A Course in Ocean Engineering by Sverre Gran, A.S. Veritas Research, Norway, Developments in Marine Technology 8 (summary)

Developments in Marine Technology, 8

P1992-1

A Course in

Ocean Engineering

A.S. Veritas Research, H0vil<, Oslo, Norway

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c Schuttersveld 2 D«tft

E L S E V I E R

p

DEVELOPMENTS IN MARINE TECHNOLOGY

Vol. 1 Marine and Offshore Safety (RA. Frieze, R.C. McGregor and I.E.

Winkle, Editors)

Vol. 2 Behaviour of Offshore Structures (J.A. Battjes, Editor)

Vol. 3 Steel in Marine Structures (C. Noordhoek and J. de Back, Editors)

Vol. 4 Floating Structures and Offshore Operations (G. van Oortmerssen,

Editor)

Vol. 5 Nonlinear Methods in Offshore Engineering (S.K. Chakrabarti)

Vol. 6 CFD and CAD in Ship Design (G. van Oortmerssen, Editor)

Vol. 7 Dynamics of Marine Vehicles and Structures in Waves (W.G. Price,

P. Temarel and A.J. Keane, Editors)

Vol. 8 A Course in Ocean Engineering (S. Gran)

Vol. 9 MARIN Jubilee 1992

O

ELSEVIER SCIENCE PUBLISHERS B.V Sara Burgerhartstraal 25

P.O. Box 211,1000 AE Amsterdam, The Netherlands

•

Library of Congress Cataloglnfl-ln-Publicatlon Data

Gran, Sverre.

193S-A course In ocean enstneerlna / Sverre Gran.

p. C B . (Developsents In aarlne technology : 6)

"Based on lectures given at the Matheiatleal I n s t i t u t e , University of Oslo, In the period froE 1982 to IQeS"—Pref.

Includes bibliographical references and Index. ISBN 0-444-88143-3

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• 92-18205 CIP

ISBN: 0-444-88143-3

© ELSEVIER SCIENCE PUBLISHERS B.V.. 1992

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PREFACE

This book is based on lectures given at the Mathern^u:al InstUute, University of Oslo, in the period from 1982 to 1986 The main objective was to offer practical and theoretical education in subjects relevant to the marine industry. The practical part was to P^l\i'^'lJj^J^''J': eround needed for consultant services, whUe the theoreti-cal part gave the combination of hydrodynamics and stathtics required in structural ^f^^^'^'^. "^^f^f^^^^^

Selected items of the work have also been used at Veritas Training Center in courses for surveyors and engineers within Det norske Veritas.

The university lectures were given to graduate and post-graduate ^^^dents m

' " " " V / has not been the intention to outline the present state-of-the-art within ocean

during a visit in Norway in 1983. . . c t / - « - / « C

book. xcfpt" 0*^' xa/'iï"^^

-S . G . May 1991

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CONTENTS

Preface

Chapter 1:

A P R E L I M I N A R Y S U R V E Y

1.1.8 Structures and Reliability 1.2 - ENVIRONMENTAL CONDITIONS

1.2.1 Wave Spectrum and Extreme Wave

Wave spectra. Extreme wave.

1.2.2 Stationary and Non-Stationary Sea Conditions

Storm models. Storm profile parameters.

1.2.3 Wave Predictions Based on Wind Statistics

Wind-driven waves. Seas from North-West. Seas from North-East.

1.2.4 Extreme Condition and Storm Statistics

Storm statistics. Wave height distributions and periodicity. lOO-years* wave in special cases. 100-years' wave through the normalised process. Distribution of sea-state maxima.

1.2.5 Extreme Waves Based on Wave Statistics

Long-term wave statistics. Optimised elementary method. Saddle-point method. Method of logarithmic moments.

1.2.6 Duration of Stationary Storms

Extreme waves and storm duration.

1.2.7 Influence of Wave Period

Dynamic sampling.

1.2.8 Joint Wave Height-Period Distribution . ^ ^.

Joint gamma distributions. Marginal wave penod distribu-tion and correladistribu-tion. Intrinsic distribudistribu-tion parameters. Short-term crest-period distribution. Long-term crest-penod distribution.

References 1 . 3 - W A V E F O R C E S

1.3.1 Limits in Offshore Loading

Persistence of sea-states.

