Finite Amplitude Deep Water Waves : a Comparison of Theoretical And

Total Page:16

File Type:pdf, Size:1020Kb

Finite Amplitude Deep Water Waves : a Comparison of Theoretical And AN ABSTRACT OF THE THESIS OF Robert Edward Jensen for the degree Master of Ocean Engineering (Name) (Degree) in Civil Engineering presented on June 9, 1978 (Major Department) Title: FINITE AMPLITUDE DEEP WATER WAVES: A COMPARISON OF THEORETICAL AND EXPERIMENTAL KINEMATICS AND DYNAMICS Redacted for Privacy Abstract approved: Dr. Charles K. Sollitt Problems associated with quantification of deepwater wave kine- matics and dynamics are examined. Summaries of Stokes' theory and Dean's stream function theory are reported. A computer algorithm is developed to solve the hydrodynamic characteristics for Stokes' fifth andthird order and linear theories. Tests are initiated to verify the results obtained from the computer routine. Experimental data are obtained for several hydrodynamic characteristics in deepwater conditions. Com- parisons between Dean's stream function, Stokes' fifthand third order and linear wave theory, and experimental resultsare made for various deep water conditions. Finite Amplitude Deep Water Waves: A Comparison of Theoretical and Experimental Kinematics and Dynamics by Robert Edward Jensen A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Ocean Engineering Completed June 9, 1978 Commencement June 1979 APPROVED: Redacted for Privacy Professor of Civil Engineering in charge of major Redacted for Privacy Head of D rtment of ivil Engineerin Redacted for Privacy/ Dean of Graduate School Date thesis is presented June 91978 Typed by Pameila Jensen for Robert Edward Jensen ACKNOWLEDGEMENTS This project would not be complete without acknowledging various individuals for their support of this thesis. Special thanks to the Sea Grant College Program for the funding of this project; to Dr. John Nath, Director of the Oregon State University Wave Research Facility for donating the needed time to conduct the experimental program; and to Dr. D.R. Basco, Associate Professor at Texas A&M University for stimulating discussions on the subject of water wave kinematics and dynamics. Two other individuals proven to be indispensable during the experimental work are Terry Dibble and Larry Crawford. Final re- view of the thesis could not have been accomplished without the support of Dr. R.T. Hudspeth, Dr. P.C. Klingeman and Dr. T. Yamamoto.A very special thanks to a professor whom I have grown to admire for his intelligence, guidance and support on this project and throughout the course of my academic career at Oregon State University goes to Dr. C.K. Sollitt. Finally, I would like to express my appreciation to my wife, Pam, not only for typing of the manuscript but also for her patience and understanding. TABLE 02 CONTENTS I. INTRODUCTION 01 1.1 Problem Statement 01 1.2 Previous Theoretical Work 02 1.3 Previous Experimental Work 04 II. THEORY 06 2.1 Boundary Valu2 Problem 06 2.2 Stokes' and Linear Wave Theories 11 2.2.1 Solution tc the Boundary Value Problem 11 2.2.2 Iteration Technique for Solving Definite Equations 17 2.2.3 Calculation of Hydrodynamic Parameters Above the Still Water Level 20 2.3 Dean's Stream Function Theory 27 III. DIGITAL COMPUTATIONS 31 3.1 Stokes' Wave Theory Computer Algorithm 31 3.2 Implementation of Dean's Stream Function 40 IV. EXPERIMENTAL APPARATUS AND PROCEDURES 41 4.1 Wave Research Facility 41 4.2 Kinematic Measurements 41 4.3 Dynamic Measurements 48 V. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS . 52 5.1 Background Information 52 5.2 Theoretical and Experimental Results 53 VI. CONCLUSIONS 08 6.1 Summary 98 6.2 Evaluation 98 6.3 Future investigations 102 Bibliography 103 Appendix A 106 Appendix B 103 Appendix C 109 Program STOKES 110 Program PERTA 126 Appendix D 141 TABLE OF CONTENTS (Continue6) Appendix E 143 Appendix F 151 Index 161 List of Figures Figure Page 1.1 Approximate Range of E:perimental Work 05 2.1 Definition Sketch 07 2.2 Definition Sketch of the Method Employed in the Taylor 22 Series Expansion 3.1 Flow Diagram of the Program STOKES 32 3.2 Definition Sketch 33 3.3 Relative Operational Range of the STOKES Program 39 4.1 Test Facility 42 4.2 Calibration Curve for the Current Meter 45 4.3 Current Meter and Pressure Transducer Set up, Locking 46 From the Wave Board 5.1 Percent Difference of Theoretical Wave Crest Amplitudes 56 in Comparison to Measured. Results 5.2 Percent Difference of Theoretical Wave Trough Amplitudes 57 in Comparison to Measured Results 5.3 Case 7.5C' Maximum Nondimensional Kinematic and Dynamic 61 Profiles 5.4 Case 7.5B Maximum Nondimensional Kinematic