they were, but at once spake out among them all:

“Listen, ye wanderers by sea, to what it befits you to know. It is the rulethat no stranger who comes to the Bebrycians should depart till he has raisedhis hands in battle against mine. Wherefore select your bravest warrior fromthe host and set him here on the spot to contend with me in boxing. But ifye pay no heed and trample my decrees under foot, assuredly to your sorrowwill stern necessity come upon you.”

Thus he spake in his pride, but fierce anger seized them when they heardit, and the challenge smote Polydeuces most of all. And quickly he stood forthhis comrades’ champion, and cried:

“Hold now, and display not to us thy brutal violence, whoever thou art; forwe will obey thy rules, as thou sayest. Willingly now do I myself undertaketo meet thee.”

Argonautica, Book II, Verses 1-24.

67

68 CHAPTER 5. IMPULSIVE WAVEMAKER MOTION

5.1 Introduction

In the study of water waves associated with surface piercing bodies, a number of linearand nonlinear analyses has indicated that the potential solution may become singular atthe point where the floating body intersects the free surface. Physically, this singularityis plausible in view of the splashing which occurs when, for instance, the bow of a shipslams the water plane. From a mathematical point of view, this singular behaviour isdue to a confluence of the boundary conditions on the free surface and the floating body.

This singular behaviour is of great interest in the numerical simulation of ship motionsin water waves. In a discrete sense the boundary conditions must be satisfied on exactboundaries, but the position of both the free surface and the floating body are not knownin advance; this explains the difficulty in determining the position of the intersection pointand the local fluid flow. Without detailed knowledge of the singular behaviour of thedisplacement of, and the velocity potential at the free surface, it is hardly possible todevelop a stable and accurate numerical algorithm.

To reduce the complexity of this hydrodynamic problem, let us concentrate only onthe local region near the intersection point. As a simplified model, the following situationis considered; in a semi-infinite wave tank, a piston-type wavemaker starts to move fromrest at time t = 0 at a specified horizontal velocity, and the subsequent motion of the freesurface is sought. Assumptions on the wavemaker motion and the fluid flow are madelater on.

The major difficulty in the nonlinear analysis of this ‘wavemaker problem’ is that boththe kinematic condition and the dynamic condition must be satisfied at the free surface.In the Eulerian description of the wave motion, the free surface position is unknown, andusually series expansions are used to overcome this difficulty. For instance, Peregrine(1972) used perturbation series in small time t; employing a moving coordinate systemattached to the wavemaker, he revealed a singularity in the free surface elevation atthe intersection point, which behaves like η ∼ t log x. In a similar way, Chwang (1983)developed a nonlinear theory to determine the hydrodynamic pressure on an acceleratingvertical plate.1 Using a fixed coordinate system, he suggested that the singularity liesoutside the fluid domain and, hence, is of no physical interest. However, the singularbehaviour in the small time solution was confirmed by Lin (1984), who employed aLagrangian description for the wavemaker problem.

The absence of gravity and the singular behaviour of the small time solutions men-tioned above, induced Roberts (1987) to carry out a rigorous analysis of free surfacewaves generated by a wavemaker, using both small time and small Froude number ex-pansions. For power-law displacement of a vertical plate, Roberts showed that the flowdevelops a disperging wavetrain which emanates instantaneously from the intersectionpoint. He also found that the solution varies considerably close to the intersection pointand presented a self-similar formulation to describe this ‘wiggling’ behaviour.

To avoid the artificial singularity at the intersection point introduced by small expan-sions, Joo, Schultz, and Messiter (1990) developed a Fourier-integral method for smallFroude number, which also allows the study of capillary effects. In the absence of surfacetension, their method reveals small-scale wiggles near the wavemaker, in agreement withRoberts’ local solution for small time. In addition, it was found that the contact angle

1Similar studies have also been carried out by Chwang and Wang (1984) for an accelerating rectangularor circular container, and by Wang and Chwang (1989) for an accelerating vertical surface-piercingcylinder.

5.2. NONLINEAR ANALYSIS 69

has a jump at t = 0; surface tension suppresses these wiggles and maintains the contactangle at its initial value. Systematic studies of capillary effects along similar lines havebeen carried out by Hocking and Mahdmina (1991) and Miles (1991).

In the past decade, the nonlinear wavemaker problem2 has also been addressed numer-ically. In particular, Lin’s (1984) doctoral thesis was devoted to this subject; earlier,Vinje and Brevig (1981) reported numerical instabilities originating from the intersec-tion of the free surface and a floating body in their ship motion computations. Followingtheir approach, Lin developed a boundary element method — based on Cauchy’s integraltheorem, see section 4.2 — for two-dimensional free surface flow simulations. To Lin’sopinion, the numerical difficulties could be solved by specifying both the velocity poten-tial and the stream function at the intersection point. This approach was applied to theproblem of an impulsively started wavemaker; the numerical results show that accurateglobal calculations of the potential along the wavemaker are obtained even when rela-tively coarse grids are used. The local behaviour of the free surface is properly describedand the singularity is accommodated.

In view of the extension of Romate’s panel method for water waves to nonlinear shipmotions, it was felt necessary to test our computer code for the impulsive wavemakerproblem. The present method differs from Lin’s approach in that our collocation pointscoincide with the panel midpoints. Hence, in our method there is no direct confluenceof boundary conditions in the intersection point, and it will appear that no special localtreatment is needed. In this study, attention is focused on the ability of the code topredict the initial behaviour of the potential along the wavemaker and the free surfaceelevation for small time, since these variables determine the hydrodynamic forces exertedby the fluid. We do not aim to reproduce the above-mentioned small-scale wiggles orcapillary effects.

The present chapter is an elaborated study, based on a recent paper written by vanDaalen and Huijsmans (1991). In section 5.2 we derive analytical solutions for the initialbehaviour of the potential and the tangential velocities on the wavemaker and the bot-tom, using small time expansions; series expressions for these solutions are transformedinto compact integral expressions which are more suitable for numerical evaluation. Ananalytical expression for the small time free surface elevation — which confirms the log-arithmic singularity — is also derived. These solutions allow a direct verification of thenumerical results obtained with the TIPHYS-code for two-dimensional water-wave simu-lations; this part is presented in section 5.3. Concluding remarks are given in section 5.4.

5.2 Nonlinear analysis

Consider the configuration depicted in Figure 5.1, where the two-dimensional fluid do-main extends to infinity to the right and is bounded in vertical direction by a horizontalbottom and a free surface; at the left end of the wave tank a piston-type wavemaker issituated. The origin O of the coordinate system x, z is initially located at the intersec-

2Some related problems are, for example: a two-dimensional cylinder in a current, see Grosenbaughand Yeung (1988); the oblique water entry of a two-dimensional profile, see Korobkin (1988); the verticalwater entry of a horizontal circular cylinder — see Greenhow (1988) — or a spherical projectile — seeMiloh (1981, 1991a). The problem of an oblique water entry of a rigid sphere has also been discussed byMiloh (1991b). Finally, we refer to the review article on water impact problems, written by Korobkinand Pukhnachov (1988).


tion of the wavemaker and the free surface. At time t = 0, the wavemaker impulsivelystarts to move towards the fluid with a constant velocity U .

Figure 5.1: Impulsive wavemaker problem: geometry definition.

5.2.1 Governing equations and small time expansions

Firstly, the governing equations for this particular problem are derived; under the as-sumptions of an inviscid, incompressible fluid and an irrotational flow, the velocity po-tential φ satisfies

∇2φ = 0 for x > Ut , −h < z < η (x; t) (5.1)

i.e. Laplace’s equation throughout the transient fluid domain.Due to the motion of the wavemaker, the free surface is changed from its undisturbed

level z = 0 to a new position z = η (x; t), where the nonlinear free surface conditionsread

φt +12

(∇φ · ∇φ) + gz = 0

ηt + ηxφx − φz = 0

at z = η (x; t) (5.2)

where a suffix denotes partial differentiation, i.e. ηt = ∂η/∂t et cetera.At the bottom the vertical velocity must vanish; this is expressed by

φz = 0 at z = −h (5.3)

The boundary condition on the moving wavemaker reads

φx = U at x = Ut (5.4)

which completes the set of governing equations for the impulsive wavemaker problem; itis assumed that φ and η vanish for x →∞ such that a well-posed problem is obtained.


In order to obtain small time solutions for the velocity potential and the vertical deviationof the free surface, φ and η are written as power series in t:

φ (x, z; t) =∞∑

m=0

φm (x, z) tm (5.5)

η (x; t) =∞∑

m=0

ηm (x) tm (5.6)

The governing equations for the φm’s and ηm’s are obtained by substitution of (5.5-5.6)into the field equation (5.1) and the boundary conditions (5.2-5.4).

Obviously, each φm is a harmonic function, i.e. it satisfies Laplace’s equation (5.1):

∇2φm = 0 , m = 0, 1, 2, . . . (5.7)

Similarly, each φm meets the zero-flux bottom condition (5.3), that is

∂φm

∂z= 0 at z = −h , m = 0, 1, 2, . . . (5.8)

Substitution of (5.5) into the wavemaker condition (5.4) and developing Taylor seriesround x = 0 up to first order gives

U =∂φ0

∂xat x = 0 , (5.9)

−U∂2φ0

∂x2=

∂φ1

∂xat x = 0 . (5.10)

In a similar way, substitution of (5.5-5.6) into the nonlinear free surface conditions (5.2)and developing Taylor series round z = 0 yields

0 = φ0 at z = 0 , (5.11)

−12

(∂φ0

∂z

)2

= φ1 at z = 0 , (5.12)

and

0 = η0 at z = 0 , (5.13)∂φ0

∂z= η1 at z = 0 , (5.14)

12

∂φ1

∂z+

12η1

∂2φ0

∂z2= η2 at z = 0 . (5.15)

In the next paragraph we deduce leading order solutions for φ and η, satisfying (5.7-5.15).

5.2.2 Leading order solutions

The leading (zeroth) order potential φ0, satisfying (5.7-5.9) and (5.11) is obtained in astraightforward manner by the Fourier series method:

φ0 (x, z) =2U

h

∞∑n=1

(−1)n

k2n

e−knx cos kn (z + h) (5.16)


where the coefficients kn are defined as

kn =(2n− 1)π

2h, n = 1, 2, 3, . . . (5.17)

The next (first order) term in the series expansion for the potential, φ1, satisfies (5.7-5.8),(5.10), and (5.12). Using the zeroth order result (5.16), and employing the method ofFourier cosine transformation, we obtain

φ1 (x, z) =2U2

h

∞∑n=1

(−1)n

kne−knx cos kn (z + h)− 4U2

π

∞∑n=2

AnBn (x, z)

(5.18)

where the coefficients An and the functions Bn (x, z) are defined as

An = h−2n−1∑

l=1

(klkn−l)−1

, n = 2, 3, 4, . . . (5.19)

Bn (x, z) =

∞∫

0

λn

λ2n + κ2

cosh κ (z + h)cosh κh

cos κx dκ , n = 2, 3, 4, . . . (5.20)

with

λn =(n− 1)π

h, n = 2, 3, 4, . . . (5.21)

The zeroth order term in the free surface elevation is, naturally, given by

η0 (x) = 0 (5.22)

The first non-trivial term is obtained from substitution of (5.16) into (5.14):

η1 (x) =2U

h

∞∑n=1

e−knx

kn. (5.23)

Differentiation with respect to x gives

dη1

dx(x) = −2U

h

∞∑n=1

e−knx = −U

hsinh−1

(πx

2h

), (5.24)

where we used the identity — see Gradshteyn and Ryzhik (1980) —∞∑

n=1

e−(2n−1)y =12

sinh−1 y for y > 0 . (5.25)

Integration with respect to x yields the well-known result

η1 (x) = −2U

πlog

[tanh

(πx

4h

)](5.26)


The second order term in the series expansion for the free surface elevation is obtainedby substitution of (5.16), (5.18), and (5.26) into (5.15):

η2 (x) =U2

h

∞∑n=1

e−knx − 2U2

π

∞∑n=2

An

∞∫

0

λnκ

λ2n + κ2

tanh κh cos κx dκ . (5.27)

When we use identity (5.25) and the following definition:

Cn (x) =∂Bn

∂z(x, 0) =

∞∫

0

λnκ

λ2n + κ2

tanh κh cosκx dκ , n = 2, 3, 4, . . .

(5.28)

we arrive at a more compact expression for η2, namely

η2 (x) =U2

2hsinh−1

(πx

2h

)− 2U2

π

∞∑n=2

AnCn (x) (5.29)

The above set of leading order solutions for φ and η will be used in the next paragraph toderive expressions for the initial and small time behaviour of the flow along the wavemakerand the bottom, and the free surface elevation.

5.2.3 Initial and small time behaviour

From (5.16) the potential on the wavemaker at t = 0 is found to be

φ0 (0, z) =2U

h

∞∑n=1

(−1)n

k2n

cos kn (z + h) . (5.30)

Differentiation with respect to z gives the initial vertical velocity along the wavemaker:

∂φ0

∂z(0, z) = −2U

h

∞∑n=1

(−1)n

knsin kn (z + h) . (5.31)

Using the identity — see Gradshteyn and Ryzhik (1980) —

∞∑n=1

(−1)n−1 sin (2n− 1) y

2n− 1=

12

log[tan

(π

4+

y

2

)]for |y| < π

2, (5.32)

gives an expression which is more suitable for numerical evaluation:

∂φ0

∂z(0, z) =

2U

πlog

[tan

π

4

(2 +

z

h

)](5.33)

It is easy to see that the initial vertical velocity at the intersection of the wavemaker andthe bottom vanishes:

∂φ0

∂z(0,−h) = 0 (5.34)


where the initial vertical velocity at the intersection of the wavemaker and the free surfacetends to infinity:

∂φ0

∂z(0, z) →∞ for z ↑ 0 (5.35)

An integral expression for φ0 (0, z) is obtained from integration of (5.33) with respect toz and using φ0 (0, 0) = 0:

φ0 (0, z) =2U

π

z∫

0

log[tan

π

4

(2 +

ζ

h

)]dζ . (5.36)

Substituting z = −h in (5.30), we find — by definition, see Gradshteyn and Ryzhik(1980) —

φ0 (0, 0) = −8UhG

π2(5.37)

where G ≈ 0.916 is Catalan’s constant. This leads us to an alternative integral expressionfor the initial potential distribution on the wavemaker:

φ0 (0, z) =2U

π

z∫

−h

log[tan

π

4

(2 +

ζ

h

)]dζ − 4hG

π

(5.38)

From (5.16) it follows that the analytical solution for the initial potential distribution onthe bottom z = −h is given by

φ0 (x,−h) =2U

h

∞∑n=1

(−1)n

k2n

e−knx . (5.39)

Differentiating this expression twice with respect to x, we obtain

∂2φ0

∂x2(x,−h) =

2U

h

∞∑n=1

(−1)ne−knx = −U

hcosh−1

(πx

2h

), (5.40)

where we used the identity — see Gradshteyn and Ryzhik (1980) —∞∑

n=1

(−1)n−1e−(2n−1)y =

12

cosh−1 y for y > 0 . (5.41)

Integrating (5.40) with respect to x and using

∂φ0

∂x(x,−h) → 0 for x →∞ (5.42)

gives

∂φ0

∂x(x,−h) =

4U

πarctan e−πx/2h (5.43)


Integration with respect to x, using (5.37), yields

φ0 (x,−h) =8Uh

π2

1∫

e−πx/2h

arctan ξ

ξdξ −G

(5.44)

The initial vertical velocity of the free surface is easily deduced from (5.14) and (5.26):

∂φ0

∂z(x, 0) = −2U

πlog

[tanh

(πx

4h

)](5.45)

In order to calculate the hydrodynamic force on the wavemaker at t = 0, we expand theBernoulli pressure p in the time t:

p (x, z; t) = −(

φt +12

(∇φ · ∇φ) + gz

)=

∞∑m=0

pm (x, z) tm . (5.46)

Substituting (5.5) and retaining zeroth order terms only, gives

p0 (x, z) = −(

φ1 +12

(∇φ0 · ∇φ0) + gz

)(5.47)

The first order potential solution φ1 along the wavemaker reads

φ1 (0, z) =2U2

h

∞∑n=1

(−1)n

kncos kn (z + h)− 4U2

π

∞∑n=2

AnBn (0, z) . (5.48)

Using the identity — see Gradshteyn and Ryzhik (1980) —

∞∑n=1

(−1)n−1 cos (2n− 1) y

2n− 1=

π

4for 0 < y <

π

2, (5.49)

the following expression for φ1 (0, z) is obtained:

φ1 (0, z) = −U2

1 +

4π

∞∑n=2

An

∞∫

0

λn

λ2n + κ2

cosh κ (z + h)cosh κh

dκ

(5.50)

With ∂φ0/∂x = U on the wavemaker, and expression (5.33) for ∂φ0/∂z at x = 0, theinitial pressure distribution on the wavemaker can be computed analytically (and hence,the initial force acting on the wavemaker).

Finally, from (5.22) and (5.26) it follows that to first order in time

η (x; t) = −Ut

πlog

[tanh

(πx

4h

)](5.51)


At the wavemaker we have x = Ut. Then, using tanh y = O (y), we find

limt↓0

η (x = Ut; t) = limt↓0

[−Ut

πlog

(πUt

4h

)]= 0 , (5.52)

since

limy→0

y log y = 0 . (5.53)

5.3 Numerical results

As stated in the introduction, the present numerical study is concerned with the impulsivemotion of a vertical wavemaker; at t = 0, the horizontal velocity is a step function andas a consequence the wavemaker acceleration is infinite. Analytically, a logarithmicsingularity is expected at the intersection of the free surface and the wavemaker. Froma numerical point of view this case is rather critical because the high gradients near theintersection point may ruin the stability of the computational scheme.

For the actual runs, the length of the wave tank is L = 10 m, the water depth ish = 1 m, the wavemaker velocity is U = 1 m/s, and the time t is from zero to 0.2 s.Comparative tests with a doubled tank length show that during this time interval thereis no significant reflection from the vertical wall downstream.3

The boundary integral equation is solved using a wide range of elements which are dis-tributed over the free surface, the wavemaker, the bottom, and the right lateral boundaryin accordance with the distribution shown in Figure 5.2.

s s s s s s s s s s s s s s s s s s s s s s s s s

sssssssssssssssssssssssssssssssss s

sssssss

NB

NF

NW NRNF : NW : NB : NR = 25 : 8 : 25 : 8

Figure 5.2: Element distributions for impulsive wavemaker problem.

For the wavemaker and the free surface special care is taken in the distribution of thecollocation points. For the wavemaker, the actual grid is obtained from an equidistantdistribution x = 0, zi = − (i− 1/2)∆z, with ∆z = 1/NW , through the power trans-formation z → z3/2. The initial grid on the free surface is similarly generated from auniform distribution.

The choice of the time step ∆t is rather critical; it is anticipated that shortly afterthe impulsive start-up a very thin fluid film is formed along the wavemaker face. Hence,

3In the shallow water limit the long (high-speed) wave components travel at a phase velocity cf =

(gh)1/2 ≈ 3.1m/s.

5.3. NUMERICAL RESULTS 77

∆t must be sufficiently small in order to prevent free surface collocation points frompenetrating the advancing wavemaker. In the actual computations a time step ∆t =0.0025 is used. Finally, we remark that the position of the intersection point is based oncubic spline extrapolations from the outer collocation points on the adjacent boundaries.

In the following paragraphs we discuss the computed results for

• the initial potential and pressure distributions on the wavemaker, and the verticalvelocity,

• the initial potential distribution on the bottom, and the horizontal velocity,

• the initial vertical velocity of the free surface, and

• the free surface elevation for small time.

5.3.1 Initial behaviour on the wavemaker

A plot of the velocity potential distribution on the wavemaker at t = 0 is presentedin Figure 5.3. It is observed that even with the smallest number of elements (eight)on the wavemaker, the computed potential agrees remarkably well with the analyticalresult (5.38).4

The analytical solution (5.33) suggests that the vertical fluid velocity near the inter-section point tends to infinity at t = 0, see also (5.35). This is confirmed by the computedresults for ∂φ/∂z at t = 0, shown in Figure 5.4. With each doubling of the number ofelements, the numerical approximation is improved significantly, especially in the vicinityof the intersection point. The small deviations in ∂φ/∂z that can be observed close tothe intersection point, are apparently due to the finite difference approximation, sincethe computed potential is in exact agreement with the analytical result over the wholewavemaker range, see Figure 5.3.

Since the local behaviour of the potential distribution near the intersection pointcan not be seen clearly from Figure 5.3, the computations have been repeated for largernumbers of elements, corresponding to NW = 48 and NW = 64. Figure 5.5 gives anenlarged view of the upper wavemaker region and the local distribution of the potential.For the case NW = 64, the distance of the upper collocation point to the free surfaceis 7 · 10−4; it is remarkable that the numerical solution still agrees very well with theanalytical prediction (5.38). From Figure 5.6 it can be observed that the computedvertical velocity ∂φ/∂z is also improved substantially when more elements are used.

The initial pressure distribution on the wavemaker is shown in Figure 5.7, where thedashed line represents the hydrostatic pressure. It can be seen that the initial dynamicpressure is constant over the entire wavemaker, apart from the region near the intersec-tion point. Globally, the computed pressure is in good agreement with the analyticalresult (5.47).

5.3.2 Initial behaviour on the bottom

A plot of the initial potential distribution on the bottom is presented in Figure 5.8. Notethat the results are shown for 0 ≤ x ≤ 5 only, i.e. for the left half of the wave tank.

4For completeness it is remarked that analytical solutions are represented by solid lines in Figures 5.3-5.13.


The analytical solution (5.44) indicates a regular behaviour which is confirmed by thecomputations. Excellent agreement is also observed for the horizontal velocity along thebottom, shown in Figure 5.9, where the analytical predictions from (5.43) are used forreference.

5.3.3 Initial behaviour on the free surface

The numerical results for the initial vertical velocity of the free surface are shown inFigure 5.10. Note that the results have been plotted for 0 ≤ x ≤ 2.5 only, that is for theregion close to the wavemaker. Even for the smallest number of elements (twenty-five)on the free surface, the computed vertical velocity is in excellent agreement with theanalytical prediction (5.45). This is also the case for the finer grid distributions withfifty and one-hundred free-surface elements.

In order to make sure that our method is stable and sufficiently accurate, the numberof elements is increased; Figure 5.11 shows the computed vertical velocity — in theregion 0 ≤ x ≤ 0.25 — for up to two-hundred free-surface elements. Since the analyticalleading order solutions for the vertical velocity and the free surface elevation are equal— see (5.14) — we have a firm confidence in the ability of our method to predict the freesurface elevation, at least for small time; these results are discussed next.

5.3.4 Free surface elevation for small time

Next, the transient shape of the free surface is calculated up to t = 0.2 s. Figures 5.12and 5.13 show the computed results — in the region 0 ≤ x ≤ 2 — compared to the ana-lytical predictions from (5.26). The highest symbol in each curve represents the positionof the upper collocation point. For small time, say for t < 0.1, excellent agreement isobserved; for larger time, the computed free surface shape deviates somewhat from theanalytical result (which, indeed, is valid for small time only).

Taking into account the uniform horizontal motion of the wavemaker, it is clearfrom Figure 5.13 that a very thin layer (or film) of fluid is formed along the advancingwavemaker. This results in extremely small distances between free surface nodes andwavemaker nodes, which destabilizes our algorithm; in the end, the computations breakdown.


THIS PAGE INTENTIONALLY LEFT BLANK


5.4 Concluding remarks

In this chapter we have made an attempt towards the numerical solution of the impulsivewavemaker problem. From both physical and mathematical considerations it can beexpected that the potential solution becomes singular at the intersection of the advancingwavemaker and the free surface. This singularity is confirmed in the small time analysisgiven in section 5.2, which shows that the free surface elevation behaves like η ∼ t log xclose to the wavemaker (which is located at x = Ut). The computed results supportthe analytical expressions for the initial distributions of the potential and the tangentialvelocities along the wavemaker and the bottom. Excellent agreement is also observed forthe analytical and numerical results for the initial vertical free surface velocity, and forthe free surface elevation for small time.

Contrary to Lin’s approach, no special treatment of the intersection point is neededin our method to control the local behaviour of the initial potential distribution onthe wavemaker and the free surface elevation for small time. Therefore, it seems thatLin’s suggestion to maintain a collocation point at the critical intersection point is notapplicable to all integral equation methods.

Although the method of small time expansions yields singular expressions for the freesurface elevation and the vertical fluid velocity at the intersection point, the leading ordersolutions appear to be very suitable for a direct verification of the numerically computedvalues. Even extremely close to the wavemaker, the predicted logarithmic singularity inthe elevation is confirmed by the computations; capillary effects may be found by addingsmall-scale physical factors — such as surface tension — to our numerical model. Sucheffects may be important locally, but will not affect the global (integral) behaviour of thesolution.

5.5 Bibliography

Chwang, A.T. 1983. Nonlinear hydrodynamic pressure on an accelerating plate. ThePhysics of Fluids 26(2):383-387.

Chwang, A.T., and Wang, K.-H. 1984. Nonlinear impulsive force on an acceleratingcontainer. Journal of Fluids Engineering 106:233-240.

van Daalen, E.F.G., and Huijsmans, R.H.M. 1991. On the impulsive motion of awavemaker. Sixth International Workshop on Water Waves and Floating Bodies, WoodsHole, Massachusetts.

Gradshteyn, I.S., and Ryzhik, I.M. 1980. Table of Integrals, Series, and Products.Academic Press.

Greenhow, M. 1988. Water-entry and -exit of a horizontal circular cylinder. AppliedOcean Research 10(4):191-198.

Greenhow, M., and Lin, W.-M. 1985. Numerical simulation of nonlinear free sur-face flows generated by wedge entry and wavemaker motion. Proceedings of the FourthInternational Conference on Numerical Ship Hydrodynamics, Washington, DC.

Grosenbaugh, M.A., and Yeung, R.W. 1988. Nonlinear bow flows – An experimen-tal and theoretical investigation. Proceedings of the Seventeenth Symposium on NavalHydrodynamics, The Hague, The Netherlands.


Hocking, L.M., and Mahdmina, D. 1991. Capillary-gravity waves produced by awavemaker. Journal of Fluid Mechanics 224:217-226.

Joo, S.W., Schultz, W.W., and Messiter, A.F. 1990. An analysis of the initial-value wavemaker problem. Journal of Fluid Mechanics 214:161-183.

Korobkin, A.A. 1988. Inclined entry of a blunt profile into an ideal fluid. Fluid Dy-namics 23:443-447.

Korobkin, A.A., and Pukhnachov, V.V. 1988. Initial stage of water impact. An-nual Review of Fluid Mechanics 20:159-185.

Lin, W.-M. 1984. Nonlinear Motion of the Free Surface near a Moving Body. Ph.D.thesis, Massachusetts Institute of Technology – Department of Ocean Engineering.

Lin, W.-M., Newman, J.N., and Yue, D.K.P. 1984. Nonlinear forced motionsof floating bodies. Proceedings of the Fifteenth Symposium on Naval Hydrodynamics,Hamburg, Germany.

Miles, J. 1991. On the initial-value problem for a wavemaker. Journal of Fluid Me-chanics 229:589-601.

Miloh, T. 1981. Wave slam on a sphere penetrating a free surface. Journal of Engi-neering Mathematics 15(3):221-240.

Miloh, T. 1991a. On the initial-stage slamming of a rigid sphere in a vertical waterentry. Applied Ocean Research 13(1):43-48.

Miloh, T. 1991b. On the oblique water-entry of a rigid sphere. Journal of EngineeringMathematics 25(1):77-92.

Peregrine, D.H. 1972. Flow due to vertical plate moving in a channel. Unpublishednote.

Roberts, A.J. 1987. Transient free-surface flows generated by a moving vertical plate.Quarterly Journal of Mechanics and Applied Mathematics 40(1):129-158.

Vinje, T., and Brevig, P. 1981. Nonlinear ship motion. Proceedings of the ThirdInternational Conference on Numerical Ship Hydrodynamics, Paris, France.

Wang, K.-H., and Chwang, A.T. 1989. Nonlinear free surface flow around animpulsively moving cylinder. Journal of Ship Research 33(3):194-202.


Now when they stood apart and were ready with their gauntlets, straightwayin front of their faces they raised their heavy hands and matched their mightin deadly strife. Hereupon the Bebrycian king5 — even as a fierce wave ofthe sea rises in a crest against a swift ship, but she by the skill of the craftypilot just escapes the shock when the billow is eager to break over the bulwark— so he followed up the son of Tyndareus6, trying to daunt him, and gavehim no respite. But the hero, ever unwounded, by his skill baffled the rushof his foe, and he quickly noted the brutal play of his fists to see where hewas invincible in strength, and where inferior, and stood unceasingly andreturned blow for blow. And as when shipwrights with their hammers smiteships’ timbers to meet the sharp clamps, fixing layer upon layer; and the blowsresound one after another; so cheeks and jaws crashed on both sides, and ahuge clattering of teeth arose, nor did they cease ever from striking theirblows until laboured gasping overcame both. And standing a little apart theywiped from their foreheads sweat in abundance, wearily panting for breath.Then back they rushed together again, as two bulls fight in furious rivalry fora grazing heifer. Next Amycus rising on tiptoe, like one who slays an ox,sprung to his full height and swung his heavy hand down upon his rival; butthe hero swerved aside from the rush, turning his head, and just received thearm on his shoulder; and coming near and slipping his knee past the king’s,with a rush he struck him above the ear, and broke the bones inside, and theking in agony fell upon his knees; and the Minyan heroes7 shouted for joy;and his life was poured forth all at once . . .


5i.e. Amycus6i.e. Polydeuces7i.e. the Argonauts


Figure 5.3: Initial potential distribution on the wavemaker.


Figure 5.4: Initial vertical velocity along the wavemaker.


Figure 5.5: Initial potential distribution close to the intersection point.


Figure 5.6: Initial vertical velocity close to the intersection point.


Figure 5.7: Initial pressure distribution on the wavemaker.


Figure 5.8: Initial potential distribution on the bottom.


Figure 5.9: Initial horizontal velocity along the bottom.


Figure 5.10: Initial vertical velocity of the free surface.


Figure 5.11: Initial vertical free surface velocity close to the intersection point.


Figure 5.12: Free surface elevation at times t = 0.0025− (0.0025)− 0.0200 s.


Figure 5.13: Free surface elevation at times t = 0.025− (0.025)− 0.200 s.


Chapter 6

Hydrodynamic Mass andDamping

But when the sun rising from far lands lighted up the dewy hills and wak-ened the shepherds, then they loosed their hawsers from the stem of the baytreeand put on board all the spoil they had need to take; and with a favour-ing wind they steered through the eddying Bosporus. Hereupon a wave like asteep mountain rose aloft in front as though rushing upon them, ever upheavedabove the clouds; nor would you say that they could escape grim death, for inits fury it hangs over the middle of the ship, like a cloud, yet it sinks awayinto calm if it meets with a skilful helmsman. So they by the steering-craft ofTiphys escaped, unhurt but sore dismayed. And on the next day they fastenedthe hawsers to the coast opposite the Bithynian land.

There Phineus, son of Agenor, had his home by the sea, Phineus whoabove all men endured most bitter woes because of the gift of prophecy whichLeto’s son had granted him aforetime. And he reverenced not a whit even Zeushimself, for he foretold unerringly to men his sacred will. Wherefore Zeus sentupon him a lingering old age, and took from his eyes the pleasant light, andsuffered him not to have joy of the dainties untold that the dwellers aroundever brought to his house, when they came to enquire the will of heaven. Buton a sudden, swooping through the clouds, the Harpies with their crooked beaksincessantly snatched the food away from his mouth and hands. And at timesnot a morsel of food was left, at others but a little, in order that he might liveand be tormented. And they poured forth over all a loathsome stench; andno one dared not merely to carry food to his mouth but even to stand at adistance; so foully reeked the remnants of the meal . . .


95

96 CHAPTER 6. HYDRODYNAMIC MASS AND DAMPING

6.1 Introduction

A solution for the ship motion problem at sea requires the determination of the hydro-dynamic equilibrium of forces and moments. It is generally accepted that for the fluidforces the influence of viscosity and surface tension are of minor importance comparedto pressure and wave effects. As far as ship motions are concerned, this proposition hasnot been disproved by model or full scale experiments — apart from the manoeuvringproblem. It has further been supposed that the whole motion problem can be regardedas linear; up to the present state of development this has been confirmed substantially,in any case for small amplitude motion and apart from very special objects.

By these circumstances the determination of the hydrodynamic forces acting on theship’s hull can be put as a linear boundary value problem in potential theory. Thesuperposition principle holds and the actual phenomenon is split up into the sum ofharmonic oscillations of the ship in still water and waves coming in on the restrained ship.The two fields can be investigated entirely separately. Considering only the first field,the problem can be stated as the oscillation of a rigid body moving with a certain speedin the surface of a heavy and ideal fluid. The solution supplies the six transfer functionsof the ship, which are composed of both rigid body characteristics and hydrodynamicquantities.

Ursell (1949ab) made the first contribution to the solution of this problem. He considereda circular cylinder which oscillates harmonically with small amplitude, while the meanposition of the cylinder axis coincides with the mean surface of the fluid. A solution tothe corresponding linear potential problem was found by superimposing suitably chosenfunctions, such that each separate function satisfies Laplace’s equation and the linearizedfree surface conditions, while a combination of these functions fulfills the remainingboundary conditions.