1.3.2 Deck Level of Platform

Combination of effects. Annual Extreme resulting surface elevation, level.

1.3.3 Loads on Multi-hull Semisubmersible

Inertia force. Split force. Twist moment.

1.3.4 Wave Forces on Toroidal Platform .

Toroidal platform hull. Inerüa forces on the torus Transfer functions.

extreme wave crest. Recommended deck Page: 3 3 4 5 6 6 8 9 9 11 12 13 14 16 20 23 24 26 30 31 32 33 35 36

v i ü Contents

"1

1.3.5 Wave and Current Force on Inclined Bracing 38

Forces on slender members. Water motion characteristics. Force and force components.

1.3.6 Wave Forces on Horizontal Bracing 41

Wave forces on bracings. Force by regular design wave. Force by narrow wave spectrum. Forces by peak-enhanced spectrum. Slamming forces.

1.3.7 Forces and Moments on Gravity Structure 46

Wave forces on a submerged caisson. Dimensions and wave forces on the caisson. Wave forces on the tower. Resulting forces.

1.3.8 Ocean Waves against a Circular Wall 48

Wave forces on circular wall. Responses in random seas. Responses in regular design wave.

References 49

1.4 - B E H A V I O U R O F S T R U C T U R E S 51 I ) 1.4.1 Natural Vibration of TLP-Tethers 52

Vibration of beams.

1.4.2 Repair of Buckled Pillar 53

Reliability of pillars against buckling.

1.4.3 Bending Moment in a Ship 55

Extremes of normal distributed still-water loads. Combina-tion of still-water and wave-induced loads. Extreme bending moment and buckling probability.

1.4.4 Ultrasonic Corrosion Test 58

Fitting data with exponential gamma distribution. Minimum thickness and perforation probability.

1.4.5 Fatigue in Ship Structure 60

Fatigue loading. Fatigue data and life-time estimates.

1.4.6 Reliability against Fatigue Failure 63

Fatigue limit-state and stochastic variables. Reliability index and design point. Parametric sensitivity factors.

1.4.7 Crack Growth in Ship Deck 66

Extreme stress condition by the saddle-point method. Crack-growth during the extreme storm.

H 1.4.8 Long-term Statistics and Reliability Methods 67 Mean time between impacts. Probability distributions of

slamming forces. Slamming conditions by a reliability approach.

References 7 j

C h a p t e r 2:

RANDOM MOTION

2.1 - T I M E AND F R E Q U E N C Y DOMAIN 7 5 2.1.1 Spectra and Correlation Functions 75

Power spectral density. Autocorrelation function. 1/3-octave band analysis. Perception in time domain. Visual observation of waves.

2.1.2 Buoy Motion in the Ume Domain 79

Equations of motion. Free motion. Forced motion in waves. Complete solution in the time domain.

2.1.3 Buoy Motion hi the Frequency Domain 82

I Frequency domain representation. Solution of the motion

Contents

2.1.4 Sampling and Frequency Resolution

Sampling theorem and uncertainty principle. Numerical Fourier transforms.

References 2.2 - RANDOM P H A S E

2.2.1 Regular Wave with Random Phase

Regular wave. Random phases. Distribution of surface level. Frequency and RMS-value.

2.2.2 Launching a Waverider

Drop height. Drop forces.

2.2.3 The Beta Distribution

Probability functions. Characteristic function. Statistical moments. Parameter estimation.

2.2.4 Generalisation of the Beta Distribution

Power transformation. Two-sided case.

2.2.5 General F-distribution.

Relation to the beta distribution. Two-sided case.

2.2.6 Transformation Properties.

Transformation of beta distribution. Transformation of F-distribution. Relation between beta- and F-distributions. Tangent distribution of uniform phase.

Appendix 2.2-A

Hyper-Geometric Functions

Gauss' hypergeometric function. Confluent hypergeometric functions.

2.3 - GAUSSIAN W A V E S

2.3.1 Lmear Superposition and Normal Probability

Principle of linear superposition. Gaussian surface and sig-nificant wave height. Surface velocity and average zero crossing wave period. Joint distribution of displacement and velocity. Average period and intensity of events.