and Dynamic 64 Profiles 5.5 Case 7.5A Maximum Nondimensional Kinematic andDynamic 66 Profiles 5.6 Case 8C' Maximum Nondimensional Kinematic andDynamic 68 Profiles 5.7 Case 8B Maximum Nondimensional Kinematic andDynamic 70 Profiles 5.8 Case 8A Maximum Nondimensional Kinematicand Dynamic 71 Profiles 5.9 Case 9C' Maximum Nondimensional Kinematic andDynamic 73 Profiles 5.10 Case 9B Maximum Nondimensional Kinematic andDynamic 74 Profiles List cf Figures (Continued) Figure Page 5.11 Case 9A Maximum Nondimensional Kinematic and Dynamic 75 Profiles 5.12 Comparison Between Theoretical and Measured Horizontal 77 Accelerations at Two Different Depths 5.13 Case 7.5C' and 7.5B Horizontal Velocity and Dynamic 80 Pressure Profiles Under Wave Trough 5.14 Case 8C' and 8B Horizontal Velocity and Dynamic 81 Pressure Profiles Under Wave Trough 5.15 Case 9C' and 9B Horizontal Velocity and Dynamic 84 Pressure Profiles Under Wave Trough 5.16 Dynamic Breaking Wave Criteria 85 5.17 Case 7.5C' and 7,5B Maximum Nondimensional Horizontal 89 Velocity. Gradient. Profiles 5.18 Case 8C' and 8B Maximum Nondimensional Horizontal 90 Velocity Gradient. Profiles 5.19Case 9C' and 9B Maximum Nondimensional Horizontal 91 Velocity Gradient Profiles 5.20Definition Sketch, the Adjustment in Theoretical 95 Velocity Profiles List of Photographs Photograph Page 4.1 Current Meter 44 4.2 Dynamic Pressure Measurement Apparatus 50 4.3 Pressure Gradient Measurement Apparatus 50 List of Tables Table Page 2.1 Comparison of Hydrodynamic Parameters Above the S.W.L. 25 3.1 Approximate Running Time and Ccst for STOKES 38 5.1 Experimental Wave a)nditions 52-53 5.2 Comparison of Crest and Trough Amplitudes 54-55 5.3 Phase Angle Relationships Between Experimental and 78-79 Theoretical Results 5.4 Wave Reflection Analysis Causing Contamination of 93 Horizontal Velocity Measurements 5.5 Adjustments in the Horizontal Velocity Distribution 96 List of Syrobols A Dean's stream function representation of wave heights which are 0.25 of the theoretical breaking wave height A.. Coefficients associated with the expansion 1J of the velocity potential to the i andjth order of Stokes' theory B Dean's stream function representation of wave heights which are 0.5 of the theoretical breaking wave height, referenced in Chapter 5 and 6 B(t) - B Bernoulli constant B Average value of all Bn's, in the least squares analysis of Dean's stream function theory B.. Coefficients associated with the expansion of 13 the water surface profile to the i and 3nth order of Stokes' theory C Wave celerity CO Linear wave theory wave celerity C. Coefficients associated with the expansion of 1 the dynamic free surface boundary condition to the ith order of Stokes' theory C Classification of wave heights which are ap- proximately equal to 0.75 of the theoretical breaking wave height F Arbitrary function in Section 2.2.3; force elsewhere H Wave height HO Wave height in computer routines Hz Hertz JJ The maximum order of Dean's stream function theory K. Shorthand notation in conjunction with the J coefficients associated with the jth order of the velocity potential for Stokes' theory List of Symbols (Continued) K . Shorthand notation in conjunction with the 3 coefficients associated with the jth order of the water surface profile for Stokes' theory L Results obtained employing linear wave theory, Chapter 5; wavelength elsewhere I, Deep water wavelength NUMETA Number of locations above the S.W.L. to the wave crest where the hydrodynamics are com- puted employing the STOKES program NUMH Number of locations from one wave height below the S.W.L. to the mudline where the hydrody- namics are computed employing the STOKESpro gram NUMH Number of locations from the S.W.L. to one wave height in depth where the hydrodynamics are computed employing the STOKES program NUMX Number of uniform phase angle locations where the hydrodynamics are computed employing the STOKES program P Pressure field P D Dynamic pressure P Total pressure T S Results obtained employing stream function theory S.W.L. Still water level T Wave period a Acceleration a Horizontal acceleration x a Vertical acceleration z c Hyperbolic cosin, Appndix A and B List of Symbol9 (Continued) d Total Derivative e Exponential . th f. Arbitary function to the 3 order f(a,a), fl(a,a), f2(a,0) Arbitrary functions in Section 2,2.2 g Gravitational acceleration h Water depth i Unit vector in the x-- direction j Order of the wave theory k Unit vector in the z-direction m Mass q Velocity vector s Hyperbolic sine, Appendix A and B t Time variable u Horizontal velocity component w Vertical velocity component x Longitudinal coordinate z Depth coordinate 3 