Water Waves and Ship Hydrodynamics An Introduction 2nd Edition A J Hermans

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Water Waves and Ship Hydrodynamics An Introduction 2nd Edition A J Hermans

Preface to the Second Edition

This book is a revision and extension of the book published by R. Timman, A.J. Hermans
and G.C. Hsiao based on the lecture notes of courses presented by Timman at
the University of Delaware in 1971 and by Hermans at the Technical University of
Delft. The main topic of the original text is based on linearised free surface water
wave theory. For many years the first edition of the book is used by Aad Hermans as
material for a course in ship hydrodynamics presented to Master students in applied
mathematics and naval architecture at the Technical University of Delft. Influenced
by the progress in the research in water waves and especially in ship hydrodynamics
the contents of the course has changed gradually. For instance in offshore engineering
the topic like the low-frequency motion of objects moored to a buoy has become
an important issue during this period. Therefore an introduction in this field has been
added. For didactic reasons the very simple rather abstract problem of the motion
of a vertical wall is added. The reason to do so is that most effects that play a role
can be treated analytically, while for a general three dimensional object some terms
can only be obtained numerically. The use of numerical programs is normal practice
in this field, therefor an introduction in the theory of integral equations is presented
and some specific problems which may arises, such how to avoid non-physical resonance
at the so called irregular frequencies may be avoided. In the first edition a
derivation of the structure of the equations of motion in all six degrees of freedom is
presented. Because the functions derived there are not easily computed in a practical
case, we restrict ourselves to the derivation of the equation of motion in one degree
of freedom.

Preface to the First Edition

In the spring of 1971, Reinier Timman visited the University of Delaware during
which time he gave a series of lectures on water waves from which these notes
grew. Those of us privileged to be present during that time will never forget the
experience. Rein Timman is not easily forgotten.
His seemingly inexhaustible energy completely overwhelmed us. Who could forget
the numbing effect of a succession of long wine-filled evenings of lively conversation
on literature, politics, education, you name it, followed early next day by
the appearance of the apparently totally refreshed red-haired giant eager to discuss
mathematical problems with keen insight and remarkable understanding, ready to
lecture on fluid dynamics and optimal control theory or a host of other subjects and
ready to work into the evening until the cycle repeated. He thought faster, knew
more, drank more and slept less than any of the mortals; he literally wore us out.
What a rare privilege indeed to have participated in this intellectual orgy. Timman’s
lively interest in almost everything coupled with his buoyant enthusiasm and infectious
optimism epitomised his approach to life, No delicate nibbling at the fringes,
he wanted every morsel of every course.
In these times of narrow specialisation, truly renaissance figures are, if not extinct,
at least a highly endangered species. But Timman was one of that rare breed.
His knowledge in virtually all areas of classical applied mathematics was prodigious.
I still marvel that while I was his doctoral student in Delft in the late fifties
working on a problem in electromagnetic scattering he had at the same time students
working in water waves, cavitation, elasticity, aerodynamics and numerical analysis.
He was a boundless source of inspiration to his students in all of these varied fields.
His inattention to detail is legendary but this did not hamper his ability to focus
on what was really important in a problem. With a wave of his large hand he
would dismiss unimportant errors while concentrating on central ideas, leaving to
us the task of setting things right mathematically. This nonchalant attitude toward
minus signs and numerical factors was probably deliberate. He wanted people to see
the forest, not the trees; to focus on the heart of the problem, not inconsequential
superficialities. He had little use for the all too prevalent penchant for examining
someone’s work looking for errors. He would read a paper looking for the gold, not
the dross; looking for what was right, not what was wrong.
Of course this did not make life easy for those around him but it did make it
interesting. This will be attested to by George Hsiao and Richard Weinacht whose
revised version of the notes from Timman’s water wave lectures appeared as a University
of Delaware report. Timman and Hsiao then planned to further revise and
expand these notes and publish them in book form, but the project came to an abrupt
halt with Reinier Timman’s untimely death in 1975. It might have remained unfinished
had not Aad Hermans’ visit of Delaware in 1980 breathed new life into it.
Together George Hsiao and Aad Hermans have completed the task of revising the
notes, reorganising the presentation, restoring the factors of 2 which Timman had
cavalierly omitted, and adding some new material. The first four chapters are based
substantially on the original notes, while the fifth chapter and appendices have been
added.
It is gratifying to see the completion of these notes. It is not unreasonable to hope
that they will provide a useful introduction to water waves for a new generation
of mathematicians and engineers. This area was perhaps first among equals in the
broad spectrum of Timman’s interests. If these notes succeed in stimulating a new
generation to concentrate on the challenging problems remaining in this field, they
will serve a fitting memorial to a remarkable man whose like will not be soon seen
again.

Contents

1 Theory ofWater Waves . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Linearised Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Small Amplitude Waves in a Steady Current . . . . . . . . 5
1.3.2 Small Amplitude Waves in a Small Velocity Flow Field . . 7
2 LinearWave Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Travelling Plane Waves . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 PlaneWaves . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 WaveEnergyTransport . . . . . . . . . . . . . . . . . . . 14
2.2 CylindricalWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Harmonic Source Singularity . . . . . . . . . . . . . . . . . . . . 20
2.4 TheMovingPressurePoint . . . . . . . . . . . . . . . . . . . . . 27
2.5 WaveFronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 WavePatterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Singularity in a Steady Current . . . . . . . . . . . . . . . . . . . 35
2.7.1 Steady Singularity . . . . . . . . . . . . . . . . . . . . . . 35
2.7.2 Oscillating Singularity . . . . . . . . . . . . . . . . . . . . 38
3 Boundary Integral Formulation and Ship Motions . . . . . . . . . . 41
3.1 Scattering of Acoustic Waves . . . . . . . . . . . . . . . . . . . . 41
3.1.1 DirectMethod . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Source Distribution . . . . . . . . . . . . . . . . . . . . . 45
3.2 ScatteringofFreeSurfaceWaves . . . . . . . . . . . . . . . . . . 46
3.2.1 FixedObject . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 DirectMethod . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.3 Source Distribution . . . . . . . . . . . . . . . . . . . . . 49
3.2.4 Motionsof aFloatingObject,ShipMotions . . . . . . . . 50
3.2.5 HeaveMotionof aFloatingObject . . . . . . . . . . . . . 53
3.3 Slow Speed Approximation . . . . . . . . . . . . . . . . . . . . . 56
4 Second-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Second-Order Wave Theory . . . . . . . . . . . . . . . . . . . . . 59
4.2 Wave-DriftForces andMoments . . . . . . . . . . . . . . . . . . 62
4.2.1 Constant and Low Frequency Drift Forces by Means of
Local Expansions . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Constant Drift Forces by Means of Far-Field Expansions . . 65
4.3 Demonstration of Second-Order Effects, a Classroom Example . . 69
4.3.1 InteractionofWaveswith aVerticalWall . . . . . . . . . . 69
4.3.2 Forceson aFixedWall . . . . . . . . . . . . . . . . . . . 71
4.3.3 MovingWall . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.4 First-OrderMotionof theWall . . . . . . . . . . . . . . . 73
4.3.5 Influence of the Motion on the Low Frequency Drift Force . 77
4.3.6 Second-Order Motion of the Wall . . . . . . . . . . . . . . 77
4.3.7 SomeObservations . . . . . . . . . . . . . . . . . . . . . 78
5 Asymptotic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Thin Ship Hydrodynamics, Michell Theory . . . . . . . . . . . . . 79
5.2 Short Wave Diffraction by a Sailing Ship . . . . . . . . . . . . . . 82
6 Flexible Floating Platform . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1 TheFiniteDraftProblem . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Semi-AnalyticSolution . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.1 Semi-InfinitePlatform . . . . . . . . . . . . . . . . . . . . 93
6.2.2 Strip of Finite Length . . . . . . . . . . . . . . . . . . . . 96
7 Irregular and Non-linearWaves . . . . . . . . . . . . . . . . . . . . . 103
7.1 Wiener Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Shallow Water Theory . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Non-linear Dispersive Waves . . . . . . . . . . . . . . . . . . . . 114
8 ShallowWater Ship Hydrodynamics . . . . . . . . . . . . . . . . . . 125
8.1 Thin Airfoil Theory . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 Slender Body Theory . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3 FreeSurfaceEffects . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4 Ships in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.5 InteractionofShips . . . . . . . . . . . . . . . . . . . . . . . . . 151
9 Appendices: Mathematical Methods . . . . . . . . . . . . . . . . . . 155
9.1 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . 155
9.2 TheMethodofCharacteristics . . . . . . . . . . . . . . . . . . . . 159
9.3 Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . 161
9.4 The Two-Dimensional Green’s Function . . . . . . . . . . . . . . 162
9.5 Simplification of the Set of Algebraic Equations . . . . . . . . . . 163
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167