2.3.2 Threshold Crossing and Slamming Impacts

Slamming forces. Threshold crossing and velocity distribu-tion. Maximum velocity and extreme force. Audible ming force. Generalised gamma distributions in wave slam-ming.

2.3.3 Statistical Distribution of Wave Heights

Distribution of velocity amplitudes and crest heights. Dura-tion of submersion.

References

Appendix 2.3-A

Numerical Calculation of the Normal Probability

Normal probability inte^al. Approximation with rational function. Asymptotic series. Inverse function.

2.4 - MOMENTS AND C H A R A C T E R I S T I C F U N C T I O N 2.4.1 Moments and Characteristic Functions

Statistical moments. Characteristic function. Cumulants and central moments. Examples of characteristic functions.

2.4.2 Four Rules for Moments and Characteristic Functions

Logarithmic transformation. Product of random variables. Sum of random variables. Additivity of central moments.

i

X Contents

t

2.5 - S P E C T R A AND C O R R E L A T I O N F U N C T I O N S 125

2.5.1 Encountering Spectra and Ship Responses 125 Empirical response spectra. Doppler frequency.

Encounter-ing wave spectrum.

2.5.2 Narrow-Banded Ship Response 129 Equation of motion. Variance of narrow-banded response.

Uniform spectrum.

2.5.3 Envelope Considerations 133 Envelope of narrow-banded signal. The envelope and its

velocity. Envelope crossing and duration of lulls.

2.5.4 Time Simulation 137 Simulating a given spectrum. Process with uniform

spec-trum.

2.5.5 Auto-correlation Functions 140 General properties. Periods and spectral width.

Crest-to-trough heights. Application to the box spectrum.

^~ 2.6 - GAMMA DISTRIBUTIONS 147

2.6.1 The Ordinary Gamma Distribution 147 Probability functions. Characteristic function and moments.

2.6.2 Generalisation of the Gamma Distribution 149 Probability functions. Moments. Most probable value.

Exceedance probability and maximum value

2.6.3 Transformation Properties 152 Power transformation. Dynamic sampling.

2.6.4 Pierson-Moskowitz' Frequency Distribution 153

2.6.5 Estimation of Parameters 154 Moments. Evaluation of parameters. Small logarithmic

skewness. Large logarithmic skewness.

2.6.6 Distribution of Products. 157 Approximate gamma distribution. Exact solution.

References 159 Appendix 2.6-A

Gamma Functions 159

1^ Complete gamma function. Poly-gamma functions.

" i Recurrence relations. Solution for skewness parameter.

2.7 - JOINT GAMMA D I S T R I B U T I O N S 163

2.7.1 General Probability Function 163 Elementary distribution. Generalised distribution. Joint

moments. Logarithmic moments. Marginal probability dis-tributions. Degenerated cases.

2.7.2 Symmetric Probability Density 169 Density and moments. Estimation of parameters. Joint

Rayleigh distribution.

2.7.3 Power Transformations 172 Distribution of a product. Connection to the F-distribution.

Joint distribution of products. Application to symmetrically distributed variables.

2.7.4 Dynamic Sampling 177 Sampling of an envelope. Dynamic sampling of two

vari-ables.

2.7.5 Conditional Gamma Distributions 180 Density and moments. Parameter estimation. Parameters of

Contents

2.7.6 Joint Log-Normal ProbabiUty Distribution 184

Probability density and moments. Estimation of parameters.

References 186 2.8 - WAVE S P E C T R A 187

2.8.1 Gamma Spectra 187

Relation to probability density. Standard form of the gamma spectrum. Discussion of parameters and alternative forms. Significant wave height and average wave period as parameters. Explicit formulae for the Pierson-Moskowitz spectrum.

2.8.2 Peak-Enhanced Wave Spectra 193

Narrow banded peak wave. Effect on spectral moments. Effect on periods. Effect on slope. Effect on spectral width.

2.8.3 JONSWAP Spectrum 196

Relation to peak-enhanced spectrum. Parametrisation of the JONSWAP spectrum. References 19i " Appendix 2.8-A: Planck's Spectrum 199

Chapter 3: ^ ^ ^ ^ ^ F O R C E S

Basic equations of motion. Velocity potentials. Boundary conditions.

3.1.2 Wave Solutions 208

Solution in two dimensions. Evanescent modes. Propagat-ing mode. Shallow water case. Deep water case.