Results obtained employing Stokes' third order wave theory 5 Results obtained employing Stokes' fifth order wave theory a Arbitrary constant equal to Kh Arbitrary constant equal to A a Specific weight of water A The change of a particular variable List of Symbols (Continued) Error term Displacement of water particles in the vertical direction Water surface profile 0 Phase angle equal to K(x Ct) K Wave number equal to 27/L Order multiplier PI equal to 3.1415926 p Mass density of water Summation Velocity potential X Stream function coefficients Stream function Vector gradient operator i 3 /3x + 173/3z 3 Partial derivative operator Percent Subscripts I Depth intervals J Order of Stokes' wave theory M Phase angle intervals i Order of Stokes' wave theory j Order of any wave theory max Maximum n Incremental phase angles fl Evaluated at the free surface List of Symbols (Continu,..d) Superscripts K Iteration number The derivative with respect to a or Kh FINITE AMPLITUDE DEEP WATER WAVES: A COMPARISON OF THEORETICAL AND EXPERIMENTAL KINEMATICS AND DYNAMICS I.
Recommended publications
  • Accurate Modelling of Uni-Directional Surface Waves
    Journal of Computational and Applied Mathematics 234 (2010) 1747–1756 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Accurate modelling of uni-directional surface waves E. van Groesen a,b,∗, Andonowati a,b,c, L. She Liam a, I. Lakhturov a a Applied Mathematics, University of Twente, The Netherlands b LabMath-Indonesia, Bandung, Indonesia c Mathematics, Institut Teknologi Bandung, Indonesia article info a b s t r a c t Article history: This paper shows the use of consistent variational modelling to obtain and verify an Received 30 November 2007 accurate model for uni-directional surface water waves. Starting from Luke's variational Received in revised form 3 July 2008 principle for inviscid irrotational water flow, Zakharov's Hamiltonian formulation is derived to obtain a description in surface variables only. Keeping the exact dispersion Keywords: properties of infinitesimal waves, the kinetic energy is approximated. Invoking a uni- AB-equation directionalization constraint leads to the recently proposed AB-equation, a KdV-type Surface waves of equation that is also valid on infinitely deep water, that is exact in dispersion for Variational modelling KdV-type of equation infinitesimal waves, and that is second order accurate in the wave height. The accuracy of the model is illustrated for two different cases. One concerns the formulation of steady periodic waves as relative equilibria; the resulting wave profiles and the speed are good approximations of Stokes waves, even for the Highest Stokes Wave on deep water. A second case shows simulations of severely distorting downstream running bi-chromatic wave groups; comparison with laboratory measurements show good agreement of propagation speeds and of wave and envelope distortions.
    [Show full text]
  • Arxiv:2002.03434V3 [Physics.Flu-Dyn] 25 Jul 2020
    APS/123-QED Modified Stokes drift due to surface waves and corrugated sea-floor interactions with and without a mean current Akanksha Gupta Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, U.P. 208016, India.∗ Anirban Guhay School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK. (Dated: July 28, 2020) arXiv:2002.03434v3 [physics.flu-dyn] 25 Jul 2020 1 Abstract In this paper, we show that Stokes drift may be significantly affected when an incident inter- mediate or shallow water surface wave travels over a corrugated sea-floor. The underlying mech- anism is Bragg resonance { reflected waves generated via nonlinear resonant interactions between an incident wave and a rippled bottom. We theoretically explain the fundamental effect of two counter-propagating Stokes waves on Stokes drift and then perform numerical simulations of Bragg resonance using High-order Spectral method. A monochromatic incident wave on interaction with a patch of bottom ripple yields a complex interference between the incident and reflected waves. When the velocity induced by the reflected waves exceeds that of the incident, particle trajectories reverse, leading to a backward drift. Lagrangian and Lagrangian-mean trajectories reveal that surface particles near the up-wave side of the patch are either trapped or reflected, implying that the rippled patch acts as a non-surface-invasive particle trap or reflector. On increasing the length and amplitude of the rippled patch; reflection, and thus the effectiveness of the patch, increases. The inclusion of realistic constant current shows noticeable differences between Lagrangian-mean trajectories with and without the rippled patch.