Grim (1953, 1955/1956) and Tasai (1959, 1961) extended this principle from thecircular to elliptic cylinders and so-called Lewis-forms, while Porter (1960) formulatedthe solution for heaving of an arbitrarily shaped cylinder. The extension of Ursell’stheory for circular cylinders to finite constant depth was given by Yu and Ursell (1961).

Now in principle the way was free to investigate the influence of form, of frequency ofmotion, and of the coupling effects between sway and roll in detail. But first the validityof the theoretical approach had to be established by experiment; naturally, this wasfirst tried for the most simple case of heaving. Tasai (1960) measured the wave heightsproduced by forced heaving cylinders, and Porter (1960) measured the total vertical forceon a heaving circular cylinder and the pressure in a number of points along the contour.Paulling and Richardson (1962) carried out extensive experiments recording the verticalforce and the pressure — both in magnitude and in phase — for different sections; waveheights were measured as well. The results of these experiments were such that thetheoretical predictions were confirmed substantially. Few years later, the attention wasshifted towards rolling and swaying by Vugts (1968), who systematically carried out forcemeasurements as to amplitude and phase, for heaving, swaying, and rolling cylinders ofvarious cross sections.

For completeness, we mention the extension of Ursell’s techniques to three dimensionsby Havelock (1955) and Barakat (1962) for a heaving sphere. Kim (1965) computed addedmass and damping coefficients for spheroids in surge (or sway), heave, and pitch (or roll)motion, for ellipsoids in surge, heave, and pitch motion, and for cylinders with ellipticcross section in all three modes of motion. Hulme (1982) computed added mass and

6.2. MATHEMATICAL MODEL FOR TWO-DIMENSIONAL MOTION 97

damping coefficients for heaving and surging hemispheres.

Here we report on the numerical determination of the hydrodynamic (added) mass anddamping coefficients for circular cylinders in forced harmonic heaving and swaying mo-tion. The computed results are compared to the measurements of Vugts (1968) and tothe analytical predictions of de Jong (1973), who in turn based his calculations on Ursell’s(1949ab) linear theory. In section 6.2 we present a mathematical model for the motionof infinitely long cylinders in a free surface; for sinusoidal motion with small amplitude,the hydrodynamic coefficients can be expressed in terms of the frequency and amplitudeof the body motion, and the amplitude and phase of the forces. The numerical resultsare discussed in section 6.3, and concluding remarks are given in section 6.4.

6.2 Mathematical model for two-dimensional motion

Let x, z be a coordinate system which is fixed in space. The x-axis coincides with thestill water level and its positive part is directed towards the right. The z-axis is verticaland positive upwards. The origin O is the intersection of the centreline of the cylindersection — which is of arbitrary shape — and the waterline, see Figure 6.1. Supposenow that the centre of mass G of the cylinder is situated in O. The cylinder positionand orientation are denoted by (xG, zG) and θG respectively. The most general way todescribe the cylinder motion as a linear system is a set of three linear coupled equationsof motion. Following Vugts (1968), this set can be put as

Fx = (M + axx) xG + bxxxG + cxxxG

+ axz zG + bxz zG + cxzzG + axθ θG + bxθ θG + cxθθG , (6.1)Fz = (M + azz) zG + bzz zG + czzzG

+ azθ θG + bzθ θG + czθθG + azxxG + bzxxG + czxxG , (6.2)Lθ = (I + aθθ) θG + bθθ θG + cθθθG

+ aθxxG + bθxxG + cθxxG + aθz zG + bθz zG + cθzzG , (6.3)


Figure 6.1: Two-dimensional cylinder floating on a free surface.

where a dot denotes differentiation with respect to time, and

• M is the mass of the cylinder section,

• I is the moment of inertia about G,

• aii is the hydrodynamic mass (or moment of inertia) coefficient in the i-mode ofmotion,

• aij is the mass coupling coefficient in the i-mode equation by j-mode motion,

• bii is the hydrodynamic damping coefficient against i-mode motion,

• bij is the damping coupling coefficient in the i-mode equation by j-mode motion,

• cii is the hydrostatic restoring coefficient against an i-mode displacement,

• cij is the restoring coupling coefficient in the i-mode equation against a j-modedisplacement,

• Fx is the horizontal external force,

• Fz is the vertical external force,

• Lθ is the external moment about G.

The coefficients cii and cij can be determined by pure hydrostatics. For example, ifa cylinder is given a downward displacement z, Archimedes’ principle1 states that therestoring force equals the weight of the displaced water volume. If the cross section widthat the waterline is denoted by B, we have czzz = ρgBz, and hence czz = ρgB.

By simple reasoning the above set of equations can be simplified considerably. Thehorizontal force is not opposed by any restoring force, so cxx = czx = cθx = 0. Thevertical motion is symmetric with respect to the z-axis and can not produce any lateralforces or moments; therefore axz = bxz = cxz = 0, and aθz = bθz = cθz = 0. A static

1Archimedes (287-212 B.C.), mathematician of Syracuse. He discovered the ratio of the radius of acircle to its circumference and the formulas for the surface and volume of a cylinder and of a sphere, anddemonstrated the hydrostatic law known as Archimedes’ principle.

6.2. MATHEMATICAL MODEL FOR TWO-DIMENSIONAL MOTION 99

roll displacement does not generate a horizontal force, which implies cxθ = 0. Thus, themathematical model is reduced to

Fx = (M + axx) xG + bxxxG + axθ θG + bxθ θG , (6.4)Fz = (M + azz) zG + bzz zG + czzzG

+ azθ θG + bzθ θG + czθθG + azxxG + bzxxG , (6.5)Lθ = (I + aθθ) θG + bθθ θG + cθθθG + aθxxG + bθxxG . (6.6)

From (6.4) and (6.6) it is seen that heave does not influence the coupled sway-roll motion.However, the reverse need not be true, see (6.5). In a potential flow the sway and rollproblem is asymmetric with respect to the z-axis and the corresponding contributionsin (6.5) vanish. Thus, heaving becomes an uncoupled motion with one degree of freedom.

The cylinders are harmonically oscillated in one of the three modes of motion:

xG = ax sin ωt and z = θ = 0 , (6.7)zG = az sin ωt and θ = x = 0 , (6.8)θG = aθ sin ωt and x = z = 0 , (6.9)

while Fx, Fz, and Lθ are computed in amplitude and phase:

Fx =∫

S

p nxds = Ax sin (ωt + αx) , (6.10)

Fz =∫

S

p nzds = Az sin (ωt + αz) , (6.11)

Lθ =∫

S

p (rxnz − rznx) ds = Aθ sin (ωt + αθ) . (6.12)

Substitution of (6.7) and (6.10) into (6.4) shows that the hydrodynamic mass and damp-ing coefficients for sway motion read

axx = −Ax cosαx

ω2ax−M , bxx =

Ax sin αx

ωax(6.13)

Similarly, the coefficients for heave motion are found from substitution of (6.8) and (6.11)into (6.5):

azz =azczz −Az cos αz

ω2az−M , bzz =

Az sin αz

ωaz(6.14)

Following the same procedure for roll, it is found that the hydrodynamic mass anddamping coefficients are given by

aθθ = −Aθ cos αθ

ω2aθ− I , bθθ =

Aθ sin αθ

ωaθ(6.15)

These expressions will be used in the next section to compare the computed results withthe analytical predictions and experimental measurements.


6.3 Numerical results

Here we discuss the numerical results for added mass and damping for two-dimensionalcircular cylinders in forced harmonic heaving and swaying motion.

The sinusoidal motions xG (i.e. sway) and zG (i.e. heave) are given in (6.7) and (6.8)respectively. The forces Fx and Fz are computed from pressure integrations over the wet-ted part of the cylinder surface, see (6.10-6.11). The imposed motions and the computedforces serve as input signals for a harmonic analysis.2 The main output usually consistsof the amplitude and phase leads of the leading (and, if possible, higher) frequency com-ponents, i.e. Ax and αx for sway, and Az and αz for heave. These data can then be usedto compute the hydrodynamic mass and damping coefficients from (6.13) for sway, andfrom (6.14) for heave.

Finally, we note that the forces MxG and MzG account for the accelerations of thecylinder. Since these contributions are not computed in the numerical tests discussedhereafter, they should be removed from (6.4-6.5). Consequently, the mass M is set equalto zero in (6.13-6.14).

6.3.1 Circular cylinder in heaving motion

First we discuss the computed results for circular cylinders undergoing a sinusoidal heav-ing motion. The tests have been carried out in a numerical wave tank with lengthL = 60 m and water depth h = 10m; the cylinder beam is B = 4m. The imposed cylin-der motion and the ensuing wave motion have been simulated for twelve different fre-quencies — ranging from ω = 0.5 rad/s to ω = 6.0 rad/s, with increment ∆ω = 0.5 rad/s— and the calculations have been sustained for about six cycles of motion with amplitudeaz = 0.1m.

The results for added mass and damping are listed in Table 6.1. The frequency andthe hydrodynamic coefficients are made nondimensional by

ω = ω

(B

2g

)1/2

, azz =azz

ρ∀ , bzz =bzz

ρ∀(

B

2g

)1/2

(6.16)

where ρ = 1000 kg/m3 is the water density, and ∀ = 18πB2 is the displaced water volume

when the cylinder is in its equilibrium position.

2We used the harmonic analysis program HARMAN, which has been developed at MARIN.


ω ω Az αz azz azz bzz bzz(×103)

(deg)(×103

) (×103)

0.5 0.226 3.759 6.17 7.471 1.189 8.080 0.5811.0 0.452 3.537 13.26 4.813 0.766 8.113 0.5841.5 0.677 3.243 21.29 4.010 0.638 7.850 0.5652.0 0.903 2.697 29.53 3.943 0.628 6.646 0.4782.5 1.129 1.749 40.34 4.145 0.660 4.529 0.3263.0 1.355 0.908 104.79 4.618 0.735 2.926 0.2113.5 1.580 2.336 163.43 5.031 0.801 1.903 0.1374.0 1.806 4.628 173.98 5.329 0.848 1.213 0.0874.5 2.032 7.390 177.77 5.584 0.889 0.639 0.0465.0 2.258 10.62 178.97 5.817 0.926 0.382 0.0275.5 2.483 14.15 179.31 5.975 0.951 0.313 0.0226.0 2.709 17.97 179.39 6.081 0.968 0.319 0.023

Table 6.1: Added mass and damping coefficients for heaving circular cylinder.

Figures 6.2 and 6.3 present the numerical results for added mass and damping re-spectively, and the comparison with the measurements of Vugts and the analyticalpredictions3 by de Jong.4 Over the whole frequency range excellent agreement is observedfor both added mass and damping, except for the lower frequencies; for ω = 0.226 andω = 0.452 the computed added mass is too low, and for the lowest frequency ω = 0.226the damping is too high.

It is very likely that these effects are caused by reflections from the lateral out-flow boundaries situated at both ends of the wave tank. For low frequency motion,the generated long wave components radiate at a high velocity towards these so-called‘Sommerfeld/Orlanski’ boundaries, where they are partially reflected. Once the reflectedwaves approach the heaving cylinder, the force registration is disturbed; this may spoilthe numerical results.

To confirm this supposition, the computations for ω = 0.452 have been repeated in anumerical wave tank with a doubled length L = 120 m. The results are listed in Table6.2; indeed, it can be observed that the computed added mass is somewhat improved— that is, a small increase can be observed. However, the damping has also increased;evidently, there are other circumstances (i.e. physical parameters) which influence thenumerical computations.

3For completeness it is remarked that analytical solutions are represented by solid lines in Figures 6.2-6.7.

4Note that the results for ω ≥ 4.5 rad/s are not shown in these figures; the computations for thesehigh frequencies have been performed just to verify that — in this case — for ω → ∞ the added massand damping tend to unity and zero respectively.


L Az αz azz azz bzz bzz(×103)

(deg)(×103

) (×103)

60 3.537 13.26 4.813 0.766 8.113 0.584120 3.528 14.48 5.081 0.809 8.821 0.635

Table 6.2: Added mass and damping coefficients for heaving circular cylinder: effectsdue to longer wave tank for ω = 0.452.

Another important parameter in this problem is the uniform water depth. Since theabove-mentioned improvement in the added mass is but marginal, it can be expected thatboth the wave tank length and the water depth must be chosen considerably larger toobtain fair agreement with the analytical predictions for both added mass and damping.This option however, is rather inefficient since the number of panels will increase, andthe computational costs will rise dramatically.

A more efficient way to improve the computed results is to choose a smaller cylinderdiameter; by decreasing the beam B and increasing the frequency ω — such that thenondimensional frequency ω remains constant — in effect a larger wave tank is obtained.5

The computed results for three combinations of ω, B, and az are listed in Table 6.3.

ω/B/az ω Az αz azz azz bzz bzz(×103)

(deg)(×103

) (×103)

1.0/4.0/0.10 0.452 3.537 13.26 4.813 0.766 8.113 0.5842.0/1.0/0.05 0.452 0.424 15.41 0.409 1.041 1.127 0.648

2.0/0.25/0.025 0.226 0.056 5.81 0.051 2.095 0.057 0.526

Table 6.3: Added mass and damping coefficients for heaving circular cylinder: effectsdue to smaller cylinders.

Figures 6.4 and 6.5 show the overall results, including the improved results for the lowerfrequency range. Excellent agreement with linear theory is observed, whereas Vugts’measurements are too low — and scattered6 — for low-frequency added mass.

5In this respect it should be mentioned that Vugts (1968) used cylinders with beam B = 0.3m in anexperimental wave tank with length L = 142 m and mean water depth h = 2m.

6Probably, this scattering of experimental results for the lower frequency range is also due to thelimited accuracy of the measurement equipment used by Vugts.


6.3.2 Circular cylinder in swaying motion

Finally, we discuss the computed results for circular cylinders in forced harmonic swayingmotion. The tank dimensions are given by L = 60 m and h = 10 m; the cylinder beam isB = 4 m. The imposed sinusoidal cylinder motion — with amplitude ax = 0.1m — andthe ensuing wave motion are computed for four different frequencies, and up to six motioncycles. The frequency and the hydrodynamic coefficients are made nondimensional by

ω = ω

(B

2g

)1/2

, axx =axx

ρ∀ , bxx =bxx

ρ∀(

B

2g

)1/2

(6.17)

The results for added mass and damping are listed in Table 6.4, where both dimensionaland nondimensional values are given.

ω ω Ax αx axx axx bxx bxx(×103)

(deg)(×103

) (×103)

1.0 0.452 0.800 167.7 7.816 1.244 1.704 0.1231.5 0.677 2.072 143.3 7.383 1.175 8.255 0.5942.0 0.903 2.572 121.8 3.388 0.539 10.93 0.7863.0 1.355 2.883 112.0 1.200 0.191 8.910 0.641

Table 6.4: Added mass and damping coefficients for swaying circular cylinder.

Figures 6.6 and 6.7 compare our results with the experimental results from Vugts andthe analytical ‘linear’ predictions. Over the whole frequency range fair agreement canbe observed. Unfortunately, the limited amount of time available did not allow us tocompute added mass and damping in sway for the higher and lower frequency range, nordid we have time to investigate the effects of the tank dimensions for the case of swayingmotion.



In this chapter we have focused our attention on the hydrodynamic mass and dampingcoefficients for circular cylinders in harmonic heaving and swaying motion. In section 6.2the mathematical model for linearized two-dimensional motion has been presented, in-cluding expressions for the added mass and damping coefficients in terms of the bodyamplitude and frequency, and the force amplitude and phase lead.

At first sight, the numerically computed results for heave seem to be in excellentagreement with both analytical and experimental results. However, in the lower frequencyrange one should be careful; the reflected parts of the long (radiated) wave componentsdisturb the registered force signal, and hence the computed added mass and damping.This problem can be by-passed by choosing a larger (numerical) wave tank — that is,both the tank length and the water depth have to be increased — or by decreasing thecylinder diameter. The latter option is more efficient, considering the number of panelsand the computational effort involved. Indeed, this technique leads to improved resultsfor both added mass and damping in heave.

The computed results for added mass and damping for a circular cylinder in swayingmotion are in fair agreement with the linear theory and Vugts’ experiments.

Of course we are well aware of the limited scope of this study. First of all, the approachsuggested above should also be applied to the higher frequency range in order to validatethe method. Secondly, for all modes of motion the wave-amplitude ratios should also becomputed. Further, we have to admit that the number of frequencies used in swayingmotion is rather small; unfortunately, the completion of this thesis did not permit us tospend more time on additional computations.

On the other hand, the results obtained so far give us some confidence in the abilityof the method to predict added mass and damping coefficients for other cross sections, inall three modes of motion. Moreover, our nonlinear method offers the unique opportunityto investigate the influence of amplitude — which is likely to introduce nonlinear effects— on ‘linear’ concepts as added mass and damping coefficients.

6.5 Bibliography

Barakat, R. 1962. Vertical motion of a floating sphere in a sine-wave sea. Journal ofFluid Mechanics 13:540-556.

Grim, O. 1953. Berechnung der durch Schwingungen eines Schiffkorpers erzeugtenhydrodynamischen Krafte. Jahrbuch der Schiffbautechnischen Gesellschaft.

Grim, O. 1955/1956. Die hydrodynamischen Krafte beim Rollversuch. Schiffstechnik,Band 3.

Havelock, T. 1955. Waves due to a floating sphere making periodic heaving oscilla-tions. Proceedings of the Royal Society of London, Series A, 231:1-7.

Hulme, A. 1982. The wave forces acting on a floating hemisphere undergoing forcedperiodic oscillations. Journal of Fluid Mechanics 121:443-463.

de Jong, B. 1973. Computation of the hydrodynamic coefficients of oscillating cylin-ders. Report 145S, Netherlands Ship Research Centre TNO, Shipbuilding Department,Delft, The Netherlands.


Kim, W.D. 1965. On the harmonic oscillations of a rigid body on a free surface. Journalof Fluid Mechanics 21(3):427-451.

Paulling, J.R., and Richardson, R.K. 1962. Measurement of pressures, forcesand radiating waves for cylinders oscillating in a free surface. University of California,Institute of Engineering Research, Berkeley.

Porter, W.R. 1960. Pressure distribution, added mass and damping coefficients forcylinders oscillating in a free surface. University of California, Institute of EngineeringResearch, Berkeley.

Tasai, F. 1959. On the damping force and added mass of ships heaving and pitching.Reports of the Research Institute for Applied Mechanics, Kuyushu University, VolumeIX, Number 26.

Tasai, F. 1960. Measurement of the wave height produced by the forced heaving of thecylinders. Reports of the Research Institute for Applied Mechanics, Kuyushu University,Volume VIII, Number 29.

Tasai, F. 1961. Hydrodynamic force and moment produced by swaying and rolling os-cillation of cylinders on the free surface. Reports of the Research Institute for AppliedMechanics, Kuyushu University, Volume IX, Number 35.

Ursell, F. 1949a. On the heaving motion of a circular cylinder on the surface of afluid. Quarterly Journal of Mechanics and Applied Mathematics 2(2):218-231.

Ursell, F. 1949b. On the rolling motion of cylinders in the free surface of a fluid.Quarterly Journal of Mechanics and Applied Mathematics 2(3):335-353.

Vugts, J.H. 1968. The hydrodynamic coefficients for swaying, heaving and rolling cylin-ders in a free surface. International Shipbuilding Progress 15(167):251-275.

Yu, S.C., and Ursell, F. 1961. Surface waves generated by an oscillating circularcylinder on water of finite depth: theory and experiment. Journal of Fluid Mechanics11:529-551.


Meanwhile the chiefs carefully cleansed the old man’s squalid skin andwith due selection sacrificed sheep which they had borne away from the spoilof Amycus. And when they had laid a huge supper in the hall, they sat downand feasted, and with them feasted Phineus ravenously, delighting his soul,as in a dream. And there, when they had taken their fill of food and drink,they kept awake all night waiting for the sons of Boreas.7 And the aged sirehimself sat in the midst, near the hearth, telling of the end of their voyageand the completion of their journey:

“Listen then. Not everything is it lawful for you to know clearly; butwhatever is heaven’s will, I will not hide. I was infatuated aforetime, when inmy folly I declared the will of Zeus in order and to the end. For he himselfwishes to deliver to men the utterances of the prophetic art incomplete, inorder that they may still have some need to know the will of heaven.

“First of all, after leaving me, ye will see the twin Cyanean rocks where thetwo seas8 meet. No one, I ween, has won his escape between them. For theyare not firmly fixed with roots beneath, but constantly clash against one an-other to one point, and above a huge mass of salt water rises in a crest, boilingup, and loudly dashes upon the hard beach. Wherefore now obey my counsel,if indeed with prudent mind and reverencing the blessed gods ye pursue yourway; and perish not foolishly by a self-sought death, or rush on following theguidance of youth. First entrust the attempt to a dove when ye have sent herforth from the ship. And if she escapes safe with her wings between the rocksto the open sea, then no more do ye refrain from the path, but grip your oarswell in your hands and cleave the sea’s narrow strait, for the light of safetywill be not so much in prayer as in strength of hands. Wherefore let all elsego and labour boldly with might and main, but ere then implore the gods asye will, I forbid you not. But if she flies onward and perishes midway, thendo ye turn back; for it is better to yield to the immortals. For ye could notescape an evil doom from the rocks, not even if Argo were of iron.”


7i.e. Zetes and Calais, who will chase away the Harpies8i.e. the Sea of Marmara and the Black Sea


Figure 6.2: Added mass coefficients for circular cylinder in heaving motion.


Figure 6.3: Damping coefficients for circular cylinder in heaving motion.


Figure 6.4: Improved added mass coefficients for circular cylinder in heaving motion.


Figure 6.5: Improved damping coefficients for circular cylinder in heaving motion.


Figure 6.6: Added mass coefficients for circular cylinder in swaying motion.


Figure 6.7: Damping coefficients for circular cylinder in swaying motion.

Chapter 7

Cylinders in Free Motion

Now when they reached the narrow strait of the winding passage, hemmedin on both sides by rugged cliffs, while an eddying current from below waswashing against the ship as she moved on, they went forward sorely in dread;and now the thud of the crashing rocks ceaselessly struck their ears, and thesea-washed shores resounded, and then Euphemus grasped the dove in hishand and started to mount the prow; and they, at the bidding of Tiphys, sonof Hagnias, rowed with good will to drive Argo between the rocks, trusting totheir strength. And as they rounded a bend they saw the rocks opening for thelast time of all. Their spirit melted within them; and Euphemus sent forth thedove to dart forward in flight; and they all together raised their heads to look;but she flew between them, and the rocks again rushed together and crashedas they met face to face. And the foam leapt up in a mass like a cloud; awfulwas the thunder of the sea; and all round them the mighty welkin roared.

The hollow caves beneath the rugged cliffs rumbled as the sea came surgingin; and the white foam of the dashing wave spurted high above the cliff. Nextthe current whirled the ship round. And the rocks shore away the end of thedove’s tail-feathers; but away she flew unscathed. And the rowers gave a loudcry; and Tiphys himself called to them to row with might and main. For therocks were again parting asunder. But as they rowed they trembled, until thetide returning drove them back within the rocks. Then most awful fear seizedupon all; for over their head was destruction without escape . . .


113

114 CHAPTER 7. CYLINDERS IN FREE MOTION

7.1 Introduction

In this chapter we report on the numerical calculations for two-dimensional cylindersin free heaving and rolling motion. The computations have been performed with ourcomputer code TIPHYS, which is based on the panel method for nonlinear ship motionsin water waves. A complete description of the numerical algorithm is given in chapter 4,where we suggested a special approach to compute the partial time derivative of thevelocity potential along the wetted part of the body. It is expected that the solutionof an extra boundary integral equation for φt, involving the hydrodynamic equationsof motion for the body, provides a good approximation of the Bernoulli pressure alongthe wetted body surface. In this way the stability of the step-by-step method for freelyfloating bodies is ensured.

In all test cases discussed here, the cylinder motion starts either from an initialdisplacement with zero initial velocity or with an initial velocity from the equilibriumposition. The ensuing motion consists of the cylinder motion and the motion of thesurrounding fluid. If possible, we shall compare our results with analytical predictions.

7.2 Circular heaving cylinder

First, consider a circular cylinder floating on the free surface of a wave tank. The cylinderradius is denoted by R, and its uniform density equals half the water density; in a stateof equilibrium the centre of mass G lies on the mean water level z = 0. The cylinder isdisturbed from its equilibrium position either by a vertical upward displacement zG (0) <R or by an upward velocity zG (0) > 0. The ensuing free motion of this wave-body systemconsists of the cylinder motion, the motion of the surrounding fluid which contributes tothe effective cylinder mass — i.e. the cylinder mass plus the virtual ‘added’ mass — anda wave motion which extracts energy from the cylinder — i.e. wave radiation ‘damping’.

The unsteady heaving motion of this particular cylinder type was first addressedanalytically by Ursell (1949, 1964) who considered the problem by superposing har-monic wave components. From the analytical behaviour of the force coefficients in thefrequency domain, Ursell obtained large time asymptotics for the response of a half-immersed cylinder. Few years later, the entire response range for initial displacementsand initial velocities was computed by Maskell and Ursell (1970). On the basis of theirlinear theory, they found that the normalized and time-scaled displacement

ζG (τ) ≡zG

(t√

g/R)

zG (0)(7.1)

7.2. CIRCULAR HEAVING CYLINDER 115

behaves for small time like

ζG (τ) = 1− 1π

τ2 +2

9π2τ4 +O (

τ6)

. (7.2)

For large τ , the following approximation was derived:

ζG (τ) = 0.9664e−0.1309τ cos (0.9117τ − 0.4805) . (7.3)

For the intermediate time range — say, for τ ∈ [1.5, 11.0] — Maskell and Ursell computedζG (τ) numerically. Though approximative, the data measured from their publicationsare used here for reference.

To start with, our method is tested for a circular cylinder with radius R = 2m inthree initial upward displacements: zG (0) = 1

8R = 0.25m, zG (0) = 14R = 0.50 m,

and zG (0) = 12R = 1.00 m; the initial cylinder positions relative to the still water level

have been sketched above. The tank length is L = 40 m, and the mean water depth ish = 10m. A Sommerfeld/Orlanski radiation condition (φt = cφx) is used at both ends ofthe wave tank, with the shallow water approximation c = (gh)1/2 ≈ 9.90m/s substitutedfor the wave celerity. The number of panels is 72 at the free surface (left plus right),30 at the wetted part of the cylinder contour, 20 at the lateral outflow boundaries (leftplus right), and 20 at the bottom. The time step is ∆t = 0.05 s, so that 200 steps areneeded for the first 10 simulation seconds. Comparative tests with a doubled tank lengthL = 80m indicate that within this time interval the reflected parts of the outgoing wavesdo not affect the solution in the neighbourhood of the cylinder.


The computed transient heaving motion of the cylinder is presented in Figure 7.1, whereboth the vertical displacements and the vertical velocities have been scaled with respectto the respective initial displacements. From these plots it can be observed that themotion is strongly damped, which indicates a high rate of energy (or momentum) transferfrom the cylinder to the fluid. Of course this effect can be attributed to the specialcircular shape, which implies a strong variance of the cross section width at the waterline.Since the heave response lines nearly coincide within drawing accuracy, we arrive at theconclusion that clear nonlinear effects are absent in these computations.

Additional tests are carried out for zG (0) = 58R = 1.25m, and zG (0) = 3

4R = 1.50m, seealso the above sketch. The results from these computations are compared with the resultsfor zG (0) = 1

2R = 1.00m in Figure 7.2. It is remarkable that clear nonlinear effects stillcan not be observed, in spite of the fact that in the last case, where zG (0) = 1.50m,the cylinder motion nearly starts from a state of total emergence. However, it shouldbe remarked that for the whole range of initial displacements the computed results arein excellent agreement with the analytical predictions from the aforementioned theoryof Ursell and Maskell; Figure 7.3 compares our results for zG (0) = 1.50m with theirnormalized and time-scaled results.

The transient heave responses for a unit initial displacement and a unit initial velocityare presented in Figure 7.4. For cylinders of arbitrary shape, Yeung (1982) has shownthat the initial-velocity response equals the time derivative of the initial-displacementresponse multiplied by one half of the infinite-fluid virtual mass of the cylinder. Apartfrom finite-fluid effects in our computations and scaling effects, this result is confirmedby the plots in Figure 7.4; compare the solid line starting from zero with the dotted linestarting from unity.

7.3. RECTANGULAR HEAVING CYLINDER 117

7.3 Rectangular heaving cylinder

Next, consider a homogeneous rectangular cylinder floating on the free surface; the equi-librium state corresponds to the situation where the centre of mass is on the mean waterlevel. In the cases discussed here, the cylinder is disturbed from its equilibrium by down-ward displacements or velocities. The cylinder width is B = 4 m, the height is H = 2 m,and the draft is T = 1m. Since sharp edges may introduce singularities in the flow ofthe surrounding fluid, the cylinder corners are rounded with radius R = 0.25m. In thisway the local fluid motion is expected to meet the requirements for potential flow as wellas possible. Neither the tank dimensions nor the panel distributions have been altered.

Three initial downward displacements were chosen: zG (0) = − 14T = −0.25m, zG (0) =

− 12T = −0.50m, and zG (0) = − 3

4T = −0.75m. The above sketch shows the initialpositions of the cylinder relative to the still water level.

The computed results for the normalized heave displacements and velocities are shownin Figure 7.5; as in the previous test series, the motion is strongly damped and appar-ently without nonlinearities. In Figure 7.6 the heave responses for an initial displace-ment zG (0) = −0.50m and an initial velocity zG (0) = −0.50m/s are presented. Again,apart from finite-fluid and scaling effects, the computations seem to support Yeung’s the-ory regarding the simple relationship between the initial-displacement response and theinitial-velocity response. Finally, it is remarked that the results presented here are in fairagreement with the computations by Chapman (1979); a discussion of this comparisonwill be given by van Daalen and Zandbergen (1993).


7.4 Rectangular rolling cylinder

The rectangular cylinder introduced in the previous section is now given an initial rolldisplacement; the sketch below gives an impression of the cylinder orientation for θG (0) =0.05 rad ≈ 2.9 deg, θG (0) = 0.15 rad ≈ 8.6 deg, θG (0) = 0.25 rad ≈ 14.3 deg, and θG (0) =0.35 rad ≈ 20.1 deg. Neither the tank dimensions nor the panel distributions have beenchanged.

The transient normalized roll responses have been gathered in Figure 7.7. Clearly, themotion is but slightly damped for all four initial roll displacements. Small nonlineareffects can be observed, since the roll period is but slightly decreased when the initialangle of roll is increased.


The above results for two-dimensional freely floating cylinders seem to indicate that clearnonlinear effects are absent in these simulations. This is not a surprising result for thecase of heaving circular and rectangular cylinders with small initial displacements, butone might expect distinct nonlinearities in the case where the cylinders are near totalemergence or submergence. Also, some nonlinear effects might have been anticipated forthe case of a rectangular rolling cylinder.

So far, we did not succeed in finding a conclusive answer to the question why nonlinearphenomena remain absent in our computations. Perhaps other simulations involvingspecial geometries — for instance, where in the equilibrium position the larger part ofthe body is submerged — will reveal nonlinearities in the coupled wave-body motion.For the time being, we have to close this discussion with the statement that these (andother) problems will be the subject of future research.


7.6 Bibliography

Chapman, R.B. 1979. Large-amplitude transient motion of two-dimensional floatingbodies. Journal of Ship Research 23(1):20-31.

van Daalen, E.F.G., and Zandbergen, P.J. 1993. A novel extension of a panelmethod for nonlinear gravity waves to interactions with freely floating bodies. To appearin the Journal of Engineering Mathematics.

Maskell, S.J., and Ursell, F. 1970. The transient motion of a floating body.Journal of Fluid Mechanics 44(2):303-313.

Ursell, F. 1949. On the heaving motion of a circular cylinder on the surface of a fluid.Quarterly Journal of Mechanics and Applied Mathematics 2(2):218-231.

Ursell, F. 1964. The decay of the free motion of a floating body. Journal of FluidMechanics 19(2):305-319.

Yeung, R.W. 1982. The transient heaving motion of floating cylinders. Journal ofEngineering Mathematics 16(2):97-119.


. . .And now to right and left broad Pontus was seen, when suddenly a hugewave rose up before them, arched, like a steep rock; and at the sight theybowed with bended heads. For it seemed about to leap down upon the ship’swhole length and to overwhelm them. But Tiphys was quick to ease the shipas she laboured with the oars; and in all its mass the wave rolled away beneaththe keel, and at the stern it raised Argo herself and drew her far away fromthe rocks; and high in air was she borne. But Euphemus strode among allhis comrades and cried to them to bend to their oars with all their might;and they with a shout smote the water. And as far as the ship yielded to therowers, twice as far did she leap back, and the oars were bent like curved bowsas the heroes used their strength.

Then a vaulted billow rushed upon them, and the ship like a cylinder ranon the furious wave plunging through the hollow sea. And the eddying currentheld her between the clashing rocks; and on each side they shook and thun-dered; and the ship’s timbers were held fast. Then Athena with her left handthrust back one mighty rock and with her right pushed the ship through; andshe, like a winged arrow, sped through the air. Nevertheless the rocks, cease-lessly clashing, shore off as she passed the extreme end of the stern-ornament.But Athena soared up to Olympus, when they had escaped unscathed. Andthe rocks in one spot at that moment were rooted fast for ever to each other,which thing had been destined by the blessed gods, when a man in his shipshould have passed between them alive. And the heroes breathed again aftertheir chilling fear, beholding at the same time the sky and the expanse of seaspreading far and wide. For they deemed that they were saved from Hades1;and Tiphys first of all began to speak:

“It is my hope that we have safely escaped this peril — we, and the ship;and none other is the cause so much as Athena, who breathed into Argodivine strength when Argus knitted her together with bolts; and she may notbe caught. Son of Aeson, no longer fear thou so much the hest of thy king,since a god hath granted us escape between the rocks; for Phineus, Agenor’sson, said that our toils hereafter would be lightly accomplished.”


1i.e. the underworld


Figure 7.1: Transient heaving motion of a circular cylinder due to moderate initial dis-placements.


Figure 7.2: Transient heaving motion of a circular cylinder due to large initial displace-ments.


Figure 7.3: Transient heaving motion of a circular cylinder due to initial displacement.Numerical versus analytical results.


Figure 7.4: Transient heaving motion of a circular cylinder due to initial unit displace-ment and initial unit velocity.


Figure 7.5: Transient heaving motion of a rectangular cylinder due to initial displace-ments.


Figure 7.6: Transient heaving motion of a rectangular cylinder due to initial displacementand initial velocity.


Figure 7.7: Transient rolling motion of a rectangular cylinder due to initial displacements.


Part III

Variational Principles andHamiltonian Formulations

129

Chapter 8

Lagrangian and HamiltonianFormulations

And they two1 by the pathway came to the sacred grove, seeking the hugeoak tree on which was hung the fleece, like to a cloud that blushes red withthe fiery beams of the rising sun. But right in front the serpent with his keensleepless eyes saw them coming, and stretched out his long neck and hissed inawful wise . . .And as he writhed, the maiden came before his eyes, with sweetvoice calling to her aid Sleep, highest of gods, to charm the monster; and shecried to the queen of the underworld, the night-wanderer, to be propitious toher enterprise. And Aeson’s son followed in fear, but the serpent, alreadycharmed by her song, was relaxing the long ridge of his giant spine, andlengthening out his myriad coils, like a dark wave, dumb and noiseless, rollingover a sluggish sea; but still he raised aloft his grisly head, eager to enclosethem both in his murderous jaws. But she with a newly cut spray of juniper,dipping and drawing untempered charms from her mystic brew, sprinkled hiseyes, while she chanted her song; and all around the potent scent of the charmcast sleep; and on the very spot he let his jaw sink down . . .

Hereupon Jason snatched the golden fleece from the oak, at the maiden’sbidding . . .Heavy it was, thickly clustered with flocks; and as he moved along,even beneath his feet the sheen rose up from the earth. And he strode on nowwith the fleece covering his left shoulder from the height of his neck to his feet,and now again he gathered it up in his hands; for he feared exceedingly, lestsome god or man should meet him and deprive him thereof . . .

Argonautica, Book IV, Fragments from Verses 123-182.

1i.e. Jason and Medea

131

132 CHAPTER 8. LAGRANGIAN AND HAMILTONIAN FORMULATIONS

8.1 Introduction

This chapter is concerned with special mathematical formulations for both the water-wave problem and the wave-body problem. The governing equations are derived fromvariational — or Lagrangian2 — principles, and it is shown that the nonlinear free surfaceconditions and the hydrodynamic equations of motion for a floating body describe so-called Hamiltonian3 systems.

The application of variational principles to problems in fluid dynamics goes back toKelvin’s (1849) minimum energy theorem and to Clebsch’s (1859) paper on the inte-gration of the equations of motion of an ideal fluid. The formulation of a Hamilton’sprinciple for an Eulerian description of the motion of an ideal fluid brought Seliger andWhitham (1968) to the conclusion that the Lagrangian density is simply the pressure;Luke (1967) discovered that in the special case of a homogeneous fluid with a free surface,the corresponding Euler-Lagrange equations comprise not only Laplace’s equation in theinterior of the fluid but also the boundary conditions on the free surface.

Zakharov (1968) was the first to note that the exact equations for waves on a perfectfluid of infinite depth constitute a dynamical system with a positive definite Hamiltonianfunctional, which represents the total energy of the fluid. The vertical displacement of,and the velocity potential at the free surface are canonical variables in Hamilton’s sense.This formalism was independently obtained for water waves on a fluid of finite depthby Broer (1974). With explicit use of this special Hamiltonian structure, a systematicaccount of the symmetries and the corresponding conservation laws for water waves wasgiven by Benjamin and Olver (1982); the extension to wave-body problems is presentedin chapter 9.

The reader who is unfamiliar with — but interested in — variational principles andHamiltonian formulations is referred to the many textbooks on these and other relatedsubjects. In chronological order we recommend: Courant and Hilbert (1953, 1962),Gelfand and Fomin (1963), Lanczos (1970), Landau and Lifshitz (1976), Finlayson (1972),Weinstock (1974), Arnol’d (1978), and Goldstein (1980).

In section 8.2 Luke’s variational principle for the classical water-wave problem isdiscussed briefly. A Hamiltonian formulation for water waves, in the form as it wasdiscovered by Zakharov and Broer, is discussed and linked with Luke’s principle viaa Legendre4 transformation in section 8.3. Then, we generalize these principles andformulations to the more complex problems of wave-body interactions in section 8.4,which is based on a paper recently written by van Daalen and van Groesen (1993).

8.2 Luke’s variation principle for water waves

In this section Luke’s (1967) variational principle for the classical water-wave problem isresumed. The two-dimensional problem is described in terms of a horizontal coordinatex and a vertical coordinate z; extension to three dimensions is straightforward and will

2Lagrange, Joseph-Louis, Comte de (1736-1813), French mathematician. His principal work wasin pure mathematics, including differential equations and the calculus of variations, and in mechanics(‘Mecanique analytique’, 1788). He presided over the committee which introduced the metric system(1793).

3Hamilton, Sir William Rowen (1805-1865), Irish mathematician and astronomer. His wide researchesinclude the origination and development of the theory of quaternions.

4Legendre, Adrien-Marie (1752-1833), French mathematician. He made important contributions tothe theory of numbers and the theory of elliptic functions.

8.2. LUKE’S VARIATION PRINCIPLE FOR WATER WAVES 133

therefore not be discussed. Only the irrotational case is considered, which allows theintroduction of a potential φ representing the velocity field ~v, that is ~v = ∇φ.

So, let φ (x, z; t) be the velocity potential of a fluid lying between a (not necessarilyeven) bottom z = −h (x) and a free surface z = η (x; t), with gravity acting in thenegative z-direction, see Figure 8.1.

Figure 8.1: Free surface potential flow.

The variational principle for water waves, as it was first proposed by Luke5, reads

δJ = δ

t2∫

t1

x2∫

x1

L dx dt = 0 (8.1)

with the Lagrangian density

L (φ, η) =

η(x;t)∫

−h(x)

p dz = −η(x;t)∫

−h(x)

(φt +

12

(φ2

x + φ2z

)+ gz

)dz (8.2)

where p denotes the Bernoulli pressure; the water density is taken as unity from now on.In (8.1-8.2), the potential φ and the free surface elevation η are allowed to vary, but aresubject to the restrictions

δφ = 0 , δη = 0 at x = x1,2 and t = t1,2 (8.3)

i.e. the variations δφ and δη must vanish at the end points of the spatial and timeintervals.

Following the usual procedure in the calculus of variations, (8.1-8.2) becomes

δJ = −t2∫

t1

x2∫

x1

[φt +

12

(φ2

x + φ2z

)+ gz

]

z=η

δη

dx dt

5From now on, ‘Luke’ refers to his 1967-paper, unless stated otherwise.


−t2∫

t1

x2∫

x1

η(x;t)∫

−h(x)

(δφt + φxδφx + φzδφz) dz

dx dt = 0 . (8.4)

The contributions from the second integral are rewritten to

−η(x;t)∫

−h(x)

δφtdz = − ∂

∂t

η(x;t)∫

−h(x)

δφ dz + ηt [δφ]z=η , (8.5)

−η(x;t)∫

−h(x)

φxδφxdz = − ∂

∂x

η(x;t)∫

−h(x)

φxδφ dz +

η(x;t)∫

−h(x)

φxxδφ dz

+ ηx [φxδφ]z=η + hx [φxδφ]z=−h , (8.6)

−η(x;t)∫

−h(x)

φzδφzdz = − ∂

∂z

η(x;t)∫

−h(x)

φzδφ dz +

η(x;t)∫

−h(x)

φzzδφ dz

= − [φzδφ]z=η + [φzδφ]z=−h +

η(x;t)∫

−h(x)

φzzδφ dz . (8.7)

The first term on the right-hand side of (8.5) integrates out to the end points of [t1, t2],and hence vanishes. The first term on the right-hand side of (8.6) integrates out to theend points of [x1, x2], and therefore equals zero. Substitution of the remaining termsinto (8.4) yields

δJ =

t2∫

t1

x2∫

x1

η(x;t)∫

−h(x)

∇2φ δφ dz −[φt +

12

(φ2

x + φ2z

)+ gz

]

z=η

δη

dx dt

+

t2∫

t1

x2∫

x1

[(ηt + ηxφx − φz) δφ]z=η + [(hxφx + φz) δφ]z=−h

dx dt

= 0 . (8.8)

First, choose δη = 0, [δφ]z=−h = 0, and [δφ]z=η = 0; since δφ is arbitrary otherwise, itfollows that

∇2φ = 0 for − h (x) < z < η (x; t) (8.9)

Then, since δη, [δφ]z=η and [δφ]z=−h may be given arbitrary independent values, weobtain

p = −(

φt +12

(φ2

x + φ2z

)+ gz

)= 0 at z = η (x; t) (8.10)

and

ηt + ηxφx − φz = 0 at z = η (x; t) (8.11)

8.2. LUKE’S VARIATION PRINCIPLE FOR WATER WAVES 135

and

hxφx + φz = 0 at z = −h (x) (8.12)

respectively.Evidently, (8.9) is Laplace’s equation for the velocity potential, i.e. the continuity

equation for incompressible, inviscid and irrotational flow. Condition (8.10) states thatthe Bernoulli pressure vanishes at the free surface, and it is known as the dynamic freesurface condition. Condition (8.11) expresses that the free surface is a material boundary— which means that fluid particles can not leave it — and it is known as the kinematic freesurface condition. Finally, (8.12) is a zero-flux condition expressing the impermeabilityof the bottom.

Thus it has been shown that (8.1-8.2), being equivalent to the field equation (8.9)and the physical boundary conditions (8.10-8.12), is a proper variational principle for theclassical water-wave problem. In other words, Luke’s variation principle describes theirrotational motion of an inviscid and incompressible fluid with a free surface under theinfluence of gravity.

Remark 1: The above variational principle is easily modified such that the effects ofsurface tension are accounted for. With σ denoting the coefficient of surface tension, thenet extra force per unit area is given by

Tσ (η) = − σηx

(1 + η2x)1/2

. (8.13)

The appropriate variation principle is then given by (8.1), where the Lagrangian densityL is replaced by

Lσ (φ, η) = L (φ, η) + Tσ (η) =

η(x;t)∫

−h(x)

p dz − σηx

(1 + η2x)1/2

. (8.14)

Variation of Tσ with respect to η, and retaining terms up to first order only, gives

δTσ (η) = − σδηx

(1 + η2x)1/2

. (8.15)

Then, integrating by parts with respect to x, and using the vanishing of δη at the endpoints x = x1,2, one arrives at the following set of free surface conditions — confer (8.10-8.11) —

p = −(

φt +12

(φ2

x + φ2z

)+ gz

)= − σηxx(

1 + η2x

)3/2

ηt + ηxφx − φz = 0

at z = η (x; t) ,

(8.16)

showing that surface tension, known to act like a stretched membrane on the free surface,indeed results in a positive pressure at the crest of a wave, where ηxx < 0, and a negativepressure at the trough of a wave, where ηxx > 0.


Linearization of (8.16) about the mean water level yields the following simplified setof free surface conditions — confer (2.35) —

φt = −gη + σηxx

ηt = φz

at z = 0 . (8.17)

For the case of small amplitude waves6 travelling over a horizontal bottom, analyti-cal solutions have been presented in section 2.4; substitution of expressions (2.37-2.38)into (8.17) yields the first order7 dispersion relation — confer (2.39) —

ω2 = (gk tanh kh)(

1 +σ

gk2

). (8.18)

For water waves under gravity, we have g = 9.81m/s2, ρ = 1000 kg/m3, and σ =0.073N/m, so that the minimum wavelength is given by

λm = 2π

(σ

ρg

)1/2

= 1.71 cm , (8.19)

and surface tension effects become negligible for wavelengths several times greater thanthis minimum value.

The special choice of the integrated Bernoulli pressure for the Lagrangian density en-abled Luke to derive the complete set of governing equations for the classical water-waveproblem from a single variational principle. Being aware of the arbitrariness in the choiceof the Lagrangian density, he noticed that this particular choice is more productive thanthe traditional form of the Lagrangian density, being

L∗ (φ, η) = K (φ, η)− P (η) =

η(x;t)∫

−h(x)

12

(φ2

x + φ2z

)dz − 1

2g

(η2 − h2

), (8.20)

i.e. kinetic minus potential energy. In Luke’s own words, the key to the difference appearsto be conservation of mass; at the outset of (8.9) and (8.11-8.12), variation of L∗ withrespect to η leads to the dynamic free surface condition (8.10). The surprising thingis that Luke, by raising this point, undoubtedly has been very close to a different, butrelated, description of the system under consideration; a Hamiltonian formulation for thewater-wave problem.

8.3 Zakharov & Broer’s Hamiltonian formulation

To the best of our knowledge, Zakharov (1968) was the first to show that the governingequations for waves on the surface of an infinitely deep fluid describe a so-called Hamil-tonian system; the evolution equations can be expressed in terms of surface variables anda suitably chosen Hamiltonian density only. Few years later, Broer (1974) independentlyestablished the proof of an equivalent theorem for water waves travelling over a horizontal

6That is, for waves with small amplitude compared to the wavelength and the mean water depth.7That is, to first order in the wave amplitude.

8.3. ZAKHAROV & BROER’S HAMILTONIAN FORMULATION 137

bottom. A corrected proof of the latter result, with applications towards approximativewave equations and conservation laws, was given two years later by Broer, van Groesen,and Timmers (1976). Other contributions with regard to Hamiltonian theories for waterwaves are due to Miles (1976, 1977).

Many approximate model equations exhibit a Hamiltonian structure analogous to theexact governing equations; examples are the shallow water equations, see Salmon (1983),the Boussinesq equations, see Whitham (1965), and the Korteweg-de Vries equations, seefor instance Broer (1975), Gardner (1971), and van Groesen, van Beckum, and Valkering(1990). Stability and bifurcation properties of water waves are directly related to theHamiltonian structure and the symmetries; see Mackay and Saffman (1986), Saffman(1985, 1988), and Zufiria (1987). Some applications of Lagrangian and Hamiltonianformulations for stratified fluids have been given by Henyey (1983) and Miles (1986ab).

In most discussions of Hamiltonian formulations for free surface waves, no reference ismade to the corresponding Lagrangian (variational) principles. It is well known that fordiscrete systems the transition from a Lagrangian to a Hamiltonian formulation is easilyestablished through a so-called Legendre transformation; in the following intermezzo thisconcept is illustrated for a simple n-particle system. Next, it will be shown that Luke’sLagrangian principle and the Hamiltonian formulation due to Zakharov and Broer arelinked through the Legendre transformation for continuous systems.8

Intermezzo: Consider a one-dimensional n-particle system under the action of a centralforce field

F (qi) = −∂V

∂qi, (8.21)

where qi is the position of particle i; the mass of each particle is denoted by mi.The evolution of this system — with n degrees of freedom — is described by n ordinary

differential equations of the form

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0 , (8.22)

which are familiar as the Euler-Lagrange equations. In (8.22) L is the system Lagrangian,defined as kinetic minus potential energy:

L = K − P =n∑

i=1

[12miq

2i − V (qi)

]. (8.23)

By substitution of (8.23) into (8.22), the explicit equations of motion are obtained:

miqi = F (qi) . (8.24)

Since all equations are of second order, the motion of this particular system is determinedfor all time only when 2n initial values are specified. For instance, one can specify then qi’s and the n qi’s at some time t1, or the n qi’s at two times t1 and t2. From theLagrangian point of view the problem is defined in terms of n independent variablesqi (t), and qi appears only as shorthand for the time derivative of qi.

8This connection between Luke’s variation principle and Zakharov’s Hamilton’s principle was alreadynoticed by van Groesen (1978).


In a Hamiltonian formulation the evolution of this n-particle system is described in termsof first-order ordinary differential equations. Since the number of conditions determiningthe motion is still 2n, there must be 2n independent first order equations expressedin 2n independent variables. In thus doubling the set of independents, it is naturalto choose half of them to be the generalized coordinates qi. The formulation becomesnearly symmetric if we choose the other half of the set to be the generalized ‘conjugate’momenta pi, defined by the so-called Legendre transformation

pi =∂L

∂qi= miqi . (8.25)

The quantities (qi, pi) are known as the canonical variables. The Hamiltonian density His defined as

H =n∑

i=1

piqi − L =n∑

i=1

[12miq

2i + V (qi)

]= K + P , (8.26)

i.e. the sum of kinetic and potential energy.The canonical equations read

d

dt

(pi

qi

)=

(0 −11 0

)(∂H/∂pi

∂H/∂qi

), (8.27)

being equivalent to (8.22).From the Hamiltonian viewpoint the problem is defined in terms of 2n independent

variables, namely n qi’s and n pi’s. From a purely mathematical point of view, thetransition from a Lagrangian to a Hamiltonian formulation corresponds to changing thevariables following the Legendre transformation (8.25).

Finally, we note that H, as a function of the generalized coordinates and momenta,is a constant of the motion; this follows directly from

dH

dt=

n∑

i=1

[∂H

∂pipi +

∂H

∂qiqi

](8.28)

and substitution of the canonical equations (8.27).

For a continuous system with generalized coordinate η, the transition from a Lagrangianto a Hamiltonian formulation is represented by

L (η, ηt) −→ H (π, η) = πηt − L (π, η) , (8.29)

where the generalized momentum is defined by

π =∂L∂ηt

. (8.30)

The infinite-dimensional quantities η and π are then known as the canonical conjugatevariables of the system.

Let us return to the classical water-wave problem, for which Luke defined the Lagrangiandensity as

L (φ, η) = −η(x;t)∫

−h(x)

(φt +

12

(φ2

x + φ2z

)+ gz

)dz . (8.31)

8.3. ZAKHAROV & BROER’S HAMILTONIAN FORMULATION 139

With the free surface elevation η in the natural role of generalized coordinate, the gen-eralized momentum (conjugate to η) follows from

−η(x;t)∫

−h(x)

φtdz = − ∂

∂t

η(x;t)∫

−h(x)

φ dz + ηt [φ]z=η . (8.32)

Clearly, application of (8.30) yields

∂L∂ηt

= [φ]z=η , (8.33)

i.e. the restriction of the velocity potential to the free surface; this variable is denotedby Φ from now on.

Then, with (8.31-8.32), the Hamiltonian density follows from the Legendre transfor-mation (8.29):

H (Φ, η) = K (Φ, η) + P (η) =

η(x;t)∫

−h(x)

12

(φ2

x + φ2z

)dz +

12g

(η2 − h2

)(8.34)

i.e. the sum of the kinetic and potential energy densities. In this particular applicationof the Legendre transformation it has been assumed that the first term on the right-handside of (8.32) — i.e. the partial time derivative of the integrated potential — integratesout to the end points of the time interval under consideration and vanishes there.

With the above definitions, and with φ satisfying the boundary value problem

∇2φ = 0 for −h (x) < z < η (x; t) , (8.35)∂φ

∂n= 0 at z = −h (x) , (8.36)

φ (x, z; t) = Φ (x; t) at z = η (x; t) , (8.37)

the following theorem can be deduced:

Theorem 3 : Hamiltonian formulation for the water-wave problem(Zakharov, 1968 / Broer, 1974)The equations of motion for gravity driven water waves describe an infinite-dimensionalHamiltonian system in the canonically conjugate variables Φ and η and with the energydensity H (Φ, η) as Hamiltonian density; the canonical equations

∂t

(Φη

)=

(0 −11 0

)(δΦHδηH

)(8.38)

are equivalent with the nonlinear free surface conditions (8.10-8.11).

The proof of the above theorem is based on the following lemma:


Lemma 1 : Variational derivatives of the kinetic energy densityThe variational derivatives of the kinetic energy density K (Φ, η) are given by

δΦK (Φ, η) =

∂φ

∂n

1 +

(∂η

∂x

)21/2

z=η

, (8.39)

δηK (Φ, η) =[12

(∇φ · ∇φ)− δΦK (Φ, η)∂φ

∂z

]

z=η

. (8.40)

Proof: First we prove (8.39); keep η fixed and let φ vary such that its variation δφcorresponds to a change δΦ in the free surface potential Φ. Then the first variation inK (Φ, η) reads

δK (Φ, η; δΦ) =

η(x;t)∫

−h(x)

∇φ · ∇δφ dz =

η(x;t)∫

−h(x)

[∇ · (∇φ δφ)−∇2φ δφ]dz . (8.41)

With Gauss’ divergence theorem and (8.35-8.36) there results

δK (Φ, η; δΦ) = [(∇φ · ~n) ds δφ]z=η , (8.42)

where ds denotes the local arclength of the free surface contour corresponding to an in-finitesimal horizontal distance dx; in terms of the generalized coordinate η this arclengthreads

ds =

1 +

(∂η

∂x

)21/2

. (8.43)

The last two results lead directly to

δK (Φ, η; δΦ) = δΦ

∂φ

∂n

1 +

(∂η

∂x

)21/2

z=η

, (8.44)

from which (8.39) follows.Next, vary η and assume that the solution φ of the boundary value problem (8.35-

8.37) is correspondingly modified for the varied fluid domain. At the modified free surfacewe have to lowest order:

φ (x, η + δη; t) = φ (x, η; t) + δη

(∂φ

∂z

)

z=η

≡ Φ(x; t) + δηΦ(x; t) , (8.45)

where the variation δηΦ corresponding to δη is given by

δηΦ = δη

(∂φ

∂z

)

z=η

. (8.46)

With the above result the total effect of a variation δη in K (Φ, η) is found to be

δK (Φ, η; δη) + δK (Φ, η; δηΦ) = δη

[12

(∇φ · ∇φ)]

z=η

, (8.47)


and hence, to lowest order

δηK (Φ, η) + δΦK (Φ, η)(

∂φ

∂z

)

z=η

=[12

(∇φ · ∇φ)]

z=η

, (8.48)

from which (8.40) follows.

Proof of theorem 3: The expressions for the variational derivatives of the potentialenergy P (η) read

δΦP (η) = 0 , (8.49)δηP (η) = gη . (8.50)

For the partial time derivative of the free surface potential Φ we write

∂Φ∂t

=(

∂φ

∂z

∂η

∂t+

∂φ

∂t

)

z=η

. (8.51)

In terms of the generalized coordinate η, the unit normal vector along the free surfacereads

~n =(−ηx, 1)T

(1 + η2x)1/2

. (8.52)

Then, with (8.39) and (8.49) the second canonical equation ηt = δΦH is easily shown tobe equivalent with

[ηt + ηxφx − φz]z=η = 0 , (8.53)

i.e. the kinematic free surface condition (8.11).Then, with (8.40) and (8.50-8.53), the first canonical equation Φt = −δηH yields

[φt +

12

(∇φ · ∇φ) + gz

]

z=η

= 0 , (8.54)

i.e. the dynamic free surface condition (8.10).

Thus it has been proven that with the free surface elevation η and the free surfacepotential Φ as canonical coordinates, with the energy density as Hamiltonian densityH (Φ, η) and with φ satisfying the boundary value problem (8.35-8.37), the Hamiltonianequations (8.38) are equivalent with the free surface conditions (8.53-8.54).

8.4 Wave-body formulations

In this section both Luke’s variation (Lagrangian) principle and the Hamiltonian for-mulation for water waves due to Zakharov and Broer are extended to the wave-bodyproblem. The complete set of governing equations for the three-dimensional problem ofnonlinear water waves in hydrodynamic interaction with bodies floating in or below thefree surface is derived from a single variational principle. In addition, we show that thenonlinear free surface conditions and the hydrodynamic equations of motion for a freely


Figure 8.2: Wave-body system.

floating body constitute an infinite-dimensional Hamiltonian system in terms of properlychosen canonical coordinates and the total energy as Hamiltonian.The system under consideration consists of a fluid, bounded by the impermeable bottomB (which is not necessarily even), the free surface F , and the wetted surface S of afloating body (either partially or totally submerged), see Figure 8.2. In the horizontaldirections x and y, the fluid domain is cut off by a cylindrical vertical surface ΣR ofinfinite radius R, extending from the bottom to the free surface. The transient fluiddomain is denoted by Ω (t).

The body mass and moments about its principal axes of inertia are denoted by Mand ~I = (I1, I2, I3)

T respectively. The position of the body is specified by its centre ofgravity ~xG = (x1, x2, x3)

T , and the body orientation is denoted by ~θG = (θ1, θ2, θ3)T .

8.4.1 A variation principle for wave-body interactions

The Lagrangian for the fluid is given by

Lf =∫∫

Ω(t)

∫p dΩ = −

∫∫

Ω(t)

∫ (∂φ

∂t+

12

(∇φ · ∇φ) + gz

)dΩ (8.55)

i.e. the Bernoulli pressure integrated over the fluid domain; the fluid density is taken asunity.

The Lagrangian for the body is defined as the kinetic energy minus the potentialenergy:

Lb = Kb − Pb (8.56)


where

Kb =12M~xG · ~xG +

12

(~I ⊗ ~θG

)· ~θG (8.57)

Pb = Mg~e3 · ~xG (8.58)

In (8.57) the symbol ⊗ is used to define the component-wise product of two vectors:~a⊗~b ≡ (a1b1, a2b2, a3b3)

T .The Lagrangian for the total system, consisting of the fluid and the body, is then

simply given as the sum of the separate Lagrangians:

Ls = Lf + Lb (8.59)

With (8.55-8.59) the proposed variational principle reads

δJ = δ

t2∫

t1

Ls dt = 0 (8.60)

for all variations in the free surface elevation η, the velocity potential φ, and the bodyposition ~xG and orientation ~θG. These variations are subject to the restrictions that theyvanish at the end points of the time interval (i.e. at times t = t1 and t = t2) and on thevertical boundary at infinity (i.e. on ΣR).

Following the standard procedure in the calculus of variations, (8.55-8.60) yields

δJ =

t2∫

t1

∫

F

∫p δη dS +

∫

S

∫p (δ~xS · ~n) dS −

∫∫

Ω(t)

∫(δφt +∇φ · ∇δφ) dΩ

dt

+

t2∫

t1

M~xG · δ~xG −Mg~e3 · δ~xG +

(~I ⊗ ~θG

)· δ~θG

dt = 0 . (8.61)

Here ~xS denotes the position of a point on the wetted body surface S; the change in ~xS

due to variations in ~xG and ~θG is given by

δ~xS = δ~xG + δ~θG × ~r , (8.62)

where ~r = ~xS − ~xG has been introduced to define the position of ~xS relative to ~xG, seeFigure 8.2.

Taking into account the motion of Ω (t) we may write∫∫

Ω(t)

∫δφtdΩ =

∂

∂t

∫∫

Ω(t)

∫δφ dΩ−

∫

F

∫∂η

∂tδφ dx dy −

∫

S

∫ (~xS · ~n

)δφ dS , (8.63)

where ~n is the unit normal vector along ∂Ω ⊃ S. The first term on the right-hand sideof (8.63) vanishes due to the restriction δφ = 0 at times t = t1 and t = t2.

With Green’s first identity we obtain∫∫

Ω(t)

∫∇φ · ∇δφ dΩ =

∫

∂Ω

∫∂φ

∂nδφ dS −

∫∫

Ω(t)

∫∇2φ δφ dΩ . (8.64)


Due to the restriction δφ = 0 on ΣR ⊂ ∂Ω the corresponding contribution to the surfaceintegral on the right-hand side of (8.64) vanishes.

Integrating by parts and using the restrictions δ~xG = ~0 and δ~θG = ~0 at times t = t1and t = t2 gives

t2∫

t1

M~xG · δ~xG dt = −t2∫

t1

M~xG · δ~xG dt , (8.65)

t2∫

t1

(~I ⊗ ~θG

)· δ~θG dt = −

t2∫

t1

(~I ⊗ ~θG

)· δ~θG dt . (8.66)

On the free surface F , the area dS corresponding to infinitesimal horizontal distances dxand dy is given by

dS = dx dy

1 +

(∂η

∂x

)2

+(

∂η

∂y

)21/2

, (8.67)

and the unit normal vector on F can be expressed as

~n =(−ηx,−ηy, 1)T

(1 + η2

x + η2y

)1/2. (8.68)

With (8.61-8.68), the proposed variational principle reads

δJ =

t2∫

t1

∫

F

∫p δη dS +

∫∫

Ω(t)

∫∇2φ δφ dΩ

dt

−t2∫

t1

∫

F

∫ (−∂η

∂x

∂φ

∂x− ∂η

∂y

∂φ

∂y+

∂φ

∂z− ∂η

∂t

)δφ dx dy

dt

−t2∫

t1

∫

S

∫ (∂φ

∂n− ~xS · ~n

)δφ dS +

∫

B

∫∂φ

∂nδφ dS

dt

+

t2∫

t1

∫

S

∫p~n dS −Mg~e3 −M~xG

· δ~xG

dt

+

t2∫

t1

∫

S

∫p (~r × ~n) dS − ~I ⊗ ~θG

· δ~θG

dt = 0 . (8.69)

From this it is clear that invariance of J with respect to a variation in the free surfaceelevation η yields the dynamic free surface condition

p = −(

∂φ

∂t+

12

(∇φ · ∇φ) + gz

)= 0 on F . (8.70)


Similarly, invariance of J with respect to a variation in the velocity potential φ yieldsthe field equation

∇2φ = 0 in Ω (t) , (8.71)

the kinematic condition on the free surface∂η

∂t+

∂η

∂x

∂φ

∂x+

∂η

∂y

∂φ

∂y=

∂φ

∂zon F , (8.72)

the ‘contact’ condition on the wetted body surface

∂φ

∂n= ~xS · ~n on S , (8.73)

and the impermeability (‘zero-flux’) condition at the bottom

∂φ

∂n= 0 on B . (8.74)

Finally, invariance of J with respect to variations in the body position ~xG and orientation~θG yields the equations of motion for the body:

∫

S

∫p~n dS −Mg~e3 = M~xG , (8.75)

∫

S

∫p (~r × ~n) dS = ~I ⊗ ~θG . (8.76)

Thus it has been proven that (8.60) with (8.55-8.59) is a proper variation principle fornonlinear gravity waves interacting with a body floating freely in or below the free surface.

For completeness, it is noted that the extension to multiple independently floatingbodies is straightforward. We shall close this paragraph with a discussion of other —less obvious — extensions which are significant from a practical point of view.

Remark 2: The above formulation is easily extended such that the effects of restoringforces and moments — acting on the body — are accounted for. Suppose that theconservative forces and moments are given by

~Fr = − ∂V∂~xG

, ~Lr = − ∂V∂~θG

, (8.77)

where V(~xG, ~θG

)denotes the ‘restoring potential’. Then, the potential energy of the

body is given by

Pb = Mg~e3 · ~xG + V(~xG, ~θG

). (8.78)

Applying the same procedure as demonstrated above, we arrive at the following set ofmodified equations of motion:

∫

S

∫p~n dS −Mg~e3 + ~Fr = M~xG , (8.79)

∫

S

∫p (~r × ~n) dS + ~Lr = ~I ⊗ ~θG . (8.80)


Remark 3: Non-conservative external forces and moments acting on the body may beincorporated in the above variational description as well. The virtual work done by suchforces and moments (denoted by ~Fe and ~Le respectively) during the time interval [t1, t2]is given by

W (t1; t2) =

t2∫

t1

dW =

t2∫

t1

(~Fe · d~xG + ~Le · d~θG

). (8.81)

Now the proposed variational principle reads

δJ = δ

t2∫

t1

Lsdt + δW (t1; t2) = 0 , (8.82)

for all variations in η, φ, ~xG, and ~θG subject to the usual restrictions.Following the standard procedure in the calculus of variations, we arrive at the mod-

ified equations of motion for the body:∫

S

∫p~n dS −Mg~e3 + ~Fe = M~xG , (8.83)

∫

S

∫p (~r × ~n) dS + ~Le = ~I ⊗ ~θG . (8.84)

Remark 4: Finally, we discuss the extension of the above variation principle to thedescription of phenomena as overturning waves. The key difference is that the single-valued free surface elevation η is replaced by the three-component position vector ~xF todescribe the free surface profile. With (8.55-8.60) unaltered, the invariance of J withrespect to a variation in φ now yields the modified kinematic free surface condition

~xF · ~n =∂φ

∂non F , (8.85)

which includes phenomena as overturning waves, and permits a Lagrangian descriptionof the free surface:

D~xF

Dt= ∇φ on F . (8.86)

8.4.2 A Hamiltonian formulation for wave-body interactions

Next, the Hamiltonian theory for the classical water-wave problem is extended to theinteraction with bodies floating in or below the free surface.

For the body we define the ‘coordinate’ vector ~ξ as the combination of the positionvector ~xG and the orientation vector ~θG:

~ξ ≡[

~xG

~θG

]. (8.87)

Similarly, the ‘normal’ vector ~ν along the wetted body surface S is defined as

~ν ≡[

~n~r × ~n

], (8.88)


and the diagonal ‘mass’ matrix M is defined as

diag (M) ≡ [M, M, M, I1, I2, I3]T

. (8.89)

With the above definitions the Lagrangian for the system can be written as — con-fer (8.55-8.59) —

Ls = −∫∫

Ω(t)

∫ (∂φ

∂t+

12

(∇φ · ∇φ) + gz

)dΩ +

12M~ξ · ~ξ −Mg~e3 · ~ξ (8.90)

Taking into account the evolution of the fluid domain, and integrating by parts, we maywrite

Ls = − ∂

∂t

∫∫

Ω(t)

∫φ dΩ +

∫

F

∫φ

∂η

∂tdx dy +M~ξ · ~ξ +

∫

S

∫φ

(~ξ · ~ν

)dS

−∫∫

Ω(t)

∫ (12

(∇φ · ∇φ) + gz

)dΩ− 1

2M~ξ · ~ξ −Mg~e3 · ~ξ , (8.91)

where ~ξ · ~ν is the normal velocity of a point on S. The first term on the right-hand sideof (8.91) integrates out to the end points of the time interval considered.

Considering (8.91) we may put formally

Ls (ηs, ηs) = πsηs −Hs (ηs, πs) (8.92)

with the following definitions: the canonical coordinates are given by

ηs = (ηf , ~ηb) with ηf ≡ η and ~ηb ≡ ~ξ (8.93)

i.e. the free surface elevation and the body position and orientation.The canonical conjugate momenta are given by

πs = (πf , ~πb) with πf ≡ Φ and ~πb ≡M~ξ +∫

S

∫φ~ν dS (8.94)

i.e. the free surface potential Φ ≡ [φ]z=η and the rigid body impulse plus a momentumcontribution of the velocity potential over the wetted body surface.9

The Hamiltonian is given by

Hs =∫∫

Ω(t)

∫ (12

(∇φ · ∇φ) + gz

)dΩ +

12M~ξ · ~ξ +Mg~e3 · ~ξ (8.95)

9In fluid mechanics this integral contribution is known as the Kelvin impulse, see also Benjamin(1987). A physical interpretation is given by Lamb (1932) in his discussion of the ‘impulsive generationof motion’. As Lamb says, a sudden alteration in the motion of a fluid may take place “when a solidimmersed in the fluid is suddenly set in motion”. The reverse case is, of course, also true; a suddenchange in the fluid motion causes a transfer of momentum from the fluid to the solid. The resultant‘impulsive pressure’ can be shown to satisfy Laplace’s equation, and from its formal definition it is clearthat, for our problem, this quantity is exactly the velocity potential. The surface integral obviouslyaccounts for the transfer of momentum (or energy) from the fluid to the floating body and vice versa.See also Lamb’s discussion of the motion of solids through an infinite liquid.


i.e. the total energy of the system.Then, with φ satisfying the boundary value problem

∇2φ = 0 in Ω (t) , (8.96)∂φ

∂n= ~ξ · ~ν on S , (8.97)

∂φ

∂n= 0 on B , (8.98)

φ (x, y, z) = Φ (x, y) on F , (8.99)

we can establish the following theorem:

Theorem 4 : Hamiltonian formulation for the wave-body problemThe equations of motion for gravity driven water waves interacting with bodies floating

in or below the free surface describe an infinite dimensional Hamiltonian system in thecanonically conjugate variables πs and ηs and with the total energy Hs as Hamiltonian;the canonical equations

∂t

(πs

ηs

)=

(0 −11 0

) (δπsHs

δηsHs

)(8.100)

are equivalent with the nonlinear free surface conditions (8.70) and (8.72) and the equa-tions of motion for the body (8.75-8.76).

The first canonical equation ∂tπs = −δηsHs is shorthand for

∂πf

∂t= −δHs

δηf, (8.101)

and

∂~πb

∂t= −δHs

δ~ηb. (8.102)

Similarly, the second canonical equation ∂tηs = δπsHs is shorthand for

∂ηf

∂t=

δHs

δπf, (8.103)

and

∂~ηb

∂t=

δHs

δ~πb. (8.104)

Proof of (8.103): Keeping η fixed and varying φ such that its variation δφ correspondsto a change in the free surface potential Φ, the first variation in Kf reads

δKf (δΦ) =∫∫

Ω(t)

∫∇φ · ∇δφ dΩ =

∫

∂Ω

∫∂φ

∂nδφ dS −

∫∫

Ω(t)

∫∇2φ δφ dΩ , (8.105)

from which we obtain

δΦKf =

∂φ

∂n

1 +

(∂η

∂x

)2

+(

∂η

∂y

)21/2

F

. (8.106)


Then, with δΦPf = 0 and δΦHb = 0, and with (8.67) it is seen that (8.103) is equivalentwith the kinematic free surface condition

∂η

∂t+

∂η

∂x

∂φ

∂x+

∂η

∂y

∂φ

∂y=

∂φ

∂zon F . (8.107)

Proof of (8.101): Here we must be aware of the fact that a variation δη in the freesurface elevation naturally leads to a variation in the free surface potential Φ. To firstorder we have

φ (η + δη) = φ (η) + δη

(∂φ

∂z

)

F

≡ Φ + δηΦ , (8.108)

where

δηΦ = δη

(∂φ

∂z

)

F

. (8.109)

Then, the effect of a variation δη in Kf is found to be

δKf (δη) + δKf (δηΦ) = δη

[12

(∇φ · ∇φ)]

F

, (8.110)

hence

δηKf + δΦKf

(∂φ

∂z

)

F

=[12

(∇φ · ∇φ)]

F

. (8.111)

With δηPf = gη and δηHb = 0, and the relation

∂Φ∂t

=(

∂φ

∂t+

∂φ

∂z

∂η

∂t

)

F

, (8.112)

it follows that (8.101) is equivalent with the dynamic free surface condition

∂φ

∂t+

12

(∇φ · ∇φ) + gz = 0 on F . (8.113)

Proof of (8.104): This is more complicated, due to the fact that the canonical momen-

tum of the body depends on both ~ξ and φ along S, see (8.94). Thus, a variation in ~πb

corresponds to variations δ~ξ and δφ according to

δ~πb = Mδ~ξ +∫

S

∫δφ~ν dS . (8.114)

Using (8.96), it can be shown that a variation in the potential φ along S yields the firstvariation in Kf , given by

δKf (δφ) =∫

S

∫∂φ

∂nδφ dS . (8.115)

With (8.97) this can be written as

δKf (δφ) =∫

S

∫~ξ · ~ν δφ dS . (8.116)


It is obvious that the first variation of Kb due to a variation in ~ξ is given by

δKb

(δ~ξ

)= M~ξ · δ~ξ . (8.117)

With the nullity of the variational derivatives of Pf and Pb with respect to φ and ~ξ, weobtain

δHs (δ~πb) =

Mδ~ξ +

∫

S

∫δφ~ν dS

· ~ξ = δ~πb · ~ηb , (8.118)

which completes the proof of (8.104).

Proof of (8.102): Note that a variation in ~ξ leads to a variation in Hf , according to

δHf

(δ~ξ

)=

∫

S

∫ (12

(∇φ · ∇φ) + gz

) (δ~ξ · ~ν

)dS . (8.119)

The first variation in Hb due to a variation in ~ξ is given by

δHb

(δ~ξ

)= Mg~e3 · δ~ξ . (8.120)

It is then easy to show that (8.102) is equivalent with the equations of motion for thefloating body:

∫

S

∫p~ν dS −Mg~e3 = M~ξ . (8.121)

This completes the proof of theorem 4.

Remark 5: The left-hand side of (8.102) represents the change of momentum of thebody, i.e. it is the force related to the change of the body velocity. The right-handside of (8.102) represents the force due to a change in the position of the body. Fora single body immersed in an infinite fluid — in the absence of external forces — thetotal energy Hs is independent of the body position; for this special case the right-handside expression equals zero. This leads us to d’Alembert’s10 paradox, stating that nohydrodynamic force acts on a body moving with constant translational velocity in aninfinite and inviscid fluid which is free of vorticity.

Remark 6: In the canonical momentum for the body — see (8.94) — the integral overthe wetted body surface S accounts for the transfer of momentum from the fluid to thebody (and vice versa) through the velocity potential. This transfer becomes more clear

10Alembert, Jean le Rond d’ (1717-1783), French mathematician and philosopher. In his ‘Traite de ladynamique’ (1743) he proved that Newton’s third law (‘to every action there is an equal and oppositereaction’) applies to moving as well as stationary bodies. This is known as d’Alembert’s principle.


if we consider the product of the canonical body coordinate and the canonical bodymomentum:

~πb · ~ηb =

M~ξ +

∫

S

∫φ~ν dS

· ~ξ

= M~ξ · ~ξ +∫

S

∫φ

∂φ

∂ndS , (8.122)

where we used (8.97). The first term on the right-hand side equals twice the kineticenergy of the body as if it moved autonomously. The second term represents the energytransferred from the fluid to the body. This contribution may be called ‘added energy’,and obviously it is related to concepts like added mass and damping for a body oscillatingwith small amplitude in a fluid.


8.5 Bibliography

Several relevant publications which have not been cited in the text are also included here.

Abarbanel, H.D.I., Brown, R., and Yang, Y.M. 1988. Hamiltonian formulationof inviscid flows with free boundaries. The Physics of Fluids 31:2802-2809.

Aranha, J.A.P., and Pesce, C.P. 1989. A variational method for water wave radi-ation and diffraction problems. Journal of Fluid Mechanics 204:135-157.

Arnol’d, V.I. 1978. Mathematical Methods of Classical Mechanics. Springer-Verlag.

Bateman, H. 1932. Partial Differential Equations of Mathematical Physics. CambridgeUniversity Press.

Becker, J.M., and Miles, J.W. 1991. Standing radial cross-waves. Journal of FluidMechanics 222:471-499.

Benjamin, T.B. 1984. Impulse, flow force and variational principles. IMA Journal ofApplied Mathematics 32:3-68.

Benjamin, T.B. 1987. Hamiltonian theory for motions of bubbles in an infinite liquid.Journal of Fluid Mechanics 181:349-379.

Benjamin, T.B., and Olver, P.J. 1982. Hamiltonian structure, symmetries andconservation laws for water waves. Journal of Fluid Mechanics 125:137-185.

Bridges, T.J., and Dias, F. 1989. Group-theoretic considerations lead to new solu-tions of the water wave problem. The Fourth International Workshop on Water Wavesand Floating Bodies, Øystese, Norway.

Broer, L.J.F. 1974. On the Hamiltonian theory of surface waves. Applied ScientificResearch 29:430-446.

Broer, L.J.F. 1975. Approximate equations for long water waves. Applied ScientificResearch 31:377-395.

Broer, L.J.F., van Groesen, E.W.C., and Timmers, J.M.W. 1976. Stable modelequations for long water waves. Applied Scientific Research 32:619-636.

Clebsch, A. 1859. Uber die Integration der hydrodynamischen Gleichungen. Journalfur Reine und Angewandte Mathematik 56:1–10.

Courant, R., and Hilbert, D. 1953. Methods of Mathematical Physics I. Inter-science Publishers.

Courant, R., and Hilbert, D. 1960. Methods of Mathematical Physics II. Inter-science Publishers.

Creamer, D.B., Henyey, F., Schult, R., and Wright, J. 1989. Improved linearrepresentation of ocean surface waves. Journal of Fluid Mechanics 205:135-161.

van Daalen, E.F.G., and van Groesen, E.W.C. 1993. A Hamiltonian formulationfor water waves and floating bodies. Submitted to the Journal of Ship Research.

Dingemans, M.W. 1993. Water Wave Propagation over Uneven Bottoms. World Sci-entific.

Finlayson, B.A. 1972. The Method of Weighted Residuals and Variational Principles.Academic Press.


Gardner, C.S. 1971. Korteweg-de Vries equation and generalizations IV. The Korteweg-de Vries equation as a Hamiltonian system. Journal of Mathematical Physics 12(8):1548-1551.

Gelfand, I.M., and Fomin, S.V. 1963. Calculus of Variations. Prentice-Hall.

Goldstein, H. 1980. Classical Mechanics. Addison-Wesley.

van Groesen, E.W.C. 1978. Variational Methods in Mathematical Physics. Ph.D.thesis, Eindhoven University. Eindhoven, The Netherlands.

van Groesen, E.W.C., van Beckum, F.P.H., and Valkering, T.P. 1990. Decayof travelling waves in dispersive Poisson systems. Journal of Applied Mathematics andPhysics (ZAMP) 41:501-523.

Henyey, F.S. 1983. Hamiltonian description of stratified fluid dynamics. The Physicsof Fluids 26:40-47.

Katopodes, N.D., and Dingemans, M.W. 1989. Hamiltonian approach to surfacewave models. HYDROCOMP’89, Dubrovnik, Yugoslavia. Elsevier.

Kelvin, Lord. 1849. On the vis-viva of a liquid in motion. Cambridge and DublinMathematical Journal.


Lanczos, C. 1970. The Variational Principles of Mechanics. University of TorontoPress.

Landau, L.D., and Lifshitz, E.M. 1976. Mechanics. Volume 1 of Course of Theo-retical Physics. Pergamon Press.

Luke, J.C. 1967. A variational principle for a fluid with a free surface. Journal of FluidMechanics 27(2):395-397.

Mackay, R.S., and Saffman, P.G. 1986. Stability of water waves. Proceedings ofthe Royal Society of London, Series A, 406:115-125.

McIntyre, M.E., and Shepherd, T.G. 1987. An exact local conservation theorem forfinite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonianstructure and on Arnol’d’s stability theorems. Journal of Fluid Mechanics 181:527-565.

Milder, D.M. 1977. A note regarding ‘On Hamilton’s principle for surface waves’.Journal of Fluid Mechanics 83(1):159-161.

Milder, D.M. 1982. Hamiltonian dynamics of internal waves. Journal of Fluid Me-chanics 119:269-282.

Milder, D.M. 1990. The effect of truncation on surface-wave Hamiltonians. Journalof Fluid Mechanics 217:249-262.

Miles, J.W. 1976. Nonlinear surface waves in closed basins. Journal of Fluid Mechanics75(3):419-448.

Miles, J.W. 1977. On Hamilton’s principle for surface waves. Journal of Fluid Me-chanics 83:153-158, with a note by D.M. Milder (1977).

Miles, J.W. 1981. Hamiltonian formulations for surface waves. Applied Scientific Re-search 37:103-110.

Miles, J.W. 1986a. Weakly nonlinear waves in a stratified fluid: a variational formu-


lation. Journal of Fluid Mechanics 172:499-512.

Miles, J.W. 1986b. Weakly nonlinear Kelvin-Helmholtz waves. Journal of Fluid Me-chanics 172:513-529.

Miles, J.W., and Henderson, D. 1990. Parametrically forced surface waves. AnnualReview of Fluid Mechanics 22:143-165.

Miloh, T. 1984. Hamilton’s principle, Lagrange’s method and ship motion theory. Jour-nal of Ship Research 28(4):229-237.

Olver, P.J. 1982. A nonlinear Hamiltonian structure for the Euler equations. Journalof Mathematical Analysis and Applications 89:233-250.

Radder, A.C. 1992. An explicit Hamiltonian formulation of surface waves in water offinite depth. Journal of Fluid Mechanics 237:435-455.

Saffman, P.G. 1985. The superharmonic instability of finite-amplitude water waves.Journal of Fluid Mechanics 159:169-174.

Saffman, P.G. 1987. Application of Hamiltonian methods to the structure and stabil-ity of water waves of permanent form. IUTAM Symposium on Nonlinear Water Waves,Tokyo, Japan. Springer-Verlag.

Salmon, R. 1983. Practical use of Hamilton’s principle. Journal of Fluid Mechanics132:431-444.

Salmon, R. 1988. Hamiltonian fluid mechanics. Annual Review of Fluid Mechanics20:225-256.

Seliger, R.L., and Whitham, G.B. 1968. Variational principles in continuum me-chanics. Proceedings of the Royal Society of London, Series A, 305:1-25.

Weinstock, R. 1974. Calculus of Variations. Dover Publications.

Whitham, G.B. 1965. A general approach to linear and nonlinear dispersive waves us-ing a Lagrangian. Journal of Fluid Mechanics 22:273-283.

Whitham, G.B. 1967. Variational methods and applications to water waves. Proceed-ings of the Royal Society of London, Series A, 299:6-25.

Zakharov, V.E. 1968. Stability of periodic waves of finite amplitude on the surface ofa deep fluid. Journal of Applied Mechanics and Technical Physics 2:190-194.

Zufiria, J.A. 1987a. Weakly nonlinear non-symmetric gravity waves on water of finitedepth. Journal of Fluid Mechanics 180:371-385.

Zufiria, J.A. 1987b. Non-symmetric gravity waves on water of infinite depth. Journalof Fluid Mechanics 181:17-39.

Zufiria, J.A. 1987c. Symmetry breaking in periodic and solitary gravity-capillary waveson water of finite depth. Journal of Fluid Mechanics 184:183-206.

Zufiria, J.A. 1988. Oscillatory spatially periodic weakly nonlinear gravity waves ondeep water. Journal of Fluid Mechanics 191:341-372.

Zufiria, J.A., and Saffman, P.G. 1986a. An example of stability exchange in aHamiltonian wave system. Studies in Applied Mathematics 74:85-91.

Zufiria, J.A., and Saffman, P.G. 1986b. The superharmonic instability of finite-amplitude surface waves on water of finite depth. Studies in Applied Mathematics 74:259-


266.

Zwartkruis, T.J.G. 1991. Computation of solitary wave profiles described by a Hamil-tonian model for surface waves. Part 1: Final report; Part 2: Appendices. ECMI-report,Eindhoven University of Technology, Eindhoven, The Netherlands.


Dawn was spreading over the earth when they reached the throng of heroes;and the youths marvelled to behold the mighty fleece, which gleamed like thelightning of Zeus. And each one started up eager to touch it and clasp it in hishands. But the son of Aeson restrained them all, and threw over it a mantlenewly-woven; and he led the maiden to the stern and seated her there, andspake to them all as follows:

“No longer now, my friends, forbear to return to your fatherland. For nowthe task for which we dared this grievous voyage, toiling with bitter sorrowof heart, has been lightly fulfilled by the maiden’s counsels. Her — for suchis her will — I will bring home to be my wedded wife; do ye preserve her,the glorious saviour of all Achaea and of yourselves. For of a surety, I ween,will Aeetes come with his host to bar our passage from the river into the sea.But do some of you toil at the oars in turn, sitting man by man; and halfof you raise your shields of oxhide, a ready defence against the darts of theenemy, and guard our return. And now in our hands we hold the fate of ourchildren and dear country and of our aged parents; and on our venture allHellas depends, to reap either the shame of failure or great renown.”

Thus he spake, and donned his armour of war; and they cried aloud,wondrously eager. And he drew his sword from the sheath and cut the hawsersat the stern. And near the maiden he took his stand ready armed by thesteersman Ancaeus, and with their rowing the ship sped on as they straineddesperately to drive her clear of the river.

Argonautica, Book IV, Verses 183-211.

Chapter 9

Symmetries and ConservationLaws

Thence they entered the deep stream of Rhodanus which flows into Eri-danus; and where they meet there is a roar of mingling waters. Now thatriver, rising from the ends of the earth, where are the portals and mansionsof Night, on one side bursts forth upon the beach of Ocean, at another poursinto the Ionian sea, and on the third through seven mouths sends its streamto the Sardinian sea and its limitless bay1. And from Rhodanus they enteredstormy lakes, which spread throughout the Celtic mainland of wondrous size;and there they would have met with an inglorious calamity; for a certainbranch of the river was bearing them towards a gulf of Ocean which in ig-norance they were about to enter, and never would they have returned fromthere in safety. But Hera leaping forth from heaven pealed her cry from theHercynian rock; and all together were shaken with fear of her cry; for ter-ribly crashed the mighty firmament. And backward they turned by reason ofthe goddess, and noted the path by which their return was ordained. Andafter a long while they came to the beach of the surging sea by the devisingof Hera, passing unharmed through countless tribes of the Celts and Ligyans.For round them the goddess poured a dread mist day by day as they fared on.


1Apollonius seems to have thought that the Po, the Rhone, and the Rhine are all connected together.

157

158 CHAPTER 9. SYMMETRIES AND CONSERVATION LAWS

9.1 Introduction

This chapter is devoted to conservation laws for the problem of gravity driven waterwaves and bodies floating freely in or below the free surface. Our main objective is tofind constants of the motion for the nonlinear wave-body problem. The present treatmentis an elaborated and more detailed version of a paper written recently by van Daalenand van Groesen (1993).

The first systematic account of symmetries2 and conservation laws for the water-waveproblem was given ten years ago in an excellent paper written by Benjamin and Olver(1982). In their quest for constants of the motion, the Hamiltonian structure of wa-ter waves3 is exploited successfully; the techniques are based on some results of Olver(1980b, 1986) for the possible sets of integral invariants of evolution equations under thetransformations of a Lie group of symmetries.4

Many of the results thus obtained were already well-known; see, for instance, Lax(1968), Miura, Gardner, and Kruskal (1968), Benjamin and Mahony (1971), Benjamin(1974), Longuet-Higgins (1974, 1975, 1980), Longuet-Higgins and Fenton (1974), andBroer, van Groesen and Timmers (1976). Considering free surface waves on a layerof water of infinite depth, Benjamin and Olver5 proved the existence of exactly eightconserved densities — i.e. quantities that are not exact differentials but whose integralsover any horizontal domain depend on boundary values only — for two-dimensional flow6

without surface tension.7 Some of these densities are related to well-known quantities,for example the mass and energy; others correspond to less familiar quantities, such asthe angular momentum and another similar-looking quantity.

Because of the generality of Benjamin and Olver’s analysis and the length of theirderivations, the particular results mentioned above were deduced in an alternative, moresimple, and direct way by Longuet-Higgins (1983) for two-dimensional, inviscid, irrota-tional flow under gravity. The method employed was already referred to in Benjaminand Olver’s paper, but full credit must be given to Longuet-Higgins for presenting aclear and simple proof of the existence of eight invariants for this particular problem. Inaddition, most conservation laws were generalized to rotational motion or vorticity (i.e.no potential) flow.

The primary aim of the present treatment is to generalize the aforementioned results— for water waves only — to the more complex problem of wave-body interactions. Inparticular, an attempt is made towards the extension of the water-wave conservationlaws to freely floating bodies in two-dimensional free surface potential flow.

For future reference, the fundamental results from Benjamin and Olver’s analysis aresummarized in section 9.2. For the deeper question of completeness — in the sense thatall basic conservation laws for the water-wave problem are covered indeed — the reader

2Throughout this chapter, the word ‘symmetry’ is used to denote an invariance of a given systemunder certain transformations. For example, if a system is completely self-contained, with only internalforces between the particles, then the system can be moved as a rigid whole without affecting the forcesor subsequent motion; the system is said to be invariant or symmetric under a rigid displacement.

3This special structure of water waves is discussed in chapter 8.4For a more recent application of these techniques to the Boussinesq model for wave motions we refer

to Benjamin (1986a).5From now on, ‘Benjamin and Olver’ refers to their 1982-paper, unless stated otherwise.6For the three-dimensional case twelve conserved densities were found.7When surface tension is operative, the number of conserved densities is reduced by one.

9.2. CONSERVED DENSITIES FOR WATER WAVES 159

will of course have to study the arguments of Benjamin and Olver, who in turn refer toa subsequent paper written by Olver (1983).

In section 9.3 Longuet-Higgins’ approach8 is adopted in the treatment of the wave-body problem; it appears that seven out of eight fluid quantities have a direct counterpartfor a rigid body. In the absence of external forces and moments, one arrives at extendedconservation laws by a straightforward method; explicit proofs, based on the equationsof motion of the body and the boundary condition on the wetted body surface, arepresented.

The remaining eighth fluid quantity and its body counterpart are extensively dealtwith in section 9.4. The main difficulty in this discussion is the lack of an unambigu-ous physical interpretation of this quantity, which seems to be related to the angularmomentum.9 Here it is surmised (and argued) that the presence of a rigid body in ef-fect breaks the underlying symmetry and, as a result, prohibits the generalization of thecorresponding conservation law to the wave-body problem.

The key assumption in Longuet-Higgins’ proofs of the conservation laws for waterwaves is the vanishing of the Bernoulli pressure along the boundary of the fluid domain.This implies the supposition that the fluid extends to infinity in downward direction,which is consistent with Benjamin and Olver’s analysis for the infinite depth case. Forpractical purposes however, this is not a reasonable assumption; therefore the effects ofimpermeable fixed boundaries are discussed in section 9.5.

Finally, in section 9.6 we demonstrate that the validity of the theory is supported bynumerical results obtained with our panel method for nonlinear wave-body interactions.The theoretical predictions with respect to the constants of the motion — including thesymmetry-breaking effect of an impermeable bottom — are confirmed by the computa-tions for a special, yet simple, test configuration.

The reader who is primarily interested in practical applications is advised to skip sec-tion 9.2 and proceed directly to section 9.3. Further, it is remarked that the approach insection 9.4 is rather theoretical, contrary to the analyses in sections 9.5 and 9.6, wherematters of a more practical nature are discussed.

9.2 Conserved densities for water waves

In this section we present a summary of Benjamin and Olver’s treatise on symmetriesand conservation laws for the water-wave problem. Only the two-dimensional infinitedepth case is considered here; let x and z be rectangular coordinates in the plane ofmotion, with z vertically upwards, see Figure 9.1.As usual, let φ denote the velocity potential, which is introduced under the assumptionsof an ideal fluid and an irrotational flow. The well-known governing equations for freesurface flow under the influence of gravity read

∇2φ = 0 for z < η (x; t) , (9.1)ηt + ηxφx = φz at z = η (x; t) , (9.2)

φt +12

(∇φ · ∇φ) + gz = σκ at z = η (x; t) , (9.3)

‖∇φ‖ → 0 for r =(x2 + z2

)1/2 →∞ , (9.4)8From now on, ‘Longuet-Higgins’ refers to his 1983-paper, unless stated otherwise.9Benjamin and Olver refer to the corresponding density as the ‘virial’ density.


Figure 9.1: Two-dimensional water waves: the infinite depth case.

where suffixes denote partial differentiation. It is assumed that the free surface elevationη is a single-valued function of x for all times t. Surface tension is included in the dynamicfree surface condition (9.3) through σ, the coefficient of surface tension, and the surfacecurvature κ, given by

κ =∂

∂x

(ηx

(1 + η2x)1/2

)=

ηxx

(1 + η2x)3/2

. (9.5)

The following two theorems due to Benjamin and Olver state that the full symmetrygroup for the two-dimensional water-wave problem is generated by nine one-parametersubgroups. This means that any symmetry with a continuous connection to the identitymap can be constructed by applying a limited number of the component symmetries insuccession.10

In the first theorem the full symmetry group — or the Lie algebra of symmetries — isdefined as the linear space spanned by nine vector fields; each vector field may be regardedas a first order differential operator acting on smooth functions f (x, z, t, φ) : IR4 → IR4.

Theorem 5 : Symmetry-generating vector fields(Benjamin and Olver, 1982)The Lie algebra of infinitesimal symmetries for the two-dimensional water-wave problemin the absence of surface tension — i.e. (9.1-9.4) with σ = 0 — is spanned by thefollowing nine vector fields:

~v1 = ∂x , (9.6)

~v2 = ∂t , (9.7)

~v3 = ∂φ , (9.8)

10This may be compared to the group of rotations in IR3 being generated by the three one-parametersubgroups of rotations about three cartesian axes. Similarly, the group of translations in IR3 is generatedby the three one-parameter subgroups of translations along three cartesian axes.


~v4 = ∂z − gt∂φ , (9.9)

~v5 = t∂x + x∂φ , (9.10)

~v6 = t∂z +(

z − 12gt2

)∂φ , (9.11)

~v7 = gt2∂z − t∂t +(

φ + 2gtz − 13g2t3

)∂φ , (9.12)

~v8 =(

z +12gt2

)∂x − x∂z + gtx∂φ , (9.13)

~v9 = x∂x + z∂z +12t∂t +

32φ∂φ . (9.14)

Each of the symmetries generated by these vector fields is unaffected by the additional sur-face tension term in the dynamic free surface condition (9.3); hence, no extra symmetryis induced when surface tension is operative.

The above vector fields are the infinitesimal generators of one-parameter groups of dif-feomorphisms, i.e. continuous maps Gi : IR4 → IR4 transforming solutions (x, z, t, φ)into new solutions to the water-wave problem (9.1-9.4). The diffeomorphisms are obtain-able by integrating a system of ordinary differential equations involving the dependentvariables x, z, t, and φ, and their derivatives with respect to the group parameters.

An alternative method introduced by Benjamin and Olver to establish symmetrygroups for the present problem is based on the so-called prolongation theory; each vectorfield ~vj generates a one-parameter symmetry group Gj , as stated in the next theorem.

Theorem 6 : Lie algebra of symmetries (Benjamin and Olver, 1982)The full symmetry group for the two-dimensional water-wave problem in the absence ofsurface tension — i.e. (9.1-9.4) with σ = 0 — is generated by the following nine one-parameter subgroups:

Horizontal translation:

G1 : (x + ε1, z, t, φ) , (9.15)

Time translation:

G2 : (x, z, t + ε2, φ) , (9.16)

Variation of base-level for potential:

G3 : (x, z, t, φ + ε3) , (9.17)

Vertical translation:

G4 : (x, z + ε4, t, φ− ε4gt) , (9.18)

Horizontal Galilean boost:

G5 :(

x + ε5t, z, t, φ + ε5x +12ε25t

), (9.19)


Vertical Galilean boost:

G6 :(

x, z + ε6t, t, φ + ε6

(z − 1

2gt2

)+

12ε26t

), (9.20)

Vertical acceleration:

G7 :(

x, z +12gt2

(1− λ−2

7

), λ−1

7 t,

λ7

φ + gtz

(1− λ−2

7

)+

16g2t3

(1− 3λ−2

7 + 2λ−47

)), (9.21)

Gravity-compensated rotation:

G8 :(

x cos ε8 +(

z +12gt2

)sin ε8,

−x sin ε8 +(

z +12gt2

)cos ε8 − 1

2gt2, t,

φ + gt

x sin ε8 +

(z +

12gt2

)(1− cos ε8)

), (9.22)

Scaling:

G9 :(λ9x, λ9z, λ

1/29 t, λ

3/29 φ

). (9.23)

When surface tension is operative (σ > 0), all the above groups except G7 and G9 remainsymmetries; in this case, the following ‘combination’ of G7 and G9 — in effect, G7 =G9 G7 with λ7 = λ−1

7 = λ−19 — remains a symmetry:

Scaled acceleration:

G7 :(

λ7x, λ7

z +

12gt2

(1− λ2

7

), λ

3/27 t,

λ1/27

φ + gtz

(1− λ2

7

)+

16g2t3

(1− 3λ2

7 + 2λ47

)). (9.24)

In the above expressions, each εi ∈ IR denotes an additive group parameter, and eachλj = eεj > 0 denotes a multiplicative group parameter.

The given subgroups may be considered to act geometrically on the four-dimensionalspace with coordinates (x, z, t, φ) for the two-dimensional problem. Each subgroup in-duces a transformation in the space of solutions, in effect transforming the graphs ofthe free surface and the velocity potential. The key point is that transforming a givensolution by any of the symmetries produces a continuous family of other solutions. Forinstance, the subgroup G1 induces a horizontal translation of the water-wave problem:

η (x; t) , φ (x, z; t) −→ η (x− ε1; t) , φ (x− ε1, z; t) . (9.25)

Similarly, G2 represents a time shift t → t − ε2; G3 has a bearing on the fact that thepotential φ is determined up to an additive constant; G4 induces a vertical translation ofthe water-wave problem. The Galilean boosts G5 and G6 represent the effects caused byframes of reference moving at a constant speed in the horizontal and vertical directionrespectively. The group G7 represents the effects due to a frame that is accelerating


uniformly in vertical direction, thus modifying the effective gravity constant.11. To un-derstand G8, consider the special case where g = 0; since there is no preferred directionthen, any solution will remain a solution after being rotated about an arbitrary point inthe (x, z)-plane. Finally, the transformations in the scaling group G9 are evident fromdimensional considerations.

Noether’s theorem12 (1918) states that every one-parameter group of symmetries for avariational problem13 determines a conservation law satisfied by solutions of the corre-sponding Euler-Lagrange equations; a more comprehensive description of this fundamen-tal result can be found in, for instance, Goldstein (1980). This principle, in an adaptedform for free-boundary problems due to Olver (1980a), was used by Benjamin and Olverto obtain the conserved densities listed in the following theorem.

Theorem 7 : Conserved densities (Benjamin and Olver, 1982)The two-dimensional water-wave problem in the absence of surface tension — i.e. (9.1-9.4) with σ = 0 — has the following eight conserved densities:

T1 = −ηxΦ , (9.26)

T2 = H =12Φηt +

12gη2 , (9.27)

T3 = η , (9.28)T4 = Φ + gtT3 , (9.29)T5 = xη − tT1 , (9.30)

T6 =12η2 − tT4 +

12gt2T3 , (9.31)

T7 = (η − xηx)Φ− t (4T2 − 7gT6) +72gt2T4 − 7

6g2t3T3 , (9.32)

T8 = (x + ηηx)Φ + gtT5 +12gt2T1 , (9.33)

where Φ(x; t) ≡ φ (x, z = η (x; t) ; t) is the restriction of the velocity potential to the freesurface, and H is the Hamiltonian density.

Each density Ti is generated by the corresponding vector field ~vi, apart from T7, whichis induced by ~v9 − (7/2)~v7. When surface tension is operative, i.e. σ > 0, then ~v9 is nolonger a symmetry, and consequently the density T7 is no longer conserved. Furthermore,the appropriate term is then added to the Hamiltonian density:

T2 = H =12Φηt +

12gη2 + σ

(1 + η2

x

)1/2 − 1

. (9.34)

Apart from these two changes, the results remain as above.

The first density is easily associated with the horizontal momentum, by noting that+∞∫

−∞T1dx = − [ηΦ]+∞−∞ +

+∞∫

−∞ηΦxdx =

+∞∫

−∞ηΦxdx , (9.35)

11Hence, for the appropriate choice of ε7, the transformed graphs of η (x; t) and φ (x, z; t) show theevolution of (fictitious) water waves on the moon!

12Noether, Emmy (1882-1935), German mathematician. She was one of the leading mathematiciansof this century, and has been properly described as ‘the greatest of women mathematicians’.

13Note that both the water-wave problem and the wave-body problem can be described by a variationalprinciple, see chapter 8.


where it is assumed that the free surface elevation η vanishes for |x| → ∞.

Clearly, T2 corresponds to the energy, and T3 is a density of mass. Further physicalconnotations due to Benjamin and Olver are as follows: T4 is to be interpreted as thevertical momentum density; T5 and T6 correspond to the horizontal and vertical positionof the mass centroid; T8 is an angular momentum density, and T7 is a related densitycorresponding to an unfamiliar quantity, also named a ‘virial’ density.

9.3 Conservation laws for the wave-body problem

The above results for water waves were deduced in a more direct and simple way byLonguet-Higgins; in the next subsections his approach is followed to construct conserva-tion laws for the more complex wave-body problem.

9.3.1 Invariants for the two-dimensional case

The notation used here differs but slightly from the notation used by Longuet-Higgins.The subscripts f , b, and s refer to the fluid, the floating body, and the fluid-body system14

respectively, see Figure 9.2.

Figure 9.2: Two-dimensional wave-body system.

As in the previous section, let x and z be rectangular coordinates in the plane of motion,with z vertically upwards, and let φ denote the velocity potential, which is introducedunder the assumptions of an ideal fluid and an irrotational flow. If C denotes any simpleclosed contour bounding a domain Df of the fluid — see Figure 9.2 — then, using Green’s

14From now on, the system consisting of the fluid and the floating body is referred to as ‘the wave-bodysystem’ or (simply) ‘the system’.

9.3. CONSERVATION LAWS FOR THE WAVE-BODY PROBLEM 165

theorem15

∫

Df

∫(Pz −Qx) dx dz =

∫

C

(P dx + Qdz) , (9.36)

where Pz = ∂P/∂z and Qx = ∂Q/∂x, Longuet-Higgins defined the following eight inte-gral quantities for the fluid:

Mf =∫

Df

∫dx dz =

∫

C

z dx , (9.37)

Mf xf =∫

Df

∫x dx dz =

∫

C

xz dx , (9.38)

Mf zf =∫

Df

∫z dx dz =

∫

C

12z2 dx , (9.39)

If =∫

Df

∫φx dx dz = −

∫

C

φdz , (9.40)

Jf =∫

Df

∫φz dx dz =

∫

C

φdx , (9.41)

Af =∫

Df

∫[(xφ)z − (zφ)x] dx dz =

∫

C

φ (x dx + z dz) , (9.42)

Bf =∫

Df

∫[(xφ)x + (zφ)z] dx dz =

∫

C

φ (z dx− x dz) , (9.43)

Hf = Kf + Pf , (9.44)

with Kf and Pf defined as

Kf =∫

Df

∫12

(φ2

x + φ2z

)dx dz =

∫

C

12φ (φz dx− φx dz) , (9.45)

Pf =∫

Df

∫gz dx dz =

∫

C

12gz2 dx = Mfgzf . (9.46)

Thus, Mf is familiar as the fluid mass (the fluid density ρf being taken as unity); xf

and zf are the coordinates of the centre of fluid mass; If and Jf are the two componentsof the fluid momentum; Af is the angular fluid momentum; Bf an unfamiliar fluidquantity, looking similar to Af ; and Hf is the fluid energy, being the sum of the kineticfluid energy Kf and the potential fluid energy Pf . In (9.45) Laplace’s equation for thevelocity potential is used, by continuity.

15Green, George (1793-1841), British mathematician. In 1828 he published for private circulationan essay on electricity and magnetism that contained the important theorem bearing his name. Thistheorem, or its analogue in three dimensions, is also known as the divergence theorem or Gauss’ theorem,for Green’s results were largely overlooked until rediscovered by Lord Kelvin in 1846. The theoremmeanwhile had been discovered by Michel Ostrogradski (1801-1861), and in Russia it bears his name tothis day.


Most of the above fluid quantities have a direct analogue for a rigid body floating in orbelow the free surface. The body mass and moment of inertia about its centre of mass —or centre of gravity — (xb, zb) are denoted by Mb and Nb respectively. The body angleof roll is denoted by θb, the bar indicating that the rolling motion is considered withrespect to the centre of gravity (see Figure 9.2). The wetted part of the body surface isdenoted by S.

With these definitions, the following quantities for the body are defined:

Ib = Mb ˙xb , (9.47)Jb = Mb ˙zb , (9.48)

Ab = Nb˙θb +Mb (xb ˙zb − zb ˙xb) , (9.49)

Hb = Kb + Pb , (9.50)

with Kb and Pb defined as:

Kb =12Mb

(˙x2b + ˙z2

b

)+

12Nb

˙θ2

b , (9.51)

Pb = Mbgzb . (9.52)

Thus, Ib and Jb are the two components of the body momentum; Ab is the angular bodymomentum with respect to the origin — in appendix C we derive expression (9.49); andHb is the body energy, being the sum of the kinetic body energy Kb and the potentialbody energy Pb.

With the fluid quantities (9.37-9.42) and (9.44-9.46), and with the body quantities (9.47-9.52), the quantities for the wave-body system are defined as

Ms = Mf +Mb , (9.53)Msxs = Mf xf +Mbxb , (9.54)Mszs = Mf zf +Mbzb , (9.55)

Is = If + Ib , (9.56)Js = Jf + Jb , (9.57)As = Af +Ab , (9.58)Hs = Hf +Hb . (9.59)

Thus, Ms is the system mass; xs and zs are the coordinates of the centre of systemmass; Is and Js are the two components of the system momentum; As is the angularsystem momentum; and Hs is the system energy, being the sum of the kinetic systemenergy Ks = Kf +Kb and the potential system energy Ps = Pf + Pb. The discussion ofBb and Bs (the analogues of Bf for the body and the system respectively) is deferred tosection 9.4.

Let p denote the pressure given by Bernoulli’s equation

p + φt +12

(φ2

x + φ2z

)+ gz = 0 . (9.60)


Then, assuming that the contour C moves with the fluid, Longuet-Higgins deduced thefollowing identities:

dMf

dt= 0 , (9.61)

Mfdxf

dt= If , (9.62)

Mfdzf

dt= Jf , (9.63)

dIf

dt=

∫

C

p dz , (9.64)

dJf

dt= −

∫

C

p dx−Mfg , (9.65)

dAf

dt= −

∫

C

p (x dx + z dz)−Mfgxf , (9.66)

dHf

dt= −

∫

C

p (φz dx− φx dz) . (9.67)

Proof: Note that since C moves with the fluid,16 which is incompressible, we have ingeneral

d

dt

∫

Df

∫f dx dz =

∫

Df

∫D

Dt[f ] dx dz , (9.68)

where D/Dt denotes material differentiation — i.e. following the motion — and f is afunction of position and time defined in Df .

The proof of (9.61-9.63) follows directly from

D

Dt[1] = 0 , (9.69)

D

Dt[x] = φx , (9.70)

D

Dt[z] = φz . (9.71)

To prove (9.64) and (9.65), note that from (9.60)

D

Dt[φx] = −px , (9.72)

D

Dt[φz] = −pz − g . (9.73)

To prove (9.66), note that from (9.60)

D

Dt[φ] = φt + φ2

x + φ2z = − (p + gz) +

12

(φ2

x + φ2z

). (9.74)

16This means that C is a Lagrangian boundary; its transient shape is determined by the trajectoriesof the outmost fluid particles.


Hence, using (9.70-9.73), it is found that

D

Dt[(xφ)z − (zφ)x] =

D

Dt(xφz − zφx)

=(

Dx

Dtφz − Dz

Dtφx

)+

(x

Dφz

Dt− z

Dφx

Dt

)

= (φxφz − φzφx)− x (pz + g) + zpx

= − (xp)z + (zp)x − gx . (9.75)

Lastly, to prove (9.67), note that since ∇2φ = 0

D

Dt

[12

(φ2

x + φ2z

)+ gz

]=

(φx

Dφx

Dt+ φz

Dφz

Dt

)+ g

Dz

Dt

= −φxpx − φz (pz + g) + gφz

= − (p φx)x − (p φz)z . (9.76)

With (9.47-9.59) and (9.61-9.67), the following identities for the wave-body system canbe proven:

dMs

dt= 0 , (9.77)

Msdxs

dt= Is , (9.78)

Msdzs

dt= Js , (9.79)

dIs

dt=

∫

C

p dz +Mb ¨xb , (9.80)

dJs

dt= −

∫

C

p dx−Mfg +Mb¨zb , (9.81)

dAs

dt= −

∫

C

p (x dx + z dz)−Mfgxf

+Nb¨θb +Mb (xb¨zb − zb ¨xb) , (9.82)

dHs

dt= −

∫

C

p (φz dx− φx dz)

+Mb ( ˙xb ¨xb + ˙zb¨zb) +Nb˙θb

¨θb +Mbg ˙zb . (9.83)

Proof: It is clear that (9.77) follows from (9.53), (9.61), and the fact that Mb is aconstant. Identity (9.78) can be deduced from (9.54), (9.56), (9.47), and (9.62). Sim-ilarly, (9.79) is obtained from (9.55), (9.57), (9.48), and (9.63). Identity (9.80) can bededuced from (9.56), (9.47), and (9.64). Similarly, (9.81) is obtained from (9.57), (9.48),and (9.65). Identity (9.82) follows from (9.58), (9.49), and (9.66). Finally, (9.83) isdeduced from (9.59), (9.50-9.52), and (9.67).


To arrive at conservation laws for the wave-body system, the dynamical interaction ofthe fluid and the body has to be stated mathematically. To be precise, an extra setof equations is needed, specifying the mutual exchange of momentum and energy, anddescribing the fluid flow on the wetted body surface.

In the absence of external forces, the only forces acting on the body are the pressureforces due to the hydrodynamic interaction with the fluid. Under these circumstancesthe equations of motion for the body read

Mb ¨xb =∫

S

p nx ds = −∫

S

p dz , (9.84)

Mb¨zb =∫

S

p nz ds−Mbg =∫

S

p dx−Mbg , (9.85)

Nb¨θb =

∫

S

p (rxnz − rznx) ds =∫

S

p (rx dx + rz dz) , (9.86)

where ~n = (nx, nz)T is the unit normal vector along the wetted body surface S; along S

we have

dx = nz ds , dz = −nx ds . (9.87)

In (9.86) the vector ~r = (rx, rz)T denotes the position of a point (x, z) on S relative to

the centre of gravity (xb, zb), that is

rx = x− xb , rz = z − zb . (9.88)

The equations of motion for the body, connecting the hydrodynamic pressure forces andmoment with the translational and rotational accelerations, are supplemented with a‘contact’ condition on the wetted body surface:

∂φ

∂n= ∇φ · ~n = ~ub · ~n on S , (9.89)

where ~ub = (ub, wb)T denotes the velocity of a point (x, z) on S:

ub = ˙xb − ˙θb (z − zb) = ˙xb − ˙θbrz , (9.90)

wb = ˙zb + ˙θb (x− xb) = ˙zb + ˙θbrx . (9.91)

The Neumann boundary condition (9.89) states that the normal fluid velocity and thenormal body surface velocity coincide for all time, thus excluding the occurrence of cav-ities on the body surface.

With the identities (9.77-9.83), the equations of motion (9.84-9.86), the boundary condi-tion (9.89), and the vanishing of p on C \ S, seven conservation laws for the wave-bodysystem can be deduced; they are listed in the following theorem.

Theorem 8 : Conservation laws for the wave-body problem (2D)The two-dimensional wave-body problem in the absence of external forces has the following


seven constants of the motion:

c1 = Ms , (9.92)c2 = Is , (9.93)c3 = Js + c1gt = Js +Msgt , (9.94)c4 = Msxs − c2t = Msxs − Ist , (9.95)

c5 = Mszs − c3t +12c1gt2 = Ms

(zs − 1

2gt2

)− Jst , (9.96)

c6 = Hs , (9.97)

c7 = As + g

(c4t +

12c2t

2

)= As + gt

(Msxs − 1

2Ist

). (9.98)

In the special case where the effect of gravity vanishes (i.e. g = 0), these quantities reduceto Ms, Is, Js, (Msxs − Ist), (Mszs − Jst), Hs, and As respectively.

Proof: Conservation law (9.92) follows directly from (9.77), expressing that

The total mass is a constant of the motion.

If the pressure p vanishes everywhere on C \ S, identity (9.80) yields

dIs

dt=

∫

S

p dz +Mb ¨xb . (9.99)

It is then obvious that (9.93) follows from (9.84), thus stating that

The total horizontal momentum is a constant of the motion.

Similarly, with the vanishing of p on C \ S, (9.81) gives

dJs

dt= −

∫

S

p dx−Mfg +Mb¨zb , (9.100)

and hence, using (9.85)

dJs

dt= − (Mf +Mb) g = −Msg . (9.101)

This leads to (9.94), expressing that in the absence of gravity

The total vertical momentum is a constant of the motion.

Next, (9.95) is obtained from (9.78) and (9.93), stating that

The horizontal velocity of the centre of massis a constant of the motion.

Similarly, (9.96) is derived from (9.79) and (9.94), expressing that in the absence ofgravity

The vertical velocity of the centre of massis a constant of the motion.


In order to prove (9.97), note that with p vanishing on C \ S, (9.83) yields

dHs

dt= −

∫

S

p (φxnx + φznz) ds

+Mb ( ˙xb ¨xb + ˙zb¨zb) +Nb˙θb

¨θb +Mbg ˙zb . (9.102)

With the Neumann condition (9.89), this can be rewritten to

dHs

dt= ˙xb

Mb ¨xb +

∫

S

p dz

+ ˙zb

Mb¨zb −

∫

S

p dx +Mbg

+ ˙θb

Nb

¨θb −∫

S

p (rx dx + rz dz)

. (9.103)

Substitution of the equations of motion (9.84-9.86) then yields (9.97), stating that

The total energy is a constant of the motion.

Finally, with p non-zero only on S, (9.82) can be written as

dAs

dt= Nb

¨θb −∫

S

p (rx dx− rz dz)−Mfgxf

+ xb

Mb¨zb −

∫

S

p dx

− zb

Mb ¨xb +

∫

S

p dz

. (9.104)

Substitution of (9.84-9.86) then yields

dAs

dt= −Mfgxf −Mbgxb = −Msgxs , (9.105)

which completes the proof of (9.98), expressing that in the absence of gravity

The total angular momentum is a constant of the motion.

This completes the proof of theorem 8.

9.3.2 Invariants for the three-dimensional case

The generalization of the above results to the three-dimensional case is straightforwardand based on Gauss’ divergence theorem

∫∫

Vf

∫∇ · ~U dV =

∫

Sf

∫~U · ~n dS , (9.106)

where the vector field ~U is defined on the fluid domain Vf , bounded by a closed surfaceSf .


In this case we have twelve quantities for the fluid, and each quantity is defined as avolume integral of the corresponding density:

Mf =∫∫

Vf

∫dV , (9.107)

Mf xf =∫∫

Vf

∫x dV , (9.108)

Mf yf =∫∫

Vf

∫y dV , (9.109)

Mf zf =∫∫

Vf

∫z dV , (9.110)

Ixf =

∫∫

Vf

∫φx dV =

∫

Sf

∫φnx dS , (9.111)

Iyf =

∫∫

Vf

∫φy dV =

∫

Sf

∫φny dS , (9.112)

Izf =

∫∫

Vf

∫φz dV =

∫

Sf

∫φnz dS , (9.113)

Axf =

∫∫

Vf

∫ [(zφ)y − (yφ)z

]dV =

∫

Sf

∫φ (zny − ynz) dS , (9.114)

Ayf =

∫∫

Vf

∫[(xφ)z − (zφ)x] dV =

∫

Sf

∫φ (xnz − znx) dS , (9.115)

Azf =

∫∫

Vf

∫ [(yφ)x − (xφ)y

]dV =

∫

Sf

∫φ (ynx − xny) dS , (9.116)

Bf =∫∫

Vf

∫ [(xφ)x + (yφ)y + (zφ)z

]dV

=∫

Sf

∫φ (xnx + yny + znz) dS , (9.117)

Hf = Kf + Pf , (9.118)

with Kf and Pf defined as

Kf =∫∫

Vf

∫12

(φ2

x + φ2y + φ2

z

)dV

=∫

Sf

∫12φ (φxnx + φyny + φznz) dS , (9.119)


Pf =∫∫

Vf

∫gz dV = Mfgzf . (9.120)

Thus, Mf is familiar as the fluid mass; xf , yf , and zf are the coordinates of the centreof fluid mass (x and y are the horizontal directions); Ix

f , Iyf , and Iz

f are the threecomponents of the fluid momentum; Ax

f , Ayf , and Az

f are the three components of theangular fluid momentum; Bf is the unfamiliar fluid quantity; and Hf is the fluid energy,being the sum of the kinetic fluid energy Kf and the potential fluid energy Pf .

The analogues for a rigid body — characterized by its mass Mb, its moment of inertia~Nb = (N x

b ,N yb ,N z

b ), its position ~xb = (xb, yb, zb), and its orientation ~θb =(θx

b , θyb , θz

b

)—

read

Ixb = Mb ˙xb , (9.121)Iy

b = Mb ˙yb , (9.122)Iz

b = Mb ˙zb , (9.123)

Axb = N x

b˙θx

b +Mb (zb ˙yb − yb ˙zb) , (9.124)

Ayb = N y

b˙θy

b +Mb (xb ˙zb − zb ˙xb) , (9.125)

Azb = N z

b˙θz

b +Mb (yb ˙xb − xb ˙yb) , (9.126)Hb = Kb + Pb , (9.127)

where Kb and Pb are defined as

Kb =12Mb

(~xb · ~xb

)+

12

(~Nb ⊗ ~θb

)· ~θb , (9.128)

Pb = Mbgzb . (9.129)

Thus, Ixb , Iy

b , and Izb are the three components of the horizontal body momentum; Ax

b ,Ay

b , and Azb are the three components of the angular body momentum; and Hb is the

body energy, being the sum of the kinetic body energy Kb and the potential body energyPb.

Next, we put the system quantities as

Ms = Mf +Mb , (9.130)Msxs = Mf xf +Mbxb , (9.131)Msys = Mf yf +Mbyb , (9.132)Mszs = Mf zf +Mbzb , (9.133)

Ixs = Ix

f + Ixb , Iy

s = Iyf + Iy

b , Izs = Iz

f + Izb , (9.134)

Axs = Ax

f +Axb , Ay

s = Ayf +Ay

b , Azs = Az

f +Azb , (9.135)

Hs = Hf +Hb , Ks = Kf +Kb , Ps = Pf + Pb . (9.136)

Then, with the equations of motion for the unrestrained body

Mb ¨xb =∫

S

∫p nxdS , (9.137)

Mb¨yb =∫

S

∫p nydS , (9.138)


Mb¨zb =∫

S

∫p nzdS −Mbg , (9.139)

N xb¨θ

x

b =∫

S

∫p (rynz − rzny) dS , (9.140)

N yb¨θ

y

b =∫

S

∫p (rznx − rxnz) dS , (9.141)

N zb¨θ

z

b =∫

S

∫p (rxny − rynx) dS , (9.142)

with the contact condition on the wetted body surface

∇φ · ~n = ~ub · ~n on S with ~ub = (ub, vb, wb)T

, (9.143)

and with the vanishing of the Bernoulli pressure

p = −(

φt +12

(φ2

x + φ2y + φ2

z

)+ gz

)(9.144)

on Sf \ S, we arrive at the conservation laws listed in the following theorem.

Theorem 9 : Conservation laws for the wave-body problem (3D)The three-dimensional wave-body problem in the absence of external forces has the fol-lowing eleven constants of the motion:

c1 = Ms , (9.145)c2 = Ix

s , (9.146)c3 = Iy

s , (9.147)c4 = Iz

s +Msgt , (9.148)c5 = Msxs − Ix

s t , (9.149)c6 = Msys − Iy

s t , (9.150)

c7 = Ms

(zs − 1

2gt2

)− Iz

s t , (9.151)

c8 = Hs , (9.152)

c9 = Axs + gt

(Msys − 1

2Iy

s t

), (9.153)

c10 = Ays + gt

(Msxs − 1

2Ix

s t

), (9.154)

c11 = Azs . (9.155)

In the special case where g = 0, these quantities reduce to Ms, Ixs , Iy

s , Izs , (Msxs − Ix

s t),(Msys − Iy

s t), (Mszs − Izs t), Hs, Ax

s , Ays , and Az

s respectively.

9.4. THE VIRIAL AND CONSERVATION LAW NO.8 175

9.4 The virial and conservation law no.8

Attention is now focused on the unfamiliar fluid quantity Bf , defined in (9.43) — for thetwo-dimensional case — and here repeated for convenience:

Bf =∫

Df

∫[(xφ)x + (zφ)z] dx dz =

∫

C

φ (z dx− x dz) . (9.156)

For reasons that will be given later on, this quantity will be named ‘virial’ hereafter. Thevirial bears a strong resemblance to the angular fluid momentumAf , see (9.42). However,it will appear later on that the existence of the velocity potential φ is a prerequisite forthe definition of Bf , while Af may as well be defined in terms of the velocity field(uf , wf ) of the fluid flow (see also Appendix C). This difference appears to be the majorobstacle in the generalization of Bf for the wave-body problem. An extensive discussion— from various points of view — of the virial, its body and system analogues, andthe corresponding conservation law (which is lacking so far) is given in the followingsubsections.

9.4.1 The virial connection

The fact that potential flow is a necessary premise for the present definition of Bf becomesmore clear when (9.156) is rewritten to

Bf =∫

Df

∫(xφx + zφz) dx dz +

∫

Df

∫2φdx dz ≡ Gf +Rf . (9.157)

The first integral can also be written in terms of the velocity field (uf , wf )T :

Gf =∫

Df

∫ρf (xuf + zwf ) dx dz , (9.158)

where the fluid density ρf , which has been taken as unity before, has been reintroduced.Thus, Gf can be defined as the integral of the inner product of the position vector (x, z)T

and the momentum vector (ρfuf , ρfwf )T , where the integration is carried out over thefluid domain Df . Defined in this way, Gf is known to play an important role in theso-called virial theorem.

The virial theorem is statistical in nature; for a large variety of systems it is concernedwith the time averages of various mechanical quantities. A brief discussion of the virialtheorem for a discrete system of mass points has been given by Goldstein (1980); equiv-alent theorems for the continuous water-wave and wave-body problems will be deducedhereafter.17

Consider a system of free surface waves, where each infinitesimal fluid particle is char-acterized by its transient position ~rf = (x, z)T and its momentum ~pf = ρf ~rf — wherethe latter vector quantity should not be confused with the Bernoulli pressure p. With a

17For a discussion of the relation between fluid mechanics and statistical physics, see Uhlenbeck’sreview article (1980).


continuous force distribution ~Ff throughout the fluid domain, the fundamental equationsof motion read

~pf = ~Ff . (9.159)

Substitution of the inertial (pressure) forces and the gravitational forces for ~Ff yields thewell-known Euler equations for incompressible and inviscid fluid flow.

With the above definitions, (9.158) is rewritten to

Gf =∫

Df

∫(~rf · ~pf ) dx dz . (9.160)

The total time derivative of this quantity is

dGf

dt=

∫

Df

∫ (~rf · ~pf

)dx dz +

∫

Df

∫ (~pf · ~rf

)dx dz . (9.161)

The first term can be transformed to∫

Df

∫ (~rf · ~pf

)dx dz =

∫

Df

∫ρf

(~rf · ~rf

)dx dz = 2Kf , (9.162)

while the second by (9.159) is∫

Df

∫ (~pf · ~rf

)dx dz =

∫

Df

∫ (~Ff · ~rf

)dx dz . (9.163)

Identity (9.161) therefore reduces to

dGf

dt= 2Kf +

∫

Df

∫ (~Ff · ~rf

)dx dz . (9.164)

Averaging this result over a time interval [0, T ] gives

1T

T∫

0

dGf

dtdt =<

dGf

dt>= < 2Kf > + <

∫

Df

∫ (~Ff · ~rf

)dx dz > , (9.165)

or

2 < Kf > + <

∫

Df

∫ (~Ff · ~rf

)dx dz > =

1T

[Gf (T )− Gf (0)] . (9.166)

If the motion is periodic, i.e. all particle positions repeat after a certain time, and if Tis chosen to be the period, then the right-hand side of (9.166) vanishes.18 It then followsthat

< Kf >= −12

<

∫

Df

∫ (~Ff · ~rf

)dx dz > . (9.167)

18A similar conclusion can be reached even if the motion is not periodic, provided that the coordinatesand velocities of all fluid particles remain finite so that there is an upper bound to Gf . Then, by choosingT sufficiently long, the right-hand side of (9.166) can be made as small as desired.


Identity (9.167) is known as the virial theorem19, and the right-hand side is called thevirial of Clausius20.

The above reasoning can also be applied to the second integral in (9.157):

Rf =∫

Df

∫2φdx dz . (9.168)

Note that, since the potential field is determinate up to an additive constant, we areallowed to choose the base-level of φ at a certain time t0 such that Rf (t0) = 0. Withthis special choice it follows that Bf (t0) = Gf (t0), which is the justification for labellingBf as the ‘virial’ quantity.

The total time derivative of Rf is

dRf

dt=

∫

Df

∫2Dφ

Dtdx dz = 2Kf − 2Pf −

∫

Df

∫2p dx dz , (9.169)

where we used the Bernoulli pressure definition (9.60).Averaging over a time interval [0, T ] gives

2 < Kf > − 2 < Pf > − <

∫

Df

∫2p dx dz > =

1T

[Rf (T )−Rf (0)] . (9.170)

Substitution of the inertial and gravitational forces

~Ff = (−px,−pz − g)T (9.171)

into (9.166) yields the identity

2 < Kf > − < Pf > − <

∫

Df

∫(xpx + zpz) dx dz >

=1T

[Gf (T )− Gf (0)] . (9.172)

By simple addition of (9.170) and (9.172), we obtain with (9.157)

4 < Kf > − 3 < Pf > − <

∫

Df

∫[(xp)x + (zp)z] dx dz >

=1T

[Bf (T )− Bf (0)] . (9.173)

Now, if the potential flow is periodic, and if T is chosen to be the period of the fluidmotion, then the right-hand side vanishes, and we arrive at

4 < Kf > − 3 < Pf > = <

∫

Cf

p (xnx + znz) ds > , (9.174)

19The discrete variant of this result is very useful in the kinetic theory of gases, in particular in theproof of Boyle’s law PV = NkT .

20Clausius, Rudolf Julius Emmanuel (1822-1888), German physicist and mathematician who madeimportant contributions to thermodynamics and the kinetic theory of gases.


where Green’s divergence theorem (9.36) has been applied.Finally, with the vanishing of p on Cf , we obtain the statistical result

< Kf >

< Pf >=

34

. (9.175)

This special partition of the averaged kinetic and potential energies may be compared toa similar result for discrete systems, wherein the forces are derivable from a potential; ifthe potential behaves like rn+1, then the force law goes as rn. Application of the virialtheorem then leads to the well-known partition < K > : < P > = (n + 1) : 2.

The extension of the virial theorem to the more complex wave-body problem is ratherstraightforward; the analogue of Gf for a floating body is

Gb =∫

Db

∫ρb (xub + zwb) dx dz = Mb (xb ˙xb + zb ˙zb) , (9.176)

where ρb is the body density and (9.90-9.91) have been used. Clearly, Gb accounts forthe translational motions of the body only, irrespective of possible rotational motion.However, the equivalent of Rf for the body may be put as

Rb = Nbθb˙θb . (9.177)

Clearly, this is not a direct generalization of Rf , but merely a way to account for therolling motion of the body.

Next, with the generalized body position ~rb =(xb, zb, θb

)T and the generalized body

momentum ~pb =(Mb ˙xb,Mb ˙zb,Nb

˙θb

)T

, the following expression for the body virial isproposed:

Bb = ~rb · ~pb = Gb +Rb . (9.178)

With the hydrodynamic pressure forces and moments represented by ~Fb, the fundamentalequations of motion for the body read

~pb = ~Fb . (9.179)

The total time derivative of the body virial is

dBb

dt= ~rb · ~pb + ~pb · ~rb . (9.180)

Substitution of (9.179) and time averaging over [0, T ] yields

2 < Kb > + < ~Fb · ~rb > =1T

[Bb (T )− Bb (0)] . (9.181)

If the body performs a periodic motion — with the roll angle θb taken in [0, 2π] — and ifT is chosen to be the period, then the right-hand side of (9.181) vanishes; it then followsthat

< Kb > = −12

< ~Fb · ~rb > . (9.182)


By simple addition of (9.167) and (9.182) one obtains the virial theorem for the wave-body problem:

< Kf +Kb > = −12

<

∫

Df

∫ (~Ff · ~rf

)dx dz + ~Fb · ~rb > . (9.183)

Given explicit expressions for the periodic inertial and gravitational forces, the aboveidentity can be used to determine the averaged kinetic energy of the wave-body system.

9.4.2 The circulation alternative

When the fluid motion is not irrotational, there no longer exists a velocity potentialφ. Nevertheless, Longuet-Higgins showed that the velocity-dependent integral quanti-ties (9.40-9.42) and (9.45) still can be defined in terms of the fluid velocity field (uf , wf )T ;he also proved that the corresponding seven conservation laws for the water-wave prob-lem remain valid. However, (quoting Longuet-Higgins) ‘there appears to be no simpleanalogue to Bf for motion with vorticity’.

For rotational flow, Longuet-Higgins introduced the fluid circulation as a substitute forBf :

Cf =∫

Df

∫ωfdx dz , (9.184)

where the fluid vorticity ωf is defined as

ωf =∂wf

∂x− ∂uf

∂z. (9.185)

Then, from the vorticity equation for two-dimensional flow

D

Dt[ωf ] = 0 , (9.186)

he obtained the well-known circulation theorem

dCf

dt= 0 , (9.187)

stating that for the two-dimensional water-wave problem the circulation is a constant ofthe motion.

The straightforward generalization of the circulation concept to the wave-body problemis doomed to failure, since the motion of a rigid body is generally rotational. However,on the analogy of Cf the ‘body circulation’ may be put as

Cb =∫

Df

∫ωfdx dz , (9.188)

where the ‘body vorticity’ is defined as

ωb =∂wb

∂x− ∂ub

∂z= 2˙θb . (9.189)


To arrive at the last expression, (9.90-9.91) have been used. Thus, one obtains thefollowing expression for Cb:

Cb = 2Mb˙θb , (9.190)

Apparently, Cb is conserved if and only if the rotational motion is uniform.21

9.4.3 Fluid-filled deformable bodies

Consider the two-dimensional wave-body problem depicted in Figure 9.3, where an im-permeable, flexible bag S filled with an ideal fluid — the floating body — is submergedin an ideal, irrotational fluid.

Figure 9.3: Free surface waves and submerged deformable body.

Assuming the fluid flow inside the bag to be irrotational, a potential φ′ can be introducedto represent the interior velocity field. For this particular configuration, the virials of thebody and the surrounding fluid are defined as

Bb =∫

Db

∫ρb [(xφ′)x + (zφ′)z] dx dz =

∫

S

ρbφ′ (z dx− x dz) , (9.191)

Bf =∫

Df

∫ρf [(xφ)x + (zφ)z] dx dz =

∫

C∪S

ρfφ (z dx− x dz) . (9.192)

To obtain expressions for the total time derivatives of these virial quantities, note thatfrom (9.60)

D

Dt[(xφ)x + (zφ)z] =

D

Dt[xφx + zφz + 2φ]

=(φ2

x + φ2z

)− xpx − z (pz + g)

− 2[(p + gz)− 1

2(φ2

x + φ2z

)]

= − (xp)x − (zp)z + 2(φ2

x + φ2z

)− 3gz . (9.193)

21Obviously, this problem is simply by-passed when ωb is replaced by ωb − 2 ˙θb; it then follows thatCb = 0 for all times.


A similar expression can be derived for the interior fluid flow represented by φ′. Then,from the same arguments that have been used in the proof of (9.61-9.67), the followingidentities can be deduced:

dBf

dt= −

∫

C∪S

p (z dx− x dz) + 4Kf − 3Pf , (9.194)

dBb

dt= −

∫

S

p′ (z dx− x dz) + 4Kb − 3Pb , (9.195)

where p′ denotes the Bernoulli pressure inside the bag; Kb and Pb denote the kineticenergy and the potential energy of the interior fluid.

Adding (9.194) and (9.195), with the notion that ~n′ = −~n along S, gives

d

dt[Bf + Bb] = 4 (Hf +Hb)− 7 (Pf + Pb)−

∫

S

(p− p′) (z dx− x dz) .

(9.196)

Permanent contact between the exterior fluid and the deformable body is guaranteed bythe boundary condition

p = p′ on S . (9.197)

Substitution into (9.196) yields

dBs

dt= 4Hs − 7Ps , (9.198)

where Bs ≡ Bf +Bb, etc. Then, by integration in time, an extra constant of the motion22

is obtained for this special configuration:

c8 = Bs − (4Hs − 7Ps) t , (9.199)

which, in the absence of gravity, reduces to the statement that the rate of growth of thevirial equals the total energy multiplied by four.

9.4.4 The ‘broken symmetry’ argument

Here an attempt is made towards a conclusive answer to the question whether thereexists an analogue to Bf for a rigid body or not. Therefore, let us return to Benjaminand Olver, and the summary of their analysis presented in section 9.2.

The vector fields ~vi from theorem 5, the symmetry groups Gi from theorem 6, thedensities Ti in theorem 7, and the integral quantities in theorem 8 are related accordingto the table below. Obviously, all integral quantities have a one-to-one correspondencewith a single vector field excepting the virial Bf , which is related to two vector fields;actually, Benjamin and Olver used a linear combination of ~v7 and ~v9 to obtain T7 as aconserved density.In theorem 7 it was stated that when surface tension is operative — i.e. σ > 0 in (9.1-9.4) — ~v9 is no longer the infinitesimal generator of a one-parameter symmetry subgroup;

22With proper definitions, it can be shown that the seven conservation laws listed in theorem 8 alsoapply to this problem.


vector symmetry conserved integral physicalfield group density quantity interpretation~v1 G1 T1 If horizontal momentum~v2 G2 T2 Hf energy~v3 G3 T3 Mf mass~v4 G4 T4 Jf vertical momentum~v5 G5 T5 Mf xf horiz. pos. mass centroid~v6 G6 T6 Mf zf vert. pos. mass centroid

~v7, ~v9 G7, G9 T7 Bf virial~v8 G8 T8 Af angular momentum

Table 9.1: Interrelation between vector fields, symmetry groups, conserved densities, andintegrals quantities.

consequently, the density T7 is no longer conserved. Hence, it can be expected that in thepresence of surface tension, any conservation law involving Bf will be lost. This, however,might be the answer to the question regarding the generalization of the virial to a wave-body system; surface tension is usually viewed at as acting like a stretched membrane,thus modifying the free surface pressure — see the dynamic free surface condition (9.3).The same effect can be expected from the presence of a rigid body, floating in or belowthe free surface; the body exerts a (positive) pressure force on the surrounding fluid.Then, from the same argument, it follows that the scaling symmetry G9 is broken, andconsequently there is no conservation of a virial-like quantity for the wave-body system.

As a final remark, it is stressed that the above reasoning should not be interpretedas a proof of the non-existence of a virial for wave-body systems. So far, it can onlybe surmised that the formal extension of a fluid quantity like Bf and the correspondingconservation law is impossible.

9.5 Symmetry-breaking boundaries

In the proof of conservation laws (9.92-9.98) it was assumed that the Bernoulli pressurep vanished on C \S. However, in practical situations this is not a reasonable assumption;in many physical applications impermeable fixed boundaries — for instance, a bottom— are present. In this section we discuss the impact of such boundaries on the previousresults.Consider the problem depicted in Figure 9.4: the fluid, with the body floating in or belowthe free surface, is bounded by an impermeable fixed bottom B (which is not necessarilyeven) and lateral boundaries V and W (which are not necessarily fixed). The free surfaceand the wetted part of the body surface are denoted by F and S respectively.

It is obvious that, with V and W either impermeable and fixed or moving in a Lagrangianmanner, the total mass is a constant of the motion, which is expressed by (9.77).

With regard to the horizontal momentum, note that from the vanishing of p on F

9.5. SYMMETRY-BREAKING BOUNDARIES 183

Figure 9.4: Wave-body system with impermeable fixed boundaries.

and (9.64)

dIf

dt=

∫

S

p dz +∫

B

p dz +∫

V ∪W

p dz . (9.200)

Using (9.84) and (9.47), one arrives at

dIs

dt=

∫

B

p dz +∫

V ∪W

p dz . (9.201)

The integral over B equals zero if the bottom is horizontal. If the problem is periodic inhorizontal direction, the integrals over V and W are equal in magnitude but of oppositesign. So, with these two conditions it is found that (d/dt) Is = 0, stating that Is is aconstant of the motion. The same result is obtained if the problem (i.e. the geometry,the fluid flow, and the body motion) is symmetric about a vertical plane x = x0.

Similarly, with regard to the vertical momentum, we deduce from p = 0 on F and (9.65)

dJf

dt= −

∫

S

p dx−∫

B

p dx−∫

V ∪W

p dx−Mfg . (9.202)

Then, using (9.85) and (9.48)

dJs

dt+Msg = −

∫

B

p dx−∫

V ∪W

p dx . (9.203)

If V and W are impermeable fixed vertical walls, then the corresponding integral con-tributions vanish. If the problem is periodic in horizontal direction, then the integralsover V and W are equal in magnitude but of opposite sign. However, symmetry abouta vertical plane x = x0 does not imply the vanishing of these integrals. Likewise, theintegral over B is time dependent in general; therefore, Js +Msgt is not a constant ofthe motion.


With the vanishing of p on F and (9.66) it follows that

dAf

dt= −

∫

S

p (x dx + z dz)−∫

B

p (x dx + z dz)

−∫

V ∪W

p (x dx + z dz)−Mfgxf . (9.204)

Using (9.84-9.86) and (9.49), one obtains

dAs

dt+Msgxs = −

∫

B

px dx−∫

V ∪W

px dx−∫

B

pz dz −∫

V ∪W

pz dz .

(9.205)

Note that a horizontal bottom combined with horizontal periodicity is not a sufficientcondition for the conservation of angular momentum, since the contribution of the firstthree integrals is not necessarily zero then. However, if the problem is symmetric aboutthe plane x = 0, then one has (d/dt)As = 0, since the integrals over B vanish and theintegrals over V and W are equal in magnitude but of opposite sign; in that particularcase As is a constant of the motion (use xs = 0).

From p = 0 on F and (9.67) it follows that

dHf

dt= −

∫

S∪B∪W

p (φz dx− φx dz) = −∫

S∪B∪W

p φnds . (9.206)

With the notion that φn = 0 along B ∪W and using (9.89), (9.84-9.86) and (9.50-9.52),it is found that

dHs

dt= 0 . (9.207)

Finally, the effects of B and W on the conservation of the virial for the original problemwithout a floating body are discussed. With p vanishing on F and (9.194) it follows that

dBf

dt= −

∫

B∪W

p (z dx− x dz) + 4Hf − 7Pf . (9.208)

In case of a horizontal bottom and/or vertical walls some integral contributions vanish.However, neither horizontal periodicity nor symmetry about a plane x = x0 is sufficientto make the total contribution of the integrals vanish; therefore, Bf − (4Hf − 7Pf ) t isno longer a constant of the motion.

Summarizing, the presence of impermeable fixed boundaries affects the horizontal, ver-tical and angular momentum, and the virial of the system. In general, these quantitiesare no longer constants of the motion; the bottom B breaks the corresponding symme-tries. In the special case of a horizontal bottom and horizontal periodicity, the horizontalmomentum is a constant of the motion. The same result is obtained if the problem issymmetric about a plane x = x0; if x0 = 0 the angular momentum is conserved too. Thepresence of an impermeable bottom breaks the vertical translation symmetry (even incase of symmetry about a plane x = x0), and consequently the vertical momentum is nolonger a constant of the motion. The mass and energy conservation laws are not affectedby the introduction of impermeable fixed boundaries.

9.6. NUMERICAL VALIDATION 185

9.6 Numerical validation

In this section the theory for conservation laws for the wave-body problem is confirmedfrom numerical results for a simple test configuration.

Figure 9.5: Wave-body system: initialconfiguration.

Figure 9.6: Wave-body system: equilib-rium state.

The results presented apply to a simple test configuration; the initial position of the bodyand the free surface are depicted in Figure 9.5, while the equilibrium state of this systemis shown in Figure 9.6. The tank length is 20 m, the still water level in the equilibriumstate is 5 m, and the initial upward displacement of the cylinder (with respect to theinitial water level) is 1 m; the cylinder diameter is 6 m. The ensuing free motion of thebody generates an outgoing wave field; a direct exchange of energy (and other quantities)from the fluid to the body and vice versa is expected.

Figure 9.7 shows Mf , Mb and Ms; of course, each of these quantities is constant, whichis confirmed (fortunately!) by the computed results. The fluid mass is computed byusing the contour integral expression in (9.37).

Since the problem is symmetric about the vertical plane through the body centre ofgravity, the horizontal momentum is a constant of the motion; this is confirmed byFigure 9.8, showing If , Ib and Is. Note that the order of magnitude is 10−5, which makesit plausible to attribute the variations to numerical discretization errors. A temporaryincrease of the amplitude is observed in the time interval [8 s, 10 s]; this is due to thereflection of the radiated waves against the impermeable fixed lateral boundaries.

With the presence of an impermeable fixed bottom, it is expected that the verticalmomentum is not a constant of the motion. From Figure 9.9 it is clear that Jf and Jb

are both approximately sinusoidal, and more or less in counter phase, but at differentamplitudes; their sum Js is nearly sinusoidal too, but at a doubled frequency. Thus, thetheoretical prediction — that the vertical momentum is not preserved — is confirmed bythe numerical computations.

Since the problem has been chosen such that it is symmetric about x = 0, the angularmomentum is a constant of the motion; this is confirmed by Figure 9.10, showing Af ,Ab and As (order of magnitude: 10−4). Again, a temporary increase in amplitude isobserved in the time interval [8 s, 10 s].


Figure 9.7: Conservation of mass.

Figure 9.11 shows the energies of the fluid, the body and the wave-body system. It isclear that Hf and Hb are both sinusoidal (with the same amplitude) and in exact counterphase. Evidently, their sum Hs is a constant of the motion.

Finally, from Figure 9.12 we note that, roughly speaking, Bf oscillates sinusoidally rounda linearly decaying function; this is in fair agreement with Longuet-Higgins’ conclusionthat, for the original problem without a floating body (in the absence of gravity), Bf −4Hf t is a constant of the motion.23

23This result has also been obtained for a submerged fluid-filled deformable body, see paragraph 9.4.3.


Figure 9.8: Conservation of horizontal momentum.

THIS PAGE INTENTIONALLY LEFT BLANK


Figure 9.9: Exchange of vertical momentum — no conservation.

Figure 9.10: Conservation of angular momentum.


Figure 9.11: Exchange and conservation of energy.

Figure 9.12: Virial — no conservation.


9.7 Bibliography

Few relevant publications which have not been cited in the text are also included here.

Arnol’d, V.I., and Khesin, B.A. 1992. Topological methods in hydrodynamics. An-nual Review of Fluid Mechanics 24:145-166.

Benjamin, T.B. 1974. Lectures on nonlinear wave motion. Lectures in Applied Math-ematics 15:3-47.

Benjamin, T.B. 1986a. On the Boussinesq model for two-dimensional wave motions inheterogeneous fluids. Journal of Fluid Mechanics 165:445-474.

Benjamin, T.B. 1986b. Note on added mass and drift. Journal of Fluid Mechanics169:251-256.

Benjamin, T.B., and Mahony, J.J. 1971. On an invariant property of water waves.Journal of Fluid Mechanics 49:385-389.

Benjamin, T.B., and Olver, P.J. 1982. Hamiltonian structure, symmetries andconservation laws for water waves. Journal of Fluid Mechanics 125:137-185.

Broer, L.J.F, van Groesen, E.W.C., and Timmers, J.M.W. 1976. Stable modelequations for long water waves. Applied Scientific Research 32:619-636.

Crapper, G.D. 1979. Energy and momentum integrals for progressive capillary-gravitywaves. Journal of Fluid Mechanics 94(1):13-24.

van Daalen, E.F.G., and van Groesen, E.W.C. 1993. Conservation laws for waterwaves and floating bodies. Submitted to the Journal of Fluid Mechanics.


Lax, P.D. 1968. Integrals of nonlinear equations of evolution and solitary waves. Com-munications on Pure and Applied Mathematics 21:467-490.

Longuet-Higgins, M.S. 1974. On the mass, momentum, energy and circulation of asolitary wave. Proceedings of the Royal Society of London, Series A, 337:1-13.

Longuet-Higgins, M.S., and Fenton, J.D. 1974. On the mass, momentum, en-ergy and circulation of a solitary wave II. Proceedings of the Royal Society of London,Series A, 340:471-493.

Longuet-Higgins, M.S. 1975. Integral properties of periodic gravity waves of finiteamplitude. Proceedings of the Royal Society of London, Series A, 342:157-174.

Longuet-Higgins, M.S. 1980. Spin and angular momentum in gravity waves. Journalof Fluid Mechanics 97:1-25.

Longuet-Higgins, M.S. 1983. On integrals and invariants for inviscid, irrotationalflow under gravity. Journal of Fluid Mechanics 134:155-159.

Longuet-Higgins, M.S. 1984. New integral relations for gravity waves of finite ampli-tude. Journal of Fluid Mechanics 149:205-215.

Miles, J.W., and Salmon, R. 1985. Weakly dispersive nonlinear gravity waves.Journal of Fluid Mechanics 157:519-531.

Miura, R.M., and Gardner, C.S. 1968. Korteweg-de Vries equation and general-izations II. Existence of conservation laws and constants of motion. Journal of Mathe-


matical Physics 9:1204-1209.

Noether, E. 1918. Invariante Invariationsprobleme. Nachrichten der Gesellschaft derWissenschaften Gottingen 2:235-237.

Olver, P.J. 1980a. On the Hamiltonian structure of evolution equations. MathematicalProceedings of the Cambridge Philosophical Society 88:71-88.

Olver, P.J. 1980b. Applications of Lie Groups to Differential Equations. Lecture Notes,Mathematical Institute, University of Oxford.

Olver, P.J. 1983. Conservation laws of free boundary problems and the classification ofconservation laws for water waves. Transactions of the American Mathematical Society277(1):353-380.

Olver, P.J. 1986. Applications of Lie Groups to Differential Equations.Springer-Verlag.

Ripa, P. 1981. Symmetries and conservation laws for internal gravity waves. Proceed-ings of the American Institute of Physics 7b:281-386.

Starr, V.P. 1949a. A momentum integral for surface waves in deep water. Journal ofMaritime Research 6:126-135.

Starr, V.P. 1949b. Momentum and energy integrals for gravity waves of finite height.Journal of Maritime Research 6:175-193.

Uhlenbeck, G.E. 1980. Some notes on the relation between fluid mechanics and sta-tistical physics. Annual Review of Fluid Mechanics 12:1-9.


Now had they left behind the gulf named after the Ambracians, now withsails wide spread the land of the Curetes, and in next order the narrow islandswith the Echinades, and the land of Pelops was just descried; even then abaleful blast of the north wind seized them in mid-course and swept themtowards the Libyan sea nine nights and as many days, until they came farwithin Syrtis24, wherefrom is no return for ships, when they are once forcedinto that gulf. For on every hand are shoals, on every hand masses of seaweedfrom the depths; and over them the light foam of the wave washes withoutnoise; and there is a stretch of sand to the dim horizon; and there movethnothing that creeps or flies. Here accordingly to the flood-tide — for this tideoften retreats from the land and bursts back again over the beach coming onwith a rush and roar — thrust them suddenly on to the innermost shore, andbut little of the keel was left in the water. And they leapt forth from the ship,and sorrow seized them when they gazed on the mist and the levels of vastland stretching far like a mist and continuous into the distance; no spot forwater, no path, no steading of herdsmen did they descry afar off, but all thescene was possessed by a dead calm. And thus did one hero, vexed in spirit,ask another:

“What land is this? Whither has the tempest hurled us? Would that,reckless of deadly fear, we had dared to rush on by that same path between theclashing rocks! Better were it to have overleapt the will of Zeus and perishedin venturing some mighty deed. But now what should we do, held back by thewinds to stay here, if ever so short a time? How desolate looms before us theedge of the limitless land!”

Thus one spake; and among them Ancaeus the helmsman, in despair attheir evil case, spoke with grieving heart: “Verily we are undone by a terribledoom; there is no escape from ruin; we must suffer the cruellest woes, havingfallen on this desolation, even though breezes should blow from the land; for,as I gaze far around, on every side do I behold a sea of shoals, and massesof water, fretted line upon line, run over the hoary sand. And miserably longago would our sacred ship have been shattered far from the shore; but the tideitself bore her high on to the land from the deep sea. But now the tide rushesback to the sea, and only the foam, whereon no ship can sail, rolls round us,just covering the land. Wherefore I deem that all hope of our voyage and ofour return is cut off. Let someone else show his skill; let him sit at the helm— the man that is eager for our deliverance. But Zeus has no will to fulfilour day of return after all our toils”.


24quicksands in Libya

Chapter 10

Radiation BoundaryConditions

. . .This is the tale the Muses told; and I sing obedient to the Pierides, andthis report have I heard most truly; that ye, O mightiest far of the sons ofkings, by your might and your valour over the desert sands of Libya raisedhigh aloft on your shoulders the ship and all that ye brought therein, and bareher twelve days and nights alike. Yet who could tell the pain and grief whichthey endured in that toil? Surely they were of the blood of the immortals, sucha task did they take on them, constrained by necessity. How forward and howfar they bore her gladly to the waters of the Tritonian lake! How they strodein and set her down from their stalwart shoulders!


10.1 Introduction

In this chapter it is shown how variational principles and conservation laws can be usedto derive radiation (or absorbing) boundary conditions for partial differential equationswhich describe wave phenomena. Such boundary conditions are desirable in order tolimit the size of the computational domain and thus computation time and data storageas much as possible, while minimizing the effects of reflected waves on the solution in thearea of interest.

Many methods for developing radiation boundary conditions have been used.1 Forwave equations, the boundary conditions proposed by Bayliss and Turkel (1980, 1982),Engquist and Halpern (1988), Engquist and Majda (1977, 1979), and Higdon (1986, 1987,1990) are well-known. These boundary conditions have in common that they are basedon (properties of) solutions to the partial differential equations under consideration.

Here, radiation boundary conditions are derived without the assumption that solu-tions are available beforehand, and therefore they are applicable to nonlinear and disper-sive systems too. Moreover, these boundary conditions render the problem well-posed

1A recent overview of non-reflecting boundary conditions in the numerical solution of wave problemsis due to Givoli (1991).

193

194 CHAPTER 10. RADIATION BOUNDARY CONDITIONS

in the sense that the integral of some appropriate density — for instance, the energydensity — over the computational domain does not increase.

The present chapter is based on a paper written by van Daalen, Broeze, and van Groe-sen (1992), with an accompanying paper by Broeze and van Daalen (1992). The outlineof this chapter is as follows: in section 10.2 we develop a theory of radiation bound-ary conditions for continuous systems that are governed by a variational (Lagrangian)principle. This theory is illustrated in section 10.3 with explicit boundary conditionsfor a nonlinear version of the one-dimensional Klein-Gordon equation; numerical testresults are also presented. In section 10.4 we apply the theory to the classical (nonlinear)water-wave problem; some of the findings in the previous chapter will be re-established.

10.2 Theory for continuous systems

Consider a continuous system described by the variable u, which is a function of thespatial coordinates ~x = (x1, x2, . . . , xn)T and the time t.

It is assumed that the evolution of the system can be derived from a variationalprinciple. With a Lagrangian density L, an expression in ~x, t, u, and partial derivativesof u up to some finite order, the action integral reads

J (u) =∫

T

∫

Ω(t)

L dΩ dt (10.1)

where T is some time interval, and the spatial domain Ω may depend on time.Restrictions on the motion of Ω will be imposed further on, but first the Euler-

Lagrange equation for L and natural boundary conditions are derived under the assump-tion that the motion of Ω is given.

For the system under consideration, Hamilton’s principle can be summarized by say-ing that the evolution of the system is such that the action integral J (u) has a stationaryvalue, see (for instance) Goldstein (1980). The first variation of J (u) with respect to avariation δu is defined as

δJ (u; δu) ≡[

d

dεJ (u + εδu)

]

ε=0

=∫

T

∫

Ω(t)

DL (u) δu dΩ dt , (10.2)

where

DL (u) δu ≡[

d

dεL (u + εδu)

]

ε=0

(10.3)

denotes the Frechet2 derivative of L, defined for the variation δu.To arrive at the Euler-Lagrange equation for L, partial integrations with respect

to ~x and t are required, leaving expressions on the boundary when δu and its partialderivatives do not vanish there. In general, the formula for partial integration reads

DL (u) δu = δL (u) δu + ∂tB0 (u; δu) + div ~B1 (u; δu) (10.4)

2Frechet, Maurice (1878-1975), French mathematician. He made notable contributions to the theoryof functionals.

10.2. THEORY FOR CONTINUOUS SYSTEMS 195

where δL (u) is the variational derivative of L, and B0 and ~B1 are expressions linearin δu.

Substitution of (10.4) into (10.2) gives

δJ (u; δu) =∫

T

∫

Ω(t)

δL (u) δu dΩ dt

+∫

T

∫

Ω(t)

[∂tB0 (u; δu) + div ~B1 (u; δu)

]dΩ dt . (10.5)

Taking into account the prescribed motion of the spatial domain Ω and applying Gauss’divergence theorem, this equation can be written as

δJ (u; δu) =∫

T

∫

Ω(t)

δL (u) δu dΩ +∫

∂Ω(t)

~B1 (u; δu) · ~n dS

dt

+∫

T

d

dt

∫

Ω(t)

B0 (u; δu) dΩ−∫

∂Ω(t)

B0 (u; δu) vn dS

dt , (10.6)

where ~n is the outward pointing normal to ∂Ω, and vn is the velocity of the boundary innormal direction.

Allowing δu and its partial derivatives to be arbitrary at the boundary ∂Ω, butvanishing at the end points of the time interval T , the second term in the right-hand sidevanishes (since B0 is linear in δu), and the first variation of J (u) with respect to δu isthen given by

δJ (u; δu) =∫

T

∫

Ω(t)

δL (u) δu dΩ dt

+∫

T

∫

∂Ω(t)

[~B1 (u; δu) · ~n− B0 (u; δu) vn

]dS dt . (10.7)

From this expression the evolution of the system follows; the Euler-Lagrange equationreads

δL (u) = 0 in Ω (10.8)

and the natural boundary condition on the moving boundary is given by

B0 (u; δu) vn − ~B1 (u; δu) · ~n = 0 on ∂Ω (10.9)

for arbitrary δu.

Example 1: If the Lagrangian density L depends on ~x, t, u, and partial derivatives upto first order only, the Frechet derivative of L with respect to a variation δu reads

DL (u) δu =∂L∂u

δu +∂L∂ut

δut +∂L

∂uxi

δuxi , (10.10)


where we used Einstein’s summation convention. This can be written as

DL (u) δu =[∂L∂u

− ∂t∂L∂ut

− ∂xi

∂L∂uxi

]δu

+ ∂t

[∂L∂ut

δu

]+ ∂xi

[∂L

∂uxi

δu

]. (10.11)

The first variation of J is given by (10.5), with the variational derivative of L given by

δL (u) =∂L∂u

− ∂t∂L∂ut

− ∂xi

∂L∂uxi

, (10.12)

and with the following boundary expressions:

B0 (u; δu) =∂L∂ut

δu , (10.13)

~B1 (u; δu) =(

∂L∂ux1

,∂L

∂ux2

, . . . ,∂L

∂uxn

)T

δu . (10.14)

The vanishing of δJ (u; δu) provides the Euler-Lagrange equation — see (10.8) —

∂L∂u

− ∂t∂L∂ut

− ∂xi

∂L∂uxi

= 0 on ∂Ω (10.15)

and the natural boundary condition — see (10.9) —

∂L∂ut

vn − ∂L∂uxi

ni = 0 on ∂Ω (10.16)

Aiming to derive boundary conditions — on a fixed domain of integration — that do notinfluence the solution in the interior of the domain, several options are available. Oneimportant a priori requirement in the development of radiation boundary conditions isthat they should not be based on (properties of) solutions to the partial differentialequations considered, since in general (for nonlinear equations) such solutions can noteasily be found. Compare this, for instance, with Higdon’s (1987) derivation of absorbingboundary conditions, that is based on an exact plane monochromatic wave.

The possibility that will be examined in this chapter is to find boundary conditionsthat simulate a moving domain of integration such that the energy is conserved as wellas possible. The naive idea is that if the correct amount of energy is fluxed through thefixed boundary, any reflection caused by the presence of the boundary will have littleenergy and therefore will be small in this sense. This reasoning may be valuable, sinceenergy is a definite quantity. However, other densities — for example, the momentumdensity — may do as well, as will be shown further on.

In the derivation above, the motion of the domain Ω was assumed to be given. Then theevolution of the system is completely determined by the Euler-Lagrange equation (10.8)and the natural boundary condition (10.9). From now on, the motion of the domain Ω isnot prescribed, but instead it is demanded that Ω moves in such a way that the integralover Ω of some density is conserved.


In order to show the dependence of the results on the choice of density later on,we shall consider quite generally a density for which a local conservation law can beobtained with Noether’s theorem. For a comprehensive description of this well-knowntheorem the reader is referred to Olver (1986) and the historic work of Emmy Noetherherself (1918). The main principle of Noether’s theorem can be formulated as follows3:if the Lagrangian density is invariant under a given transformation of the variables, thenthere is a corresponding functional that is conserved.

So, let the Lagrangian density L be invariant under a variation ϕ (u), then it can beproven that for some scalar density χ0 (u;ϕ (u)) and some vector density ~χ1 (u;ϕ (u)) itholds that for all u

DL (u) ϕ (u) = ∂tχ0 (u; ϕ (u)) + div~χ1 (u; ϕ (u)) (10.17)

A combination of this equation and the integration-by-part formula (10.4) yields, takingδu = ϕ (u)

δL (u)ϕ (u) = ∂t

[χ0 (u; ϕ (u))− B0 (u;ϕ (u))

]

+ div[~χ1 (u; ϕ (u))− ~B1 (u; ϕ (u))

], (10.18)

leading to a local conservation law of the form

∂te (u) + div~fe (u) = 0 (10.19)

for solutions of the Euler-Lagrange equation (10.8). In this equation the density e andthe corresponding flux density ~fe are given by

e (u) ≡ χ0 (u;ϕ (u))− B0 (u; ϕ (u)) (10.20)

~fe (u) ≡ ~χ1 (u; ϕ (u))− ~B1 (u;ϕ (u)) (10.21)

As motivated above, let the motion of Ω be restricted in such a way that the quantity

E (u) ≡∫

Ω(t)

e (u) dΩ (10.22)

is conserved.Differentiation of (10.22) with respect to the time gives

dE

dt=

∫

Ω(t)

∂te (u) dΩ +∫

∂Ω(t)

e (u) vn dS . (10.23)

Substitution of (10.19) and application of Gauss’ divergence theorem yields

dE

dt=

∫

Ω(t)

div[e (u)~v − ~fe (u)

]dΩ , (10.24)

where ~v represents a velocity field on Ω, such that

vn = ~v · ~n on ∂Ω . (10.25)3See also, for instance, Goldstein (1980).


If the quantity E (u) is conserved, then ~v must satisfy∫

Ω(t)

div[e (u)~v − ~fe (u)

]dΩ = 0 . (10.26)

Since the initial domain Ω can be taken arbitrarily, ~v necessarily equals the local fluxvelocity ~vf , defined by

~fe (u) = ~vf (u) e (u) . (10.27)

So, in particular, the normal velocity vn on the boundary equals the local flux velocityin normal direction:

vn = ~vf (u) · ~n =~fe (u) · ~n

e (u)on ∂Ω (10.28)

at points where e (u) does not vanish. The latter condition can also be obtained directlyfrom (10.23); substitution of (10.19) and application of Gauss’ divergence theorem gives

dE

dt=

∫

∂Ω(t)

[e (u) vn − ~fe (u) · ~n

]dS . (10.29)

If the quantity E is conserved, then vn is determined by

e (u) vn − ~fe (u) · ~n = 0 on ∂Ω (10.30)

in accordance with (10.28).

Remark 1: Substitution of (10.27) into (10.19) gives

∂te (u) + ~vf (u) · ∇e (u) + e (u) div~vf (u) = 0 , (10.31)

which in case of a divergence-free flux velocity field ~vf (u) reduces to

De (u)Dt

= ∂te (u) + ~vf (u) · ∇e (u) = 0 , (10.32)

expressing that e (u) is constant along the streamlines of the corresponding flux velocityfield ~vf (u).

Remark 2: If e (u) equals the energy density, then the local flux velocity ~vf (u) definedby (10.27) is well-known, in particular for linear systems for which it equals the groupvelocity for monochromatic waves; see, for instance, Broer (1951), Biot (1957), andLighthill (1965). It is the pointwise version of the centro-velocity — i.e. the velocity ofthe centre of gravity of the density e (u) — which is given by

~vC (u) =~Fe (u)E (u)

, (10.33)

where

~Fe (u) ≡∫

Ω

~fe (u) dΩ , E (u) ≡∫

Ω

e (u) dΩ , (10.34)


see also Wehausen and Laitone (1960), van Groesen (1980), and van Groesen and Mainardi(1990).

Remark 3: If e equals the energy density, then the ideas above are related to some partsof the earlier work of Whitham (1974). In his treatment of energy propagation for theKlein-Gordon equation a slowly varying wave train is considered. It is found that theaveraged energy density E and the averaged energy flux density F (i.e. averaged overone period) satisfy

F = C (k) E , (10.35)

where C (k) denotes the group velocity depending on the wave number k. Compare thisaveraged result with the definition of the local flux velocity in (10.27).

Next, an averaged energy equation is proposed:

∂tE + ∂x

(C (k) E

)= 0 , (10.36)

which is the differential form of the statement that

The total energy between any two group lines remains constant.

Evidently, (10.36) in combination with (10.35) can be regarded as a global conservationlaw. Compare this with our local conservation law (10.19), and note that the presentderivation of radiation boundary conditions started from a similar, yet more general,principle, namely the conservation of an appropriate density on a moving domain ofintegration.

With the explicit expressions (10.20) and (10.21) for e and ~fe respectively, condition (10.30)reads

[χ0 (u; ϕ (u))− B0 (u;ϕ (u))

]vn −

[~χ1 (u; ϕ (u))− ~B1 (u;ϕ (u))

]· ~n = 0 .

(10.37)

However, the natural boundary condition, given in (10.9), already implies, taking δu =ϕ (u)

B0 (u;ϕ (u)) vn − ~B1 (u; ϕ (u)) · ~n = 0 on ∂Ω , (10.38)

and so the additional boundary condition for conservation of the quantity E (u) is givenby

χ0 (u; ϕ (u)) vn − ~χ1 (u; ϕ (u)) · ~n = 0 on ∂Ω (10.39)

where χ0 (u; ϕ (u)) and ~χ1 (u; ϕ (u)) are obtained from (10.17).

Summarizing, the proposed boundary conditions for a domain Ω, chosen in such a waythat the total amount of a density e (u) is preserved, are given by the natural boundarycondition (10.9) and the additional boundary condition (10.39).


Example 2: If the first order Lagrangian density L does not depend explicitly on ~xand t, then the energy density can be found with Noether’s theorem from a variationcorresponding to the invariance of L with respect to time translations:

ϕ (u) = ut . (10.40)

Similarly, the momentum density in a given direction ~σ is obtained with a variationcorresponding to the invariance of L with respect to spatial translations:

ψ (u) = ∇u · ~σ with ~σ 6= ~0) . (10.41)

Substitution of δu = ϕ (u) into (10.10) yields

DL (u) ut =∂L∂u

ut +∂L∂ut

utt +∂L

∂uxi

utxi

= ∂tL (u) , (10.42)

and therefore (10.17) holds with

χ0 (u; ϕ (u)) = L (u) , (10.43)~χ1 (u; ϕ (u)) = ~0 . (10.44)

Then, indeed, substitution of (10.13) and (10.43) into (10.20) yields the energy density

e (u) = L (u)− ∂L∂ut

ut , (10.45)

with the energy flux density — see (10.14), (10.21), and (10.44) —

~fe (u) = −(

∂L∂ux1

,∂L

∂ux2

, . . . ,∂L

∂uxn

)T

ut . (10.46)

Similarly, substitution of δu = ψ (u) into (10.10) gives

DL (u) (∇u · ~σ) =∂L∂u

(∇u · ~σ) +∂L∂ut

(∇u · ~σ)t +∂L

∂uxi

(∇u · ~σ)xi

= ~σ · ∇L (u)= div (~σL (u)) , (10.47)

and (10.17) holds with

χ0 (u; ψ (u)) = 0 , (10.48)~χ1 (u; ψ (u)) = ~σL (u) . (10.49)

The momentum density m (u) in the direction ~σ reads

m (u) = − ∂L∂ut

(∇u · ~σ) , (10.50)

with momentum flux density

~fm (u) = ~σL (u)−(

∂L∂ux1

,∂L

∂ux2

, . . . ,∂L

∂uxn

)T

(∇u · ~σ) . (10.51)


Substitution of (10.43) and (10.44) into (10.39) yields the additional boundary conditionfor energy conservation

L (u) vn = 0 on ∂Ω . (10.52)

Similarly, the additional boundary condition for momentum conservation reads

L (u) (~σ · ~n) = 0 on ∂Ω . (10.53)

Consequently, taking

L (u) = 0 on ∂Ω , (10.54)

it follows that both energy and momentum (in any direction) are conserved in this specialcase.

Remark 4: Whitham (1970) has considered dispersive wave problems in which theEuler-Lagrange equation has approximate solutions of the form

u = U (θ, a) , θ = kx− ωt , (10.55)

where, in the nonlinear case, the amplitude a, the wave number k and the frequency ωwill generally vary in time; k and ω are generalized by defining them as

k (x, t) =∂θ

∂x, ω (x, t) = −∂θ

∂t. (10.56)

It is assumed that ω, k and a are slowly varying functions of x and t, corresponding tothe slow modulation of the wave train considered. The averaged Lagrangian L is thendefined as

L (ω, k, a) =12π

2π∫

0

L (u) dθ , (10.57)

and is calculated by substitution of the uniform periodic solution u = U (θ, a) in L.The equations for ω, k, and a are then obtained from the averaged variational principle

δ

∫ ∫L (ω, k, a) dt dx = 0 . (10.58)

In the linear case we have

L (ω, k, a) = G (ω, k) a2 , (10.59)

and a variation of L with respect to a yields the linear dispersion relation4

G (ω, k) = 0 . (10.60)

It is then observed that the stationary value of L is zero. Compare this global condition— under the assumptions of slowly varying wave trains and periodic solutions — to thepointwise boundary condition (10.54).

In the following sections the theory will be applied to a nonlinear one-dimensional waveequation, and to the nonlinear three-dimensional water-wave problem.

4Note that this relation does not involve the amplitude a, and compare this result with the discussionin paragraph 2.4.3, in particular the dispersion relation (2.39).


10.3 Application to one-dimensional wave equations

In this section the theory is applied to a nonlinear one-dimensional wave equation. Theboundary conditions obtained will be used to simulate ‘open’ boundaries, i.e. boundarieswhich give low reflections for radiating waves.

Consider a nonlinear version of the Klein-Gordon equation:

utt = c2uxx − V ′ (u) , (10.61)

where V ′ (u) is some nonlinear function of u — the prime denoting differentiation withrespect to u — and c is the wave velocity. This hyperbolic equation arises in variousphysical problems; for example, if V ′ (u) = sin u, then (10.61) is known as the sine-Gordon equation.5

The action integral for this problem is given by

J (u) =∫

T

x1∫

x0

L (u) dx dt , (10.62)

where the Lagrangian density L is a function of u and its first order derivatives — notethat L does not depend explicitly on t:

L (u) = −12

(u2

t − c2u2x

)+ V (u) . (10.63)

Here the potential function V (u) is given by

V (u) =12αu2 +

14βu4 , (10.64)

which is an appropriate expansion for even functions V . In our test cases α and β arerestricted to positive values to guarantee the positive definiteness of the energy density,which for this problem reads — see (10.45) and (10.63-10.64) —

e (u) =12

(u2

t + c2u2x

)+ V (u) . (10.65)

The natural boundary condition for this system is obtained from (10.16):

c2un + vnut = 0 at x = x0,1 , (10.66)

and (10.52) yields the additional boundary condition:[12

(u2

t − c2u2x

)− V (u)]

vn = 0 at x = x0,1 . (10.67)

5Clearly, the sine-Gordon equation is nonlinear, with a nonlinear amplitude-dependent dispersionrelation. It can have solutions with properties shared by only a few other nonlinear equations. Thesesolutions are travelling wave disturbances that can pass through each other and emerge with unchangedshape, apart perhaps from a phase shift. Such solutions are also found, for example, for the nonlinearKorteweg-de Vries equation ut + αuux + νuxxx = 0, where α and ν are constants. These waves thatpreserve their shape even through interactions have been termed ‘solitons’ or ‘solitary’ waves and arefinding an expanding area of application throughout physics, from elementary particles to solid-statephysics.

10.3. APPLICATION TO ONE-DIMENSIONAL WAVE EQUATIONS 203

From (10.66) and (10.67) it can be deduced that either vn = un = 0 must hold or

vn = c

(−cun

ut

)at x = x0,1 , (10.68)

where ut and un are related by L (u) = 0, that is

u2t − c2u2

x = 2V (u) at x = x0,1 . (10.69)

The sign of vn has to be chosen such as to make sure that no energy flows into the spatialdomain, since energy inflow might spoil the stability of the problem; if an increasingspatial domain is simulated, and conservation of energy is required on this domain, noenergy inflow will occur on the fixed domain Ω. It is for this reason that vn is chosenpositive. From (10.66) it can be seen that this is enforced by requiring

unut ≤ 0 . (10.70)

If β = 0, then (10.61) and (10.64) reduce to the linear Klein-Gordon equation

utt = c2uxx − αu . (10.71)

Solutions to (10.71) can be written as

u = aeiθ , θ (x, t) = kx− ωt , (10.72)

where k is the wave number and ω is the frequency.Substitution of (10.72) into (10.71) yields the (linear) dispersion relation6

ω2 = c2k2 + α ⇒ ω =(c2k2 + α

)1/2, (10.73)

from which the phase velocity cf and the group velocity cg can be calculated:

cf ≡ ω

k= c

(1 +

α

c2k2

)1/2

, cg ≡ dω

dk= c

(1 +

α

c2k2

)−1/2

. (10.74)

Substitution of (10.72) into (10.68) gives (n is taken in positive x-direction)

vn =c2k

ω= cg , (10.75)

thus arriving at a result that might have been expected for the linear case; conservation ofenergy on a moving domain leads to a boundary velocity which equals the group velocity.

Finally, note that if both α and β equal zero, condition (10.69) reduces to Sommer-feld’s radiation condition — see Orlanski (1976) —

ut = −cun , (10.76)

corresponding to a boundary velocity which equals the local phase velocity cf if and onlyif α = 0:

vSommerfeld =ω

c2k= cf . (10.77)

6See also remark 4.


Next, we discuss the results from these numerical tests that were done for this one-dimensional wave problem; the field equation is given by the nonlinear Klein-Gordonequation

utt = uxx − αu− βu3 with α ≥ 0 , β ≥ 0 . (10.78)

Three-point central discretizations have been used for the second order derivatives in (10.78).An explicit Euler scheme has been used here, which is stable if the time step is chosensufficiently small.

The spatial domain and the time domain are given by

Ω = [−π, π] , (10.79)T = [0, 4π] . (10.80)

The initial conditions have been chosen as

u (x, 0) =

cos2 x for |x| ≤ π/20 for |x| > π/2 , (10.81)

∂u

∂t(x, 0) =

(1 + α)1/2 sin 2x for |x| ≤ π/20 for |x| > π/2

. (10.82)

These conditions represent a wave travelling in positive direction. It is obvious that (10.81-10.82) does not represent a steady wave for (10.78) if (α, β) 6= (0, 0). In the smallamplitude limit, these initial conditions may be regarded as a superposition of manymonochromatic solutions, each of them travelling at a different phase velocity. There-fore, it can be expected that from the very beginning some energy will propagate to theleft. This means that the radiation boundary conditions have to be implemented in bothend points x = ±π.

The new energy-transmitting radiation boundary condition is given by

ut = −un

(1 +

αu2 + (β/2)u4

u2n

)1/2

, (10.83)

and Sommerfeld’s radiation condition reads

ut = −un . (10.84)

Condition (10.83) can be written as

ut = −c (u) un , (10.85)

which can be regarded as a u-dependent Sommerfeld radiation condition — see (10.76)— where the correction factor (or velocity) c (u) is given by

c (u) =(

1 +αu2 + (β/2)u4

u2n

)1/2

. (10.86)

If un approaches zero, which corresponds to a local extremum of the solution u nearthe boundary, then (10.83) would give a significant value for ut, involving an undesiredchange in u at the boundary. Therefore, an upper limit is set to the correction velocity:

c∗ (u) = min(c (u) , (1 + α)1/2

). (10.87)


The new radiation boundary condition then reads

ut = −c∗ (u)un . (10.88)

Since an explicit discretization of the field equation has been chosen, conditions (10.84)and (10.88) are implemented explicitly too.

For want of analytical solutions to (10.78) with initial conditions (10.81-10.82), thenumerical solutions obtained with (10.84) and (10.88) will be compared with the solutionobtained on an extended spatial domain Ω′ ⊃ Ω. It is important that Ω′ is chosen largeenough, so that reflections from the boundaries of Ω′ do not reach the smaller domain Ωduring the time interval T . Test results have indicated that this requirement is met bythe choice

Ω′ = [−3π, 3π] . (10.89)

On this extended domain the initial conditions (10.81-10.82) are imposed. Since thenumerical errors on the interior of Ω and those on the interior of Ω′ are equal, a goodimpression of the errors due to the radiation boundary conditions is obtained in this way.

Three configurations have been used for the calculation of the numerical solutionof (10.78) with initial conditions (10.81-10.82):

1. The numerical solution is determined on the spatial domain Ω with the new energy-transmitting boundary condition (10.88) implemented in the end points x = ±π,where the correction velocity c∗ (u) is determined by (10.86-10.87).

2. The numerical solution is determined on the spatial domain Ω with Sommerfeld’sradiation condition (10.84) implemented in the end points x = ±π.

3. The numerical solution is determined on the extended spatial domain Ω′ withSommerfeld’s radiation condition (10.84) implemented in the end points x = ±3π.In this way we obtain an ‘undisturbed’ — i.e. not affected by artificial boundaries— numerical solution on the smaller domain Ω, which can be used to calculate theerrors due to the radiation boundary conditions used in configurations 1 and 2.

In order to get an impression of the error ε = u − u, where u denotes the numericalapproximation to u, two norms are used. The first norm is the L2-norm of ε, which isdefined as

‖ε‖2 ≡

π∫

−π

ε2dx

1/2

. (10.90)

The second norm is an energy norm of ε:

‖ε‖E ≡π∫

−π

12

(ε2t + ε2

x

)dx . (10.91)

We also compute the transient energy of the solutions on the restricted domain Ω, definedby

E (u) ≡π∫

−π

e (u) dx , (10.92)


with

e (u) =12

(u2

t + u2x

)+

12αu2 +

14βu4 , (10.93)

and the transient momentum of the solutions, defined by

M (u) ≡π∫

−π

m (u) dx , (10.94)

with

m (u) = uxut . (10.95)

Five tests have been done for different combinations of α and β. In the first three testsβ equals zero, yielding a linear field equation. However, note that boundary condi-tion (10.88) (configuration 1) is nonlinear if either α or β is nonzero. In the last twotests the values of both α and β are nonzero, yielding a nonlinear field equation.

Test 1: Take α = 0.1, β = 0. In this case the linear disturbance term in (10.78) israther small. Figure 10.2 shows the L2-norm of the errors in the solutions obtained withconfigurations 1 and 2. In Figure 10.3 the energy norms of the errors are presented. Theerrors obtained with the new radiation boundary condition (10.88) are about the sameas the errors obtained with Sommerfeld’s radiation condition (10.84). An explanation isdeduced from (10.86); if both α and β are small, then the correction factor c∗ (u) will bebut slightly larger than unity.

It is convenient to combine Figures 10.4-10.5 and 10.1. The first two figures show theenergy and momentum of the numerical solution obtained with configurations 1 and 2during the first two periods7 — i.e. from t = 0 to t = 4π. In these graphs the partsof rapid decrease in energy correspond to high gradients in the solution passing throughthe right boundary x = π, see Figure 10.1.8 Similarly, the inflexion points at t = π inFigures 10.4-10.5 correspond to the crest of the wave arriving at the right boundary.

Test 2: Take α = 1, β = 0. Due to the larger value of α — which implies a strongerlinear disturbance — the errors are expected to be larger than in test 1. This is confirmedby the graphs in Figures 10.6-10.7, which show the L2-norm and the energy norm of theerrors in the solutions obtained with configurations 1 and 2.

For 0 ≤ t ≤ 3 the errors obtained with the first configuration are somewhat largerthan the errors obtained with configuration 2. For t > 3 there is a clear advantage ofthe first method over the second. Apparently, α is sufficiently large now to make thecorrection velocity c∗ (u) in (10.88) effective. After two periods the energy norms of theerrors are 0.01 for configuration 1 and 0.03 for configuration 2. Figures 10.8-10.9 showthe energy and the momentum of the computed solutions in all three configurations.Note that there are two inflexion points in these graphs now, indicating two successivewave crests travelling through the ‘open’ boundary.

Test 3: Take α = 5, β = 0. Figures 10.10- 10.11 show a great advantage of thenew radiation boundary condition over Sommerfeld’s radiation condition. After two

7Note from Figures 10.4-10.5 that E (u) ≈ M (u); it can easily be shown that — with the presentinitial conditions — in the case where α = β = 0 the equality E (u) = M (u) holds.

8A close look at Figure 10.1 reveals a small wave disturbance travelling to the left.


periods the L2-norms of the errors obtained with configurations 1 and 2 are 0.13 and 0.32respectively. The energy norms of the errors are 0.10 for configuration 1 and 0.45 forconfiguration 2.

It is clear that the new radiation boundary condition provides better results for largervalues of α. As mentioned before, this is due to the correction factor c∗ (u) in (10.88).Figures 10.12-10.13 show the energy and momentum of the solutions; note that the graphsrepresenting configurations 1 and 3 nearly coincide.

From the above three tests it is clear that the new radiation condition (10.88) pro-vides much better results than Sommerfeld’s condition (10.84), especially for large val-ues of α. As mentioned earlier, this is due to the effectiveness of the correction fac-tor c∗ (u) in (10.86). In the last two tests we shall investigate the influence of a nonlineardisturbance in the field equation on the performance of boundary conditions (10.88)and (10.84).

Test 4: Take α = 1, β = 0.5. Since β is nonzero, the field equation (10.78) is nonlinear.The results obtained with configuration 1 are better than those obtained with config-uration 2, as can be seen in Figures 10.14 and 10.15. The L2-norms after two periodsare 0.21 for configuration 1 and 0.30 for configuration 2. The maximum values of the en-ergy norms are 0.02 for configuration 1 and 0.03 for configuration 2. Figures 10.16-10.17present the energy and momentum of the computed solutions.

Note that for both configurations the errors are but slightly larger than in test 2(α = 1.0 , β = 0), confer Figures 10.6-10.7. This indicates a dominant linear term αuin the disturbance in the field equation for this combination of α and β. Therefore, weshall increase the impact of the nonlinear term βu3 by raising the value of β in the nexttest.

Test 5: Take α = 1, β = 2. The expected growth in the errors (see test 4) can onlybe found in Figure 10.19, which shows the energy norm of the errors. The L2-normof the error is (for both configurations) of the same order as in test 2 (β = 0). Thisindicates a stronger variation in the derivatives of the errors due to the stronger nonlineardisturbance, see (10.91).

For this combination of α and β configuration 1 provides better results than config-uration 2 (see Figures 10.18 and 10.19). The L2-norms of the errors after two periodsare 0.17 for configuration 1 and 0.19 for configuration 2. The average values howevershow better results for configuration 1 than for configuration 2. The energy norms aftertwo periods are 0.04 for configuration 1 and 0.09 for configuration 2. Figures 10.20-10.21show the energy and momentum of the solutions.


Figure 10.1: Test 1: α = 0.1, β = 0. Solution u in configuration 1.


Figure 10.2: Test 1: α = 0.1, β = 0. Euclidean norm of error in u.

dashed line = Sommerfeld condition (10.84)dotted line = energy-transmitting condition (10.88)


Figure 10.3: Test 1: α = 0.1, β = 0. Energy norm of error in u.

solid line = extended domain solutiondashed line = Sommerfeld condition (10.84)dotted line = energy-transmitting condition (10.88)


Figure 10.4: Test 1: α = 0.1, β = 0. Energy of u.



Figure 10.5: Test 1: α = 0.1, β = 0. Momentum of u.



Figure 10.6: Test 2: α = 1, β = 0. Euclidean norm of error in u.



Figure 10.7: Test 2: α = 1, β = 0. Energy norm of error in u.



Figure 10.8: Test 2: α = 1, β = 0. Energy of u.



Figure 10.9: Test 2: α = 1, β = 0. Momentum of u.








10.4 Application to the water-wave problem

In this section we apply the above theory for radiation boundary conditions to the three-dimensional problem of nonlinear water waves travelling over a bottom.

In the numerical simulation of free surface waves under gravity, the quest for effectiveradiation boundary conditions has not been completed to full satisfaction so far. In orderto obtain a good approximation of the moving free surface, relatively dense grids haveto be used in general; consequently, the number of data to be stored is extremely high.Then, given the limited amount of available computer memory, it is clear that the fluiddomain must be cut off by an artificial boundary. With an appropriate implementationof suitable radiation conditions on these boundaries, any outgoing wave will be fullytransmitted; there will be no reflection at all, and the solution in the area of interest isnot spoiled.

The majority of the radiation conditions proposed hitherto are based on approxima-tive models, such as linear outgoing wave fields. In practical calculations however, theoutgoing waves are generated by diffraction round a floating structure or by radiationfrom an oscillating body. As a consequence, these waves generally consist of an infinitenumber of components interacting in a nonlinear fashion.

Here radiation boundary conditions are derived that simulate a moving fluid domainfor which the total energy, or the total horizontal momentum, is conserved. This meansthat the greater part of the energy of the outgoing waves will be transmitted, and hencethe reflected wave will be negligible in the sense that its energy is small. In order to

10.4. APPLICATION TO THE WATER-WAVE PROBLEM 219


demonstrate the significance of a proper variational principle for the problem underconsideration, two different approaches will be followed to arrive at such conditions. Thefirst approach is based on Luke’s (1967) variation principle and the related Hamiltonianformulation for the water wave problem described in sections 8.2 and 8.3. The secondapproach starts from the Eulerian description for an incompressible, inviscid fluid and anirrotational flow, without emphasis on the presence of a free surface. Both routes lead toradiation boundary conditions which are directly accessible to physical interpretations,but it will appear that only the first approach yields significant boundary conditions forfree surface flow.

10.4.1 The Lagrangian-Hamiltonian approach

In chapter 8 it has been demonstrated that a system of water waves travelling overan uneven bottom can be described by surface variables only. With the free surfaceelevation η in the role of canonical coordinate, and the restriction Φ = [φ]z=η of thevelocity potential to the free surface as the canonical conjugate momentum, it has beenshown that the nonlinear free surface conditions describe a so-called Hamiltonian system;it has been proven that the canonical equations, involving the energy as Hamiltoniandensity, are equivalent with the nonlinear free surface conditions.

Thus we have the composed variable (Φ, η) characterizing the system of water wavesunder gravity; in the three-dimensional case, Φ and η are both functions of the horizontalcoordinate vector ~x = (x, y)T and the time t. The action integral is given by

J (Φ, η) =∫

T

∫

Ω(t)

∫L (Φ, η) dΩ dt , (10.96)

where the two-dimensional spatial domain Ω (t) consists of all admissible horizontal po-



sition vectors ~x.In chapter 8 it was also shown that the evolution equations for water waves can be

derived from a variational principle, with the Lagrangian density L chosen as

L (Φ, η) =

η(~x;t)∫

−h(~x)

p dz = −η(~x;t)∫

−h(~x)

(φt +

12

(∇φ · ∇φ) + gz

)dz , (10.97)

where p denotes the Bernoulli pressure; the water density is taken as unity.Further, it is assumed that φ satisfies the following boundary value problem:

∇2φ = 0 for −h (~x) < z < η (~x; t) , (10.98)∂φ

∂n= 0 at z = −h (~x) , (10.99)

φ (~x, z; t) = Φ (~x; t) at z = η (~x; t) . (10.100)

Formally, the Frechet derivative of L with respect to a variation δΦ is defined as

DΦL (Φ, η) δΦ =[

d

dεL (Φ + ε δΦ, η)

]

ε=0

. (10.101)

So, let φ vary such that its variation δφ corresponds to a change δΦ in the free surfacepotential Φ. Then (10.101) can be worked out to

DΦL (Φ, η) δΦ = −η(~x;t)∫

−h(~x)

(δφt +∇φ · ∇δφ) dz


Figure 10.14: Test 4: α = 1, β = 0.5. Euclidean norm of error in u.

= − ∂t

η(~x;t)∫

−h(~x)

δφ dz −∇~x

η(~x;t)∫

−h(~x)

∇~xφ δφ dz

+

η(~x;t)∫

−h(~x)

(∇2~xφ + ∂2

zφ)δφ dz

+ [(ηt +∇~xη · ∇~xφ− φz) δφ]z=η

+ [(∇~xh · ∇~xφ + φz) δφ]z=−h , (10.102)

where ∇~x = (∂x, ∂y)T is the two-dimensional restriction of the gradient operator ∇ =(∂x, ∂y, ∂z)

T .Using (10.98-10.99), the following expression for the Frechet derivative of L with

respect to a variation in Φ is obtained:

DΦL (Φ, η) δΦ = δΦL (Φ, η) δΦ

+ ∂tB0Φ (Φ, η; δΦ) + div ~B1

Φ (Φ, η; δΦ) , (10.103)

where the variational derivative of L with respect to Φ is given by

δΦL (Φ, η) = [ηt +∇~xη · ∇~xφ− φz]z=η , (10.104)

and the corresponding boundary expressions read

B0Φ (Φ, η; δΦ) = −

η(~x;t)∫

−h(~x)

δφ dz , (10.105)


Figure 10.15: Test 4: α = 1, β = 0.5. Energy norm of error in u.

~B1Φ (Φ, η; δΦ) = −

η(~x;t)∫

−h(~x)

∇~xφ δφ dz . (10.106)

The formal definition of the Frechet derivative of L with respect to a variation in η reads

DηL (Φ, η) δη =[

d

dεL (Φ, η + ε δη)

]

ε=0

. (10.107)

However, a variation δη leads to a change in the free surface potential Φ. In first order,this change is given by — see also (8.45-8.46) —

δηΦ = δη [φz]z=η . (10.108)

Taking into account the effect of δη on Φ, the correct expression for the Frechet-derivativeof L with respect to a variation in η is

DηL (Φ, η) δη = δηL (Φ, η) δη

+ ∂tB0η (Φ, η; δη) + div ~B1

η (Φ, η; δη) , (10.109)

where the variational derivative of L with respect to η is given by

δηL (Φ, η) = −[φt +

12

(∇φ · ∇φ) + gz

]

z=η

− δΦL (Φ, η) [φz]z=η , (10.110)

and the corresponding boundary expressions read

B0η (Φ, η; δη) = 0 , (10.111)

~B1η (Φ, η; δη) = ~0 . (10.112)


Figure 10.16: Test 4: α = 1, β = 0.5. Energy of u.

With these results, it is straightforward to show that the Euler-Lagrange equation (10.8)yields

ηt +∇~xη · ∇~xφ− φz = 0 at z = η (~x; t) , (10.113)

φt +12

(∇φ · ∇φ) + gz = 0 at z = η (~x; t) . (10.114)

The so-called natural boundary condition (10.9) gives

η(~x;t)∫

−h(~x)

(∇~xφ · ~n− vn) δφ dz = 0 , (10.115)

where ~n is the unit normal vector on ∂Ω(t) and vn = ~v ·~n is the normal velocity of ∂Ω(t).Note that ~n is independent of the vertical position, whereas the boundary velocity ~v maydepend on z.

Since (10.115) must hold for arbitrary δΦ, and thus for arbitrary δφ, the naturalboundary condition reads

(∇~xφ− ~v) · ~n = 0 for − h (~x) < z < η (~x; t) . (10.116)

To obtain an additional, density conserving, boundary condition, it is worthwhile to havea closer look at the Lagrangian density, defined in (10.97). It is then observed that incase of a horizontal bottom z = −h, L does not depend explicitly on the horizontalposition ~x and the time t, and that L depends on derivatives of u = (Φ, η) up to firstorder only. As a consequence, the results of example 2 apply, and hence it is concludedthat a necessary and sufficient condition for conservation of energy and for conservation


Figure 10.17: Test 4: α = 1, β = 0.5. Momentum of u.

of momentum — in any horizontal direction — is given by

L (Φ, η) = −η(~x;t)∫

−h(~x)

(φt +

12

(∇φ · ∇φ) + gz

)dz = 0 on ∂Ω , (10.117)

stating that for all ~x on the boundary of Ω (t), the integral of the Bernoulli pressure fromthe bottom to the free surface must equal zero.

Remark 5: Since the Lagrangian density L (Φ, η) does not depend explicitly on t, theenergy density can be obtained through Noether’s theorem from the variations — seeexample 2 —

δΦ ∼ δφ = φt , δη = ηt . (10.118)

The corresponding densities are given by — see (10.43-10.44) —

χ0 (Φ, η) = L (Φ, η) = −η(~x;t)∫

−h(~x)

(φt +

12

(∇φ · ∇φ) + gz

)dz , (10.119)

~χ1 (Φ, η) = ~0 . (10.120)

With the boundary expressions — see (10.105-10.106) and (10.111-10.112) —

B0 (Φ, η) = −η(~x;t)∫

−h(~x)

φtdz , (10.121)



~B1 (Φ, η) = −η(~x;t)∫

−h(~x)

φt∇~xφdz , (10.122)

the density e = χ0 − B0 and the corresponding flux density ~fe = ~χ1 − ~B1 — see (10.20-10.21) — emerge as

e (Φ, η) = −η(~x;t)∫

−h(~x)

12

(∇φ · ∇φ) dz − 12g

(η2 − h2

), (10.123)

~fe (Φ, η) =

η(~x;t)∫

−h(~x)

φt∇~xφdz . (10.124)

Note that — apart from the minus sign — e (Φ, η) equals the energy density indeed.Then, for solutions (Φ, η) to the Euler-Lagrange equations (10.113-10.114) the follow-

ing local energy conservation law holds:

∂te +∇~x~fe = 0 . (10.125)

The proof directly follows from substitution of (10.123-10.124).

Remark 6: The above natural and additional conditions for conservation of energy andhorizontal momentum are related to some results obtained in chapter 9, in particularidentities (9.64) and (9.67), repeated here for convenience:

dIf

dt=

∫

C

p dz , (10.126)



dHf

dt=

∫

C

p (φxdz − φzdx) . (10.127)

Here If and Hf denote the horizontal fluid momentum and the fluid energy respectively.Integration is over the contour C bounding the fluid domain; with p naturally vanishingon the free surface, and with dz = 0 on a horizontal bottom, the necessary and sufficientcondition for conservation of horizontal momentum is then found to be

η∫

−h

p dz = 0 , (10.128)

at the vertical boundaries.Similarly, with φx = U and dx = 0 on the vertical boundaries, it is found that a

necessary and sufficient condition for conservation of energy is

U

η∫

−h

p dz = 0 , (10.129)

i.e. either U = 0, in which case we have fixed vertical boundaries, or the integratedpressure equals zero. Hence, both horizontal momentum and energy are conserved if theadditional boundary condition (10.117) is satisfied.

10.4.2 The Eulerian approach

The previous expressions for the Euler-Lagrange equations, the natural boundary condi-tions, and the additional boundary conditions for conservation of energy (or horizontal



momentum) were obtained at the outset of a velocity potential φ satisfying the bound-ary value problem (10.98-10.100). This approach is along the lines of the Hamiltonianformulation in chapter 8, where it is shown that the system of water waves is governedby the free surface elevation η and the free surface potential Φ.

Perhaps a more obvious and simple approach would be to consider the classical water-wave problem as a potential flow problem, without emphasizing the presence of a freesurface. From this point of view, the system is characterized by the velocity potential φ,defined throughout the transient three-dimensional fluid domain Ω (t). The boundingsurface ∂Ω(t) consists of the free surface, the bottom, and lateral boundaries which areeither physical or artificial. The proposed variational principle is

δJ (φ) = δ

∫

T

∫∫

Ω(t)

∫L (φ) dΩ dt = 0 , (10.130)

where the Lagrangian density equals the Bernoulli pressure:

L (φ) = p = −(

φt +12

(∇φ · ∇φ) + gz

). (10.131)

The Frechet derivative of L with respect to a variation δφ is

DL (φ) δφ = −δφt −∇φ · ∇δφ

= −∂tδφ−∇ · (∇φ δφ) +∇2φ δφ

= δL (φ) δφ + ∂tB0 (φ; δφ) + div ~B1 (φ; δφ) . (10.132)

Hence, the Euler-Lagrange equation becomes Laplace’s equation:

δL (φ) = ∇2φ = 0 , (10.133)



and the natural boundary condition states that the normal velocity of each boundaryparticle coincides with the local boundary velocity in normal direction:

∂φ

∂n= vn . (10.134)

For an impermeable bottom z = −h (x, y) we have vn = 0, and (10.134) reduces tothe zero-flux condition (10.99). For the free surface z = η (x, y; t), condition (10.134)becomes (10.113), so that — in this formulation — the kinematic free surface conditionplays the role of a natural boundary condition, while in the Lagrangian-Hamiltonianformulation it was part of the Euler-Lagrange equations.

Note that the second (dynamic) free surface condition (10.114) has not been obtainedfrom (10.130-10.131) so far. In this respect, the present variation principle does notprovide the complete set of governing equations for nonlinear free surface flow under theaction of gravity. It does provide, however, the field equation for bounded or unboundedpotential flow — for instance, the flow between two parallel horizontal plates.

Since the Lagrangian density L does not depend explicitly on the time t, conservation ofenergy is ensured by

L (φ) vn = p vn = 0 . (10.135)

At the bottom we have vn = 0, so the additional condition is automatically satisfiedthere. At a moving free surface, we have vn 6= 0; hence, the additional condition yieldsp = 0, which is exactly the dynamic condition (10.114).

Since the Lagrangian density does not depend explicitly on the horizontal coordinatesx and y, conservation of momentum in some horizontal direction ~σ = (σx, σy, 0)T isprovided by

L (φ) (~σ · ~n) = p (~σ · ~n) = 0 . (10.136)


Note that this condition is automatically satisfied at an even (horizontal) bottom, whilein the presence of an uneven bottom the horizontal momentum will not be preserved. Atthe free surface the additional condition reduces to the dynamic condition (10.114).

With respect to the lateral boundaries, we arrive at the vanishing of the Bernoulli pres-sure as a sufficient condition for conservation of energy and horizontal momentum; this isthe point-wise version of condition (10.117) obtained with the Lagrangian-Hamiltonianapproach. This local condition, however, is insignificant from a physical point of view,since any lateral outflow boundary connects the free surface, where p = 0, with thebottom, where p > 0.

This example shows the necessity of an appropriate variation principle for the problemunder consideration, in the application of our theory for radiation boundary conditions.


10.5 Bibliography

Bayliss, A., and Turkel, E. 1980. Radiation boundary conditions for wave-likeequations. Communications on Pure and Applied Mathematics 33:707-725.

Bayliss, A., and Turkel, E. 1982. Far field boundary conditions for compressibleflows. Journal of Computational Physics 48:182-199.

Biot, M.A. 1957. General theorem on the equivalence of group velocity and energytransport. Physical Reviews 105:1129-1137.

Broer, L.J.F. 1951. On the propagation of energy in linear conservative waves. Ap-plied Scientific Research, Series A, 329-344.

Broeze, J., and van Daalen, E.F.G. 1992. Radiation boundary conditions for thetwo-dimensional wave equation from a variational principle. Mathematics of Computa-tion 58(197):73-82.

van Daalen, E.F.G., Broeze, J., and van Groesen, E.W.C. 1992. Variationalmethods and conservation laws in the derivation of radiation boundary conditions forwave equations. Mathematics of Computation 58(197):55-71.

Engquist, B., and Halpern, L. 1988. Far field boundary conditions for computationover long time. Applied Numerical Mathematics 4:21-45.

Engquist, B., and Majda, A. 1977. Absorbing boundary conditions for the numericalsimulation of waves. Mathematics of Computation 31(139):629-651.

Engquist, B., and Majda, A. 1979. Radiation boundary conditions for acoustic andelastic wave calculations. Communications on Pure and Applied Mathematics 32:313-357.

Givoli, D. 1991. Non-reflecting boundary conditions. Journal of Computational Physics94(1):1-29.


van Groesen, E.W.C. 1980. Unidirectional wave propagation in one dimensional, firstorder Hamiltonian systems. Journal of Mathematical Physics 21:1646-1655.

van Groesen, E.W.C., and Mainardi, F. 1990. Balance laws and centro velocityin dissipative systems. Journal of Mathematical Physics 31(9):2136-2140.

Higdon, R.L. 1986. Absorbing boundary conditions for difference approximations to themulti-dimensional wave equation. Mathematics of Computation 47(176):437-459.

Higdon, R.L. 1987. Numerical absorbing boundary conditions for the wave equation.Mathematics of Computation 49(179):65-90.

Higdon, R.L. 1990. Radiation boundary conditions for elastic wave propagation. SIAMJournal on Numerical Analysis 27(4):831-870.

Lighthill, M.J. 1965. Group velocity. J. Inst. Math. Appl. 1:1-28.

Noether, E. 1918. Invariante Variationsprobleme. Nachrichten der Gesellschaft derWissenschaften Gottingen 2:235-257.

Olver, P.J. 1986. Application of Lie Groups to Differential Equations. Springer-Verlag.

Orlanski, I. 1976. A simple boundary condition for unbounded hyperbolic flows. Jour-nal of Computational Physics 21:251-269.


Wehausen, J.V., and Laitone, E.V. 1960. Surface Waves. Volume IX of Encyclo-pedia of Physics. Springer-Verlag.

Whitham, G.B. 1970. Two-timing, variational principles and waves. Journal of FluidMechanics 44(2):373-395.

Whitham, G.B. 1974. Linear and Nonlinear Waves. Wiley-Interscience.


. . .And the god9 rose up from the depths in form such as he really was. Andas when a man trains a swift steed for the broad race-course, and runs along,grasping the bushy mane, while the steed follows obeying his master, and rearshis neck aloft in his pride, and the gleaming bit rings loud as he champs itin his jaws from side to side; so the god, seizing hollow Argo’s keel, guidedher onward to the sea. And his body, from the crown of his head, round hisback and waist as far as the belly, was wondrously like that of the blessed onesin form; but below his sides the tail of a sea monster lengthened far, forkingto this side and that; and he smote the surface of the waves with the spines,which below parted into curving fins, like the horns of the new moon. Andhe guided Argo on until he sped her into the sea on her course; and quicklyhe plunged into the vast abyss; and the heroes shouted when they gazed withtheir eyes on that dread portent. There is the harbour of Argo and there arethe signs of her stay, and altars to Poseidon and Triton; for during that daythey tarried . . .

Be gracious, race of blessed chieftains! And may these songs year afteryear be sweeter to sing among men. For now have I come to the gloriousend of your toils; for no adventure befell you as ye came home from Aegina,and no tempest of winds opposed you; but quietly did ye skirt the Cecropianland and Aulis inside of Euboea and the Opuntian cities of the Locrians, andgladly did ye step forth upon the beach of Pagasae.

Argonautica, Book IV, Verses 1602-1622 and 1773-1781.

9i.e. Triton

Chapter 11

Conclusions andRecommendations

In parts I and II of this thesis we have described the development of a numerical algorithmfor the time domain simulation of nonlinear wave-body interactions. This method is basedon Romate’s higher order panel method1 for nonlinear free surface wave simulations.

The theory of nonlinear waves interacting with rigid bodies at zero forward speed— that is, both the mathematical statement of the problem and the formulation interms of boundary integral equations — was treated in chapters 2-3. An outline of thenumerical algorithm was presented in chapter 4. The most important feature of thispanel method is that the dynamic equilibrium of the waves and the body is preserved inthe time marching, by solving Laplace’s equation both for the velocity potential and forits partial time derivative.

Numerical test results for a wide range of applications were presented in chapters 5-7.The main conclusion is that the present panel method is very accurate — in view ofthe excellent agreement of the numerical results with the analytical predictions for animpulsively started vertical wavemaker, see chapter 5 — and very robust — in view ofthe results on cylinders in free heaving and rolling motion, see chapter 7. Fair agreementwith linear theory was observed for the hydrodynamic mass and damping coefficients ofcircular cylinders in forced sinusoidal heaving and swaying motion.

With respect to the further development of this panel method — and the futureextension to three dimensions — we recommend an extensive validation of the two-dimensional method. In particular, the response of cylinders on incoming wave fields —including the computation of (mean) drift forces — and an elaborated study of addedmass and damping for various cross sections — in all three modes of motion — deservemuch more attention than we have been able to give so far. Furthermore, this nonlinearmodel enables the investigation of the behaviour of hydrodynamic mass and damping forlarge amplitude motion, and of nonlinear effects for cross sections of special shape.

Part III of this thesis is concerned with the use of variation principles and Hamiltonianformulations in problems involving water waves and floating bodies.

In chapter 8 we have presented a variation principle and a Hamiltonian formulation for

1Romate, J.E. 1989. The Numerical Simulation of Nonlinear Gravity Waves in Three Dimensionsusing a Higher Order Panel Method. Ph.D. thesis, University of Twente. Enschede, The Netherlands.

233

234 CHAPTER 11. CONCLUSIONS AND RECOMMENDATIONS

the nonlinear wave-body problem, using the Legendre transformation as a tool to transitfrom the first description to the latter. The main advantage of these formulations is thatthey clearly describe the physical mechanisms that are active in wave-body systems, suchas the transfer of momentum and energy.

Based on earlier results for water waves only, a full account of symmetries and thecorresponding conservation laws for the wave-body problem was given in chapter 9. Itappears that all constants of the motion but one are preserved when a surface piercingbody is introduced. The remaining water-wave invariant and the possibility of its gen-eralization to the wave-body problem was discussed in detail. The validity of the theorywas supported with numerical results — obtained with our panel method — for a simplewave-body configuration.

Finally, in chapter 10 we demonstrated the optimal use of variation principles inthe quest for radiation boundary conditions for general wave-like problems. Based onthe conservation of a chosen density — for instance, the energy density — transmittingboundary conditions are obtained that simulate a moving domain for which the integrateddensity is conserved. An application to a nonlinear one-dimensional wave equation wasgiven, and the corresponding numerical results indicate that these density-conservingboundary conditions are superior to standard type boundary conditions, such as Som-merfeld/Orlanski conditions. The theory was also applied to the classical water waveproblem, and some of the findings in the related chapters 8 and 9 were re-established.For the near future we recommend the stable implementation of these radiation boundaryconditions in our panel method for nonlinear free surface waves and ship motions.

Part IV

Appendices

235

Appendix A

An Expression for φtn on theBody

In this appendix we shall derive an expression for

φtn =∂2φ

∂t∂n= ∇φt · ~n on S (A.1)

in terms of the body velocities and accelerations, tangential derivatives of φ up to secondorder, and first order tangential derivatives of φn. The strategy is to derive an expressionfor the normal component of the acceleration of a point on the wetted body surface S,since this variable is directly related to φtn.

Figure A.1: Definition of three-dimensional Cartesian and orthogonal curvilinear coor-dinate systems.

237

238 APPENDIX A. AN EXPRESSION FOR φTN ON THE BODY

The description is in terms of a Cartesian coordinate system x1, x2, x3, where x1 andx2 denote the horizontal directions and x3 denotes the vertical direction, and in termsof an orthogonal curvilinear coordinate system s1, s2, s3 which has its origin on S; wechoose s1 and s2 to be the tangential coordinates (i.e. the coordinates on S) and s3 = nas the normal coordinate, see Figure A.1. If ~x is the position of a point on S, its localacceleration reads

~x =d

dt

[~x]

=d

dt

[3∑

k=1

(~x · ~sk

)~sk

]

=3∑

k=1

d

dt

[~x · ~sk

]~sk +

(~x · ~sk

) d

dt[~sk]

. (A.2)

Using the orthogonality of ~s1, ~s2, ~s3, the following expressions for the material deriva-tives of ~s1, ~s2 and ~s3 can be derived:

d

dt[~s1] = ~θG × ~s1 = ~θG × (~s2 × ~s3) =

(~θG · ~s3

)~s2 −

(~θG · ~s2

)~s3 , (A.3)

d

dt[~s2] = ~θG × ~s2 = ~θG × (~s3 × ~s1) =

(~θG · ~s1

)~s3 −

(~θG · ~s3

)~s1 , (A.4)

d

dt[~s3] = ~θG × ~s3 = ~θG × (~s1 × ~s2) =

(~θG · ~s2

)~s1 −

(~θG · ~s1

)~s2 , (A.5)

where ~θG is the body rotation vector with respect to the principal axes, correspondingto roll, pitch and yaw motions respectively.

Substitution of (A.3-A.5) into (A.2) yields

~x =

d

dt

[~x · ~s1

]+

(~θG · ~s2

)(~x · ~s3

)−

(~θG · ~s3

)(~x · ~s2

)~s1

+

d

dt

[~x · ~s2

]+

(~θG · ~s3

)(~x · ~s1

)−

(~θG · ~s1

)(~x · ~s3

)~s2

+

d

dt

[~x · ~s3

]+

(~θG · ~s1

)(~x · ~s2

)−

(~θG · ~s2

)(~x · ~s1

)~s3 . (A.6)

Taking the inner product with the unit normal vector ~n = ~s3 gives the normal componentof the local acceleration:

~x · ~s3 =d

dt

[~x · ~s3

]+

(~θG · ~s1

)(~x · ~s2

)−

(~θG · ~s2

)(~x · ~s1

). (A.7)

Substituting ∇φ in terms of curvilinear coordinates for ~x, the first term on the right-handside of (A.7) reads:

d

dt

[~x · ~s3

]=

d

dt[∇φ · ~s3] =

d

dt

[(3∑

k=1

1hk

∂φ

∂sk~sk

)· ~s3

], (A.8)

where the scale factors hk arise from the transformation from Cartesian to curvilinearcoordinates.

Next, equation (A.8) is elaborated to

d

dt

[~x · ~s3

]=

d

dt

[3∑

k=1

1hk

∂φ

∂sk~sk

]· ~s3 +

(3∑

k=1

1hk

∂φ

∂sk~sk

)· d

dt[~s3] . (A.9)

239

Then, using (A.5), it is found that

d

dt

[~x · ~s3

]=

d

dt

[3∑

k=1

1hk

∂φ

∂sk~sk

]· ~s3

+1h1

∂φ

∂s1

(~θG · ~s2

)− 1

h2

∂φ

∂s2

(~θG · ~s1

). (A.10)

Substitution of (A.10) into (A.7) gives the first interim expression for the local accelera-tion in normal direction:

~x · ~s3 =d

dt

[1h1

∂φ

∂s1~s1

]· ~s3 +

d

dt

[1h2

∂φ

∂s2~s2

]· ~s3 +

d

dt

[1h3

∂φ

∂s3~s3

]· ~s3

+(~θG · ~s1

) ((~x · ~s2

)− 1

h2

∂φ

∂s2

)

−(~θG · ~s2

) ((~x · ~s1

)− 1

h1

∂φ

∂s1

). (A.11)

Attention is now focused on the first three terms on the right-hand side of (A.11); applyingthe definition of the material derivative of a scalar function f in ~x, that is

d

dt[f ] =

∂

∂t[f ] +∇ [f ] · ~x =

∂

∂t[f ] +

3∑

k=1

1hk

(~x · ~sk

) ∂

∂sk[f ] , (A.12)

and using the equations for the derivatives of the unit base vectors with respect to thecurvilinear coordinates — see, for instance, Malvern (1969) —

∂~sm

∂sm= − 1

hn

∂hm

∂sn~sn − 1

hr

∂hm

∂sr~sr for m,n, r all different , (A.13)

∂~sm

∂sn= +

1hm

∂hn

∂sm~sn for m 6= n , (A.14)

the first vector in the first term on the right-hand side of (A.11) is rewritten to

d

dt

[1h1

∂φ

∂s1~s1

]=

1h1

∂2φ

∂t∂s1+

3∑

k=1

1hk

(~x · ~sk

) ∂

∂sk

[1h1

∂φ

∂s1

]~s1

+

1h2

1h2

∂φ

∂s1

((~x · ~s2

) ∂h2

∂s1−

(~x · ~s1

) ∂h1

∂s2

)~s2

+

1h2

1h3

∂φ

∂s1

((~x · ~s3

) ∂h3

∂s1−

(~x · ~s1

) ∂h1

∂s3

)~s3 . (A.15)

It then follows that the first term on the right-hand side of (A.11) gives

d

dt

[1h1

∂φ

∂s1~s1

]· ~s3 =

1h2

1h3

∂φ

∂s1

((~x · ~s3

) ∂h3

∂s1−

(~x · ~s1

) ∂h1

∂s3

). (A.16)

Similarly, it is shown that the second term yields

d

dt

[1h2

∂φ

∂s2~s2

]· ~s3 =

1h2

2h3

∂φ

∂s2

((~x · ~s3

) ∂h3

∂s2−

(~x · ~s2

) ∂h2

∂s3

), (A.17)


and the third term

d

dt

[1h3

∂φ

∂s3~s3

]· ~s3 =

1h3

∂2φ

∂t∂s3+

3∑

k=1

1hk

(~x · ~sk

) ∂

∂sk

[1h3

∂φ

∂s3

]. (A.18)

Substitution of (A.16-A.18) into (A.11) yields the second interim expression for the localacceleration in normal direction:

~x · ~s3 =1

h21h3

∂φ

∂s1

((~x · ~s3

) ∂h3

∂s1−

(~x · ~s1

) ∂h1

∂s3

)

+1

h22h3

∂φ

∂s2

((~x · ~s3

) ∂h3

∂s2−

(~x · ~s2

) ∂h2

∂s3

)

+1h3

∂2φ

∂t∂s3+

3∑

k=1

1hk

(~x · ~sk

) ∂

∂sk

[1h3

∂φ

∂s3

]

+(~θG · ~s1

) ((~x · ~s2

)− 1

h2

∂φ

∂s2

)

−(~θG · ~s2

) ((~x · ~s1

)− 1

h1

∂φ

∂s1

). (A.19)

In general, curvilinear coordinates sk are defined in terms of rectangular Cartesian coor-dinates xm by functional equations of the form

sk = sk (x1, x2, x3) (A.20)

Reversely, the Cartesian coordinates can formally be written as functions of the curvi-linear coordinates:

xm = xm (s1, s2, s3) (A.21)

The scale factors hk are then defined by

h2k =

3∑m=1

(∂xm

∂sk

)2

, k = 1, 2, 3 (A.22)

With ~x restricted to S, it follows that

~x = ~x (s1, s2) , (A.23)

since s1 and s2 are the tangential coordinates on S.

From (A.22) and (A.23) it follows that h3 is constant along S, and therefore

∂h3

∂s1=

∂h3

∂s2= 0 . (A.24)

Contrary to h3, the scale factors h1 and h2 are not constant, since S is curved in general.The radii of curvature are defined as

1R1

= − 1h1

∂h1

∂s3,

1R2

= − 1h2

∂h2

∂s3(A.25)

241

With these simplifications and definitions the third — and final — expression for thelocal acceleration in normal direction is obtained:

~x · ~s3 =∂2φ

∂t∂s3+

(~x · ~s1

) ∂2φ

∂s1∂s3+

(~x · ~s2

) ∂2φ

∂s2∂s3+

(~x · ~s3

) ∂2φ

∂s23

+(~θG · ~s1

) ((~x · ~s2

)− ∂φ

∂s2

)−

(~θG · ~s2

) ((~x · ~s1

)− ∂φ

∂s1

)

+1

R1

∂φ

∂s1

(~x · ~s1

)+

1R2

∂φ

∂s2

(~x · ~s2

). (A.26)

Using Laplace’s equation in terms of the curvilinear coordinates, that is

∇2φ = h1h2h3

[∂

∂s1

(h1

h2h3

∂φ

∂s1

)+

∂

∂s2

(h2

h1h3

∂φ

∂s2

)+

∂

∂s3

(h3

h1h2

∂φ

∂s3

)],

(A.27)

the term ∂2φ/∂s23 can be eliminated from (A.26). Rewriting this equation then yields

the following expression for φtn on S:

∂2φ

∂t∂n= ~x · ~n

+(~θG · ~s2

) ((~x · ~s1

)− ∂φ

∂s1

)−

(~θG · ~s1

)((~x · ~s2

)− ∂φ

∂s2

)

−(

1R1

∂φ

∂s1+

∂2φ

∂s1∂n

) (~x · ~s1

)−

(1

R2

∂φ

∂s2+

∂2φ

∂s2∂n

) (~x · ~s2

)

+(

∂2φ

∂s21

+∂2φ

∂s22

−(

1R1

+1

R2

)∂φ

∂n

) (~x · ~n

)

(A.28)

where s3 has been replaced by n.

Bibliography

Malvern, L.E. 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall.


Appendix B

A Body Surface IntegralCondition

In this appendix a boundary integral condition

Γ (φt, φtn, ~x) = 0 on S (B.1)

is derived, using the expression for φtn derived in the previous appendix, and the pressureintegral expressions for the hydrodynamic forces and moments acting on the body. Itwill appear that these equalities enable us to transform the hydrodynamic equationsof motion into a boundary integral equation over the wetted part of the body surfaceS. Discretization of this boundary integral equation gives a system of linear equationsconnecting φt and φtn in the NS collocation points on S.

Expression (A.28), applied in a point ~x on S, reads — in condensed notation —

φtn (~x) = ~x · ~n (~x) + ν (~x) (B.2)

where the ‘rest term’ ν (~x) is defined as

ν (~x) =(~θG · ~s2 (~x)

)((~x · ~s1 (~x)

)− ∂φ

∂s1(~x)

)

−(~θG · ~s1 (~x)

)((~x · ~s2 (~x)

)− ∂φ

∂s2(~x)

)

−(

1R1 (~x)

∂φ

∂s1(~x) +

∂2φ

∂s1∂n(~x)

) (~x · ~s1 (~x)

)

−(

1R2 (~x)

∂φ

∂s2(~x) +

∂2φ

∂s2∂n(~x)

) (~x · ~s2 (~x)

)

+(

∂2φ

∂s21

(~x) +∂2φ

∂s22

(~x)−(

1R1 (~x)

+1

R2 (~x)

)∂φ

∂n(~x)

) (~x · ~n (~x)

).

(B.3)

The acceleration of a point ~x on S can be expressed in terms of the acceleration of thecentre of mass G and the angular acceleration:

~x = ~xG + ~θG × ~r (~x) + ~θG ×(~θG × ~r (~x)

), (B.4)

243

244 APPENDIX B. A BODY SURFACE INTEGRAL CONDITION

where ~r (~x) ≡ ~x− ~xG denotes the position of ~x relative to G.Substitution of (B.4) into (B.2) gives

φtn (~x) = ~xG · ~n (~x) +(~θG × ~r (~x)

)· ~n (~x)

+((

~θG · ~r (~x))

~θG −(~θG · ~θG

)~r (~x)

)· ~n (~x) + ν (~x) , (B.5)

where we used the identity

~a×(~b× ~c

)= (~a · ~c)~b−

(~a ·~b

)~c . (B.6)

The accelerations ~xG and ~θG are given by the equations of motion:

M~xG = ~F −Mg~e3 , ~I ⊗ ~θG = ~L , (B.7)

and the hydrodynamic (inertial) forces and moments (exerted by the fluid) are obtainedfrom pressure integrations over S:

~F =∫

S

∫p

(~ξ)

~n(~ξ)

dSξ , ~L =∫

S

∫p

(~ξ)(

~r(~ξ)× ~n

(~ξ))

dSξ . (B.8)

Substitution of (B.7) and (B.8) into (B.5) yields — after a little elementary algebra —

φtn (~x) =∫

S

∫1M

p(~ξ)(

~n (~x) · ~n(~ξ))

dSξ

+∫

S

∫p

(~ξ)

(~r (~x)× ~n (~x)) ·((

~r(~ξ)× ~n

(~ξ))

® ~I)

dSξ

+(~θG · ~r (~x)

)(~θG · ~n (~x)

)−

(~θG · ~θG

)(~r (~x) · ~n (~x))

+ ν (~x)− gn3 (~x) , (B.9)

where

~a®~b ≡(

a1

b1,a2

b2,a3

b3

)T

and ~a⊗~b ≡ (a1b1, a2b2, a3b3)T (B.10)

define a component-wise vector division and product respectively. In the derivationof (B.9) we used the identity

~a ·(~b× ~c

)=

(~a×~b

)· ~c . (B.11)

The pressure along S is obtained from Bernoulli’s equation

p(~ξ)

= −ρ

(φt

(~ξ)

+12

(∇φ

(~ξ)· ∇φ

(~ξ))

+ gξ3

). (B.12)

Substitution of (B.12) into (B.9) yields a boundary integral equation, connecting φtn

with φt along S:

φtn (~x) +∫

S

∫K

(~x, ~ξ

)φt

(~ξ)

dSξ = γ (~x) (B.13)

245

where the kernel function K(~x, ~ξ

)is regular and symmetric, and depends on the geom-

etry of the body only:

K(~x, ~ξ

)= ρ

1M

(~n (~x) · ~n

(~ξ))

+ (~r (~x)× ~n (~x)) ·((

~r (~y)× ~n(~ξ))

® ~I)

(B.14)

The discretized version of (B.13), applied in collocation point i on S, can be written as

φitn +

NS∑

j=1

Cijk φj

t = γi (B.15)

where summation is over all NS collocation points on S.The coefficients Cij

k read

Cijk = ρ∆Sj

1M

(~ni · ~nj

)+

(~ri × ~ni

) ·((

~rj × ~nj)® ~I

)(B.16)

where ∆Sj is the area of panel j.Finally, the right-hand side terms γi are defined as

γi =(~θG · ~ri

)(~θG · ~ni

)−

(~θG · ~θG

) (~ri · ~ni

)+ νi − gni

3

+NS∑

j=1

Cijk

(12

(∇φj · ∇φj)

+ gxj3

), (B.17)

with the rest terms νi given by

νi =(~θG · ~si

2

) (~x

i · ~si1 − φi

s1

)−

(~θG · ~si

1

)(~x

i · ~si2 − φi

s2

)

−(

1Ri

1

φis1

+ φis1n

) (~x

i · ~si1

)−

(1

Ri2

φis2

+ φis2n

) (~x

i · ~si2

)

+(

φis1s1

+ φis2s2

−(

1Ri

1

+1

Ri2

)φi

n

) (~x

i · ~ni)

. (B.18)

246 APPENDIX B. A BODY SURFACE INTEGRAL CONDITION

Appendix C

The Angular Body Momentum

In this appendix we derive an expression for the angular body momentum in terms ofthe mass Mb, the moment of inertia Nb, the position of the centre of gravity (xb, zb),and the angle of roll θb.

Figure C.1: Two-dimensional wave-body system.

The angular fluid momentum is defined as

Af =∫

Df

∫ρf (xφz − zφx) dx dz =

∫

Df

∫ρf (xwf − zuf ) dx dz , (C.1)

where ρf is the fluid density and (uf , wf ) is the local fluid velocity in (x, z).

247

248 APPENDIX C. THE ANGULAR BODY MOMENTUM

The analogue of Af for the body is put as

Ab =∫

Db

∫ρb (xwb − zub) dx dz (C.2)

Here ρb is the body density, and (ub, wb) denotes the velocity of a point (x, z) in the rigidbody domain Db:

ub = ˙xb − ˙θbrz , wb = ˙zb + ˙θbrx , (C.3)

where the coordinates relative to G are — see Figure C.1 —

rx = x− xb , rz = z − zb , (C.4)

Substitution of (C.3-C.4) into (C.2) yields

Ab =(

˙zb + ˙θbxb

) ∫

Db

∫ρb rx dx dz −

(˙xb − ˙θbzb

) ∫

Db

∫ρb rz dx dz

+ ˙θb

∫

Db

∫ρb

(r2x + r2

z

)dx dz + (xb ˙zb − zb ˙xb)

∫

Db

∫ρb dx dz . (C.5)

From the definition of the centre of gravity (xb, zb) it follows that∫

Db

∫ρb rx dx dz =

∫

Db

∫ρb rz dx dz = 0 . (C.6)

The body mass and moment of inertia are defined as

Mb =∫

Db

∫ρb dx dz , Nb =

∫

Db

∫ρb

(r2x + r2

z

)dx dz (C.7)

Substitution of (C.6-C.7) into (C.5) yields the following expression for the angular bodymomentum:

Ab = Nb˙θb +Mb (xb ˙zb − zb ˙xb) (C.8)

List of Figures

2.1 Fluid domain Ω and bounding surfaces. . . . . . . . . . . . . . . . . . . . 192.2 Free surface with a floating body. . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Definition of Ω, ∂Ω, and ~nξ. . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Boundary ∂Ω consisting of two smooth subsurfaces S1 and S2. . . . . . . 44

4.1 Structure of matrix equation. . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Body motion and geometry definitions. . . . . . . . . . . . . . . . . . . . . 554.3 Structure of extra matrix equation. . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Impulsive wavemaker problem: geometry definition. . . . . . . . . . . . . 705.2 Element distributions for impulsive wavemaker problem. . . . . . . . . . . 765.3 Initial potential distribution on the wavemaker. . . . . . . . . . . . . . . . 835.4 Initial vertical velocity along the wavemaker. . . . . . . . . . . . . . . . . 845.5 Initial potential distribution close to the intersection point. . . . . . . . . 855.6 Initial vertical velocity close to the intersection point. . . . . . . . . . . . 865.7 Initial pressure distribution on the wavemaker. . . . . . . . . . . . . . . . 875.8 Initial potential distribution on the bottom. . . . . . . . . . . . . . . . . . 885.9 Initial horizontal velocity along the bottom. . . . . . . . . . . . . . . . . . 895.10 Initial vertical velocity of the free surface. . . . . . . . . . . . . . . . . . . 905.11 Initial vertical free surface velocity close to the intersection point. . . . . . 915.12 Free surface elevation at times t = 0.0025− (0.0025)− 0.0200 s. . . . . . . 925.13 Free surface elevation at times t = 0.025− (0.025)− 0.200 s. . . . . . . . . 93

6.1 Two-dimensional cylinder floating on a free surface. . . . . . . . . . . . . . 986.2 Added mass coefficients for circular cylinder in heaving motion. . . . . . . 1076.3 Damping coefficients for circular cylinder in heaving motion. . . . . . . . . 1086.4 Improved added mass coefficients for circular cylinder in heaving motion. 1096.5 Improved damping coefficients for circular cylinder in heaving motion. . . 1106.6 Added mass coefficients for circular cylinder in swaying motion. . . . . . . 1116.7 Damping coefficients for circular cylinder in swaying motion. . . . . . . . 112

7.1 Transient heaving motion of a circular cylinder due to moderate initialdisplacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 Transient heaving motion of a circular cylinder due to large initial dis-placements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3 Transient heaving motion of a circular cylinder due to initial displacement.Numerical versus analytical results. . . . . . . . . . . . . . . . . . . . . . . 123

249

7.4 Transient heaving motion of a circular cylinder due to initial unit displace-ment and initial unit velocity. . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.5 Transient heaving motion of a rectangular cylinder due to initial displace-ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.6 Transient heaving motion of a rectangular cylinder due to initial displace-ment and initial velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.7 Transient rolling motion of a rectangular cylinder due to initial displace-ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.1 Free surface potential flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.2 Wave-body system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.1 Two-dimensional water waves: the infinite depth case. . . . . . . . . . . . 1609.2 Two-dimensional wave-body system. . . . . . . . . . . . . . . . . . . . . . 1649.3 Free surface waves and submerged deformable body. . . . . . . . . . . . . 1809.4 Wave-body system with impermeable fixed boundaries. . . . . . . . . . . . 1839.5 Wave-body system: initial configuration. . . . . . . . . . . . . . . . . . . . 1859.6 Wave-body system: equilibrium state. . . . . . . . . . . . . . . . . . . . . 1859.7 Conservation of mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1869.8 Conservation of horizontal momentum. . . . . . . . . . . . . . . . . . . . . 1879.9 Exchange of vertical momentum — no conservation. . . . . . . . . . . . . 1889.10 Conservation of angular momentum. . . . . . . . . . . . . . . . . . . . . . 1889.11 Exchange and conservation of energy. . . . . . . . . . . . . . . . . . . . . . 1899.12 Virial — no conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10.1 Test 1: α = 0.1, β = 0. Solution u in configuration 1. . . . . . . . . . . . . 20810.2 Test 1: α = 0.1, β = 0. Euclidean norm of error in u. . . . . . . . . . . . . 20910.3 Test 1: α = 0.1, β = 0. Energy norm of error in u. . . . . . . . . . . . . . 21010.4 Test 1: α = 0.1, β = 0. Energy of u. . . . . . . . . . . . . . . . . . . . . . 21110.5 Test 1: α = 0.1, β = 0. Momentum of u. . . . . . . . . . . . . . . . . . . . 21210.6 Test 2: α = 1, β = 0. Euclidean norm of error in u. . . . . . . . . . . . . . 21310.7 Test 2: α = 1, β = 0. Energy norm of error in u. . . . . . . . . . . . . . . 21410.8 Test 2: α = 1, β = 0. Energy of u. . . . . . . . . . . . . . . . . . . . . . . 21510.9 Test 2: α = 1, β = 0. Momentum of u. . . . . . . . . . . . . . . . . . . . . 21610.10Test 3: α = 5, β = 0. Euclidean norm of error in u. . . . . . . . . . . . . . 21710.11Test 3: α = 5, β = 0. Energy norm of error in u. . . . . . . . . . . . . . . 21810.12Test 3: α = 5, β = 0. Energy of u. . . . . . . . . . . . . . . . . . . . . . . 21910.13Test 3: α = 5, β = 0. Momentum of u. . . . . . . . . . . . . . . . . . . . . 22010.14Test 4: α = 1, β = 0.5. Euclidean norm of error in u. . . . . . . . . . . . . 22110.15Test 4: α = 1, β = 0.5. Energy norm of error in u. . . . . . . . . . . . . . 22210.16Test 4: α = 1, β = 0.5. Energy of u. . . . . . . . . . . . . . . . . . . . . . 22310.17Test 4: α = 1, β = 0.5. Momentum of u. . . . . . . . . . . . . . . . . . . . 22410.18Test 5: α = 1, β = 2. Euclidean norm of error in u. . . . . . . . . . . . . . 22510.19Test 5: α = 1, β = 2. Energy norm of error in u. . . . . . . . . . . . . . . 22610.20Test 5: α = 1, β = 2. Energy of u. . . . . . . . . . . . . . . . . . . . . . . 22710.21Test 5: α = 1, β = 2. Momentum of u. . . . . . . . . . . . . . . . . . . . . 228

A.1 Definition of three-dimensional Cartesian and orthogonal curvilinear co-ordinate systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

250

C.1 Two-dimensional wave-body system. . . . . . . . . . . . . . . . . . . . . . 247

251

252

List of Tables

2.1 Order estimation of force magnitudes. . . . . . . . . . . . . . . . . . . . . 14

3.1 Uniqueness for solutions of (3.7-3.8) with various boundary conditions. . . 433.2 Uniqueness for various surface distributions. . . . . . . . . . . . . . . . . . 43

4.1 Numerical methods for nonlinear ship motion simulations. . . . . . . . . . 52

6.1 Added mass and damping coefficients for heaving circular cylinder. . . . . 1016.2 Added mass and damping coefficients for heaving circular cylinder: effects

due to longer wave tank for ω = 0.452. . . . . . . . . . . . . . . . . . . . . 1026.3 Added mass and damping coefficients for heaving circular cylinder: effects

due to smaller cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4 Added mass and damping coefficients for swaying circular cylinder. . . . . 103

9.1 Interrelation between vector fields, symmetry groups, conserved densities,and integrals quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

253

254

Abstract

In ocean engineering many problems involve the analysis of free surface waves interact-ing with large fixed or floating bodies — either partially or totally submerged — wheresurface tension, viscosity, and compressibility effects are of minor importance and maybe neglected. Under these assumptions the fluid flow is governed by a velocity potentialwhich satisfies the time-independent Laplace equation throughout the fluid domain. Thewave evolution is described by the dynamic and kinematic free surface conditions, whichare time-dependent partial differential equations. The motion of a floating body is de-scribed by the hydrodynamic equations of motion, involving pressure integrations overthe wetted body surface.

The first and second parts of this thesis are devoted to the development of a numericalmodel for the simulation of these water-wave and wave-body systems. The algorithmis based on a boundary integral equation method for the spatial discretization, and afourth order Runge-Kutta method for the discrete time marching.

First, the formulations of both problems are considered, as well as the transition tothe boundary integral equation formulation through Green’s theorem. A brief literaturesurvey of numerical solution procedures for the nonlinear wave-body problem is presented.The most important feature of the approach followed here is that the dynamic equilibriumof the fluid and the body is preserved for all times, through the solution of integralequations for both the velocity potential and its partial time derivative.

Numerical results for various water-wave and wave-body problems show fair to excel-lent agreement with analytical predictions and experimental measurements. The presentmethod appears to be quite able to reveal nonlinear effects.

In the third part we discuss the description of water-wave and wave-body problems interms of variational principles and Hamiltonian equations of motion. Based on earlierresults for water waves only, it is demonstrated that the equations of motion for freesurface waves interacting with freely floating bodies constitute an infinite-dimensionalHamiltonian system. Explicit expressions for the canonical variables and equations arepresented.

The complete set of constants of the motion for the wave-body problem is given forboth the two-dimensional and the three-dimensional problem; explicit proofs are given,and this theory is supported with numerical results obtained with the panel methoddescribed above.

Finally, the use of variational principles and conservation laws is demonstrated inthe development of absorbing boundary conditions for general wave-like problems, andapplications to a nonlinear one-dimensional Klein-Gordon equation are discussed. Thistheory is also applied to the nonlinear water-wave problem.

255

Documents

1993 PhD VanDaalen