3.1.3 Wave Impedance and Reflection Factor 213

Specific wave impedance. ReOection factor. Admittance and conformal mapping.

3.1.4 Orthogonality Relationships 2ir fl

Variational properties. Orthogonality. Definite integrals.

3.1.5 Generation of Waves 218

Wavemakers. Coupling with wave modes. Added mass and damping.

3.1.6 Reflection and Absorption of Waves 222

Impedance relationships. ReOection factor. Absorption of energy.

References

3.2 - C O M P L E X POTENTIAL 225 3.2.1 Complex Variables and the Hilbert Transform 225

Complex variables. Contour integral. Harmonic solution. Transformation in time. Velocity components.

3.2.2 Waves from Local Disturbance 229

Initial disturbance of the surface. Dimensional considera-tion. Solution by confiuent hypergeometric functions. Ini-tial surface conditions. Regular waves with phase restric-tions.

Contents

•

3.2.3 Complex Frequency 233 Complex dispersion relationship. Integration by the method

of the steepest descent.

References 236 3.3 - G R O U P S AND E N E R G Y 237

3.3.1 Groups and Pulses 238 Phase velocity. Group velocity.

3.3.2 Wave Motion and Hilbert Transforms 240 Measurement of horizontal displacement. Radial particle

displacement. Velocity components. Local wave frequency.

3.3.3 Energy Relationships 244 Potential and kinetic energy. Energy transport.

3.3.4 Distribution of Wave Amplitudes 247 Transformation to polar coordinates. Distribution of wave

crests.

References 249 Appendix 3.3-A:

Classification of Wave Phenomena after Dispersion 249

3.4 - WIND-DRIVEN W A V E S 253

3.4.1 Free Surface Condition 253 Boundary condition on the sea surface. Dispersion

relation-ships. Gravity and capillarity. Driving force.

3.4.2 Growing and Fading Waves 255 Application to a regular wave. Regular waves growing in

time. Regular waves growing in space. Fading waves.

3.4.3 Interaction between Wind and Waves 257 A i r flow over a wavy surface. A i r pressure. Numerical

value of the interaction coefficient. Modified gravity.

3.4.4 Effect of the Wind on the Wave Spectrum 260 Random waves. A i r pressure by random waves. Energy

equation. Fetch and time limitations.

3.4.5 Growth of Seas in a Storm 263 One-parameter wave spectrum. Differential equation for the

peak frequency. Solutions in special cases. Actual sea state variables. Approximation by short fetch and time.

3.4.6 Discharge of Waves after a Storm 267 Vanishing of spectral lines. Differential equation. Sea state

history for limited storm duration. Transformation of wind speed.

References 270 3.5 - C I R C U L A R W A V E S AND T O W E R S 271

3.5.1 Plane and Circular Waves 272 Expansion of regular waves in Bessel functions. Swirling,

divergent and convergent waves. Wave on finite depth.

3.5.2 Forces and Moments on Bodies 274 General formulae. The Froude-Kriloff force on a circular

dock.

3.5.3 Scattering of Waves around a Circular Tower 277 Scattered velocity potential. Force on the tower.

3.5.4 Scattering around a Submerged Caisson 280 Basic equations. Boundary conditions. Horizontal force.

Vertical force. Moments. Comparison with accurate methods.

Contents xiü

3.5.5 Very Long Wave Approximations

Approximated formulae. Approximation to forces. Moments.

References

3.6 - MORlSON's EQUATION

3.6.1 Theoretical Considerations

Change of momentum. Inertia force.

3.6.2 Dimensional Considerations

Dimensional force equation. Drag coefficients.

3.6.3 Force on Slender Elements

Basic vector formulation. Member in transverse, vertical plane. Force on vertical member. Force on horizontal, transverse member. Member in the longitudinal, vertical plane. Member in the horizontal plane.

3.6.4 Force on Vertical Pile

Current and finite depth.

3.6.5 Forces on a Toroidal Platform

Description of concept. Force components. Total forces. Internal loads.

3.6.6 Forces on Multi-Hull Semisubmersibles

Split force. Twist moment on two-pontoon semisubmersi-ble. References 3.7 - FORCE ON BRACINGS 3.7.1 3.7.2 3.7.3 3.7.4 3.7.5 3.7.6 3.7.7

Sub-Surface Spectrum of Narrow-Banded Wave

Representation of wave components. Narrow-banded wave component. Spectrum and moments below the surface.

Sub-Surface Spectrum of a Broad-Banded Wave

Frequency and period spectra. Spectra below the surface. Spectral moments. Integration by the method of steepest descent.

Sub-Surface Moments of Peak-Enhanced Wave

Spectral moments. Spectral width.

Force due to Narrow-Banded Wave

Force components. Force amplitude distribution. Max-imum force.

Representation by WeibuU Distribution.

Estimation of parameters.

Force due to Broad-Banded Waves

Force amplitude distributions. Inertia-to-drag force ratio.

Engineering Approach to Global Forces

Basic drag. Oblique member. Inertia force. Resulting local force. Global, horizontal maximum force. Elements at the surface.

3.8 - BEAM MODELS

3.8.1 Material Properties

Stress-strain relationship. Hooke's law.

3.8.2 Curve Length Integrals . , » j .

Wired ropes. Energy m a stretched fibre. Bendmg ot a rod over a wheel. Bending of a wire over a wheel.

285 287 289 290 291 294 300 301

4

309

xiv Contents

3.8.3 Vibration and Buckling of a Pillar 339

Deformation modes. Natural frequencies. Buckling modes.

3.8.4 Equations of Motion 341

Lagrange's equations. Boundary conditions. Reconsidera-tion of the support pillar. Flexural waves.

3.8.5 Wave Induced Stresses in a Bracing 344

Quasi-static solution. Vibration modes.

3.8.6 Ship Hull Beams 348

Reference functions for moment and shear force. Bending moment and shear force in a ship hull. Statistical considera-tions.

3.8.7 Rigid Body Ship Motion 353

Rigid body solutions. Eulerian angles.

References 357

Chapter 4:

E X T R E M E S AND S A F E T Y

4.1 - E X C U R S I O N S AND L O C A L MAXIMA 361 4.1.1 Joint Normal Probability 362

Probability density. Characteristic function and moments.

4.1.2 Transformation to Polar Coordinates 364

Resultant ship rotation. Angular distribution. Radial distri-bution. Maximum excursion.

4.1.3 Displacement and Acceleration 370

Joint probability density. Conditional acceleration by given displacement. Linear transformations. Displacement, velo-city and acceleration.

4.1.4 Distribution of Local Maxima 373

Criteria for a local maximum. Local maxima in a Gaussian process. Largest local maximum.

4.1.5 Distribution of Positive Maxima 377

Truncated Rice distribution. Representation by gamma dis-tributions.

References 379 4.2 - E X P O N E N T I A L GAMMA D I S T R I B U T I O N S 381

4.2.1 The Double Exponential Distribution 381

Extreme of the exponential distribution. Double exponential probability density. Characteristic function and moments.

4.2.2 Exponential Gamma Distribution 384

Probability distribution functions. Characteristic function and moments. Approximation to Rice's distribution.

4.2.3 Mode of the Exponential Gamma Distribution 389

Curvature and asymmetry. Integration and curve fitting. Integration by second-order method.

4.2.4 Application to the Pierson-Moskowitz Spectrum 392

Mode of the spectrum. Evaluation of moments. Alternative formula for the P-M spectrum.

4.2.5 Transformation Properties 394

Linear transformations. Dynamic sampling. Sum of vari-ables. Exponential generalised F-distribution.

Contents

4.2.6 Joint Exponential Gamma Probability

Probability density. Characteristic function and moments. Marginal probability distributions.

4.3 - E X T R E M E V A L U E DISTRIBUTIONS

4.3.1 Exponential Gamma Distribution for the Extreme Value General formula and exact extreme value distributions. Exponential gamma distribution parameters. Approximation by double exponential distribution.

4.3.2 Extreme Value Distributions m Special Cases

Extreme of the normal distribution. Extreme of the general gamma distribution. Extreme of Rayleigh distribution. Extreme of Rice's distribution. Extreme of the exponential gamma distribution. ExUeme of the log-normal distribution. 4.3.3 General Gamma Distribution for the Extreme Value

Extreme gamma distribution for the log-normal distribution. Extreme of Frechet-like gamma distribution.

4.3.4 Extremes under Non-Stationary Conditions

Extreme wave in a storm. Relationship to error function. Other storm forms. Other responses.

4.4 - S T O R M S T A T I S T I C S

4.4.1 Sea-State and Related Gaussian Process

Empirical evidence. Transformation of gamma distributed wave height to a Gaussian variable. Log-normal wave height.

4.4.2 Storm Maxima

Relationship to the Rice distribution. Limttmg cases. Log-normal wave height. Application of gamma distributions. 4.4.3 Duration of Storms

Definition of storm duration. Tentative formulae. 4.4.4 Persistence of Sea-States

Regularity of marine activities. Average threshold crossmg period.

4.4.5 Time Simulation of Sea-States

Simulation method. Seasonal variations.

References

4.5 - L O N G - T E R M DISTRIBUTIONS

4.5.1 Elementary Approach « • ^ Maximum amplitude and extreme condition. Optimised extreme value. Distribution of individual amplitudes.

4.5.2 Amplitude Distributions and Period Variations Distribution over time. Distribution over cycles. 4 5 3 Long-Term Distribution by the Method of Moments

Moments of long-term ampUtude disü-ibution. Approximate gamma distribution. Long-term extremes.

4.5.4 Exact Long-Term Distributions , Long-Term distributions in integral form. Solution m terms

of F-distributions. Solution by error functions. Large Fr'echet-distributed amplitudes. Long-term distribution of periods.

4.5.5 Design Condition by Saddle-Point . Approximation for the exceedance probabihty integral.

General gamma distiibution. Extreme ampUtude. Relation-ship to the elementary procedure.

xvi Contents

4.6 - J O I N T L O N G - T E R M DISTRIBUTIONS 457 4.6.1 Joint Distributions and Moment Equations 457

Specification of probability functions. Joint moments. Log-arithmic moment equations.

4.6.2 Marginal Distribution Relationships 461

Marginal distributions in separate cases. Relationship to uni-variate long-term distributions.

4.6.3 Joint Conditional Distributions 463

Conditional probability functions. Joint long-term distribu-tion.

4.6.4 Joint Wave Crest-Period Distribution 466

Short-term crest-period distribution. Conditional period dis-tribution by given crest height. Long-term H,—re-distribution. Long-term crest-period H,—re-distribution.

References 474

4.7 - F A T I G U E 475 4.7.1 Fatigue Loading 476

Sources of cyclic forces. Statistical stress distributions. Long-term stress-range distribution. Stress concentration factor.

4.7.2 Fatigue Data 481

S-N Curves in general. Welded steel connections.

4.7.3 Closed-Form Fatigue Life Formulae 485

General considerations. Basic, double-logarithmic S-N curve. S-N curves with fatigue threshold. Bilinear S-N curves. Semi-logarithmic S-N curves. Fatigue by transient loading.

4.7.4 Natural Dispersion 491

Distribution of individual steps. Equation of motion for the usage factor. Moments and approximate solutions. A ran-dom walk model.

4.7.5 Fracture Mechanics Approach 497

Crack origin and stress singularities. Crack growth. Crack size probability.

4.7.6 Life-Time ProbabUity 503

Initial condition. Constant growth rate. Linear crack growth rate. Growth rate proportional to x*.

References 506

4.8 - S A F E T Y 507 4.8.1 Reliability and Damage Reports 508

Reliability indices. Damage reports. Optimal codes. Relia-bility and information flow.

4.8.2 Structural Reliability 515

Supporting pillar in ship. Limit-state function and estimated reliability. Levels of reliability methods.

4.8.3 Design Point and Sensitivity Factors 519

Palmgren-Miner's formula as a limit-state function. Design point. Sensitivity factors.

Contents

4.8.4 Failure Intensity _

Inverse index and related variables. Continuous case. Overloading by repeated loads. Double exponential load -exponential gamma strengtii. Deteriorating processes. Multi-modal system failure.

4.8.5 ReHabUity and Casualty Messages , ^ . ,

Structural losses. Fatal accidents. Seventy of fatal accidenU. Acceptable risk and perceived information.

References A P P E N D I C E S AND INDEX

Appendix A

Wind Scale and Wave Charts Appendix B

Table of Gamma and Poly-Gamma Functions Index