    [Show full text]
  • Part II-1 Water Wave Mechanics
    Chapter 1 EM 1110-2-1100 WATER WAVE MECHANICS (Part II) 1 August 2008 (Change 2) Table of Contents Page II-1-1. Introduction ............................................................II-1-1 II-1-2. Regular Waves .........................................................II-1-3 a. Introduction ...........................................................II-1-3 b. Definition of wave parameters .............................................II-1-4 c. Linear wave theory ......................................................II-1-5 (1) Introduction .......................................................II-1-5 (2) Wave celerity, length, and period.......................................II-1-6 (3) The sinusoidal wave profile...........................................II-1-9 (4) Some useful functions ...............................................II-1-9 (5) Local fluid velocities and accelerations .................................II-1-12 (6) Water particle displacements .........................................II-1-13 (7) Subsurface pressure ................................................II-1-21 (8) Group velocity ....................................................II-1-22 (9) Wave energy and power.............................................II-1-26 (10)Summary of linear wave theory.......................................II-1-29 d. Nonlinear wave theories .................................................II-1-30 (1) Introduction ......................................................II-1-30 (2) Stokes finite-amplitude wave theory ...................................II-1-32
    [Show full text]
  • Waves and Weather
    Waves and Weather 1. Where do waves come from? 2. What storms produce good surfing waves? 3. Where do these storms frequently form? 4. Where are the good areas for receiving swells? Where do waves come from? ==> Wind! Any two fluids (with different density) moving at different speeds can produce waves. In our case, air is one fluid and the water is the other. • Start with perfectly glassy conditions (no waves) and no wind. • As wind starts, will first get very small capillary waves (ripples). • Once ripples form, now wind can push against the surface and waves can grow faster. Within Wave Source Region: - all wavelengths and heights mixed together - looks like washing machine ("Victory at Sea") But this is what we want our surfing waves to look like: How do we get from this To this ???? DISPERSION !! In deep water, wave speed (celerity) c= gT/2π Long period waves travel faster. Short period waves travel slower Waves begin to separate as they move away from generation area ===> This is Dispersion How Big Will the Waves Get? Height and Period of waves depends primarily on: - Wind speed - Duration (how long the wind blows over the waves) - Fetch (distance that wind blows over the waves) "SMB" Tables How Big Will the Waves Get? Assume Duration = 24 hours Fetch Length = 500 miles Significant Significant Wind Speed Wave Height Wave Period 10 mph 2 ft 3.5 sec 20 mph 6 ft 5.5 sec 30 mph 12 ft 7.5 sec 40 mph 19 ft 10.0 sec 50 mph 27 ft 11.5 sec 60 mph 35 ft 13.0 sec Wave height will decay as waves move away from source region!!! Map of Mean Wind
    [Show full text]
  • Variational Principles for Water Waves from the Viewpoint of a Time-Dependent Moving Mesh
    Variational principles for water waves from the viewpoint of a time-dependent moving mesh Thomas J. Bridges & Neil M. Donaldson Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK May 21, 2010 Abstract The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves. 1 Introduction In the theory of water waves, the fluid domain shape is changing with time. Therefore it is appealing to transform the moving domain to a fixed domain. This approach, first for steady waves and later for unsteady waves, has been widely used in the study of water waves. However, in all cases the transformation is either algebraic (explicit) or a conformal mapping. In this paper we look at the equations governing the motion of water waves from the viewpoint of an arbitrary time-dependent transformation of the form x(µ,ν,τ) (µ,ν,τ) y(µ,ν,τ) , (1.1) 7→ t(τ) where (µ,ν) are coordinates for a fixed time-independent rectangle D := (µ,ν) : 0 µ L and h ν 0 , (1.2) { ≤ ≤ − ≤ ≤ } with h, L given positive parameters.
    [Show full text]
  • Waves on Deep Water, I
    Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the effect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (flat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ.
    [Show full text]
  • 0123737354Fluidmechanics.Pdf
    Fluid Mechanics, Fourth Edition Founders of Modern Fluid Dynamics Ludwig Prandtl G. I. Taylor (1875–1953) (1886–1975) (Biographical sketches of Prandtl and Taylor are given in Appendix C.) Photograph of Ludwig Prandtl is reprinted with permission from the Annual Review of Fluid Mechanics, Vol. 19, Copyright 1987 by Annual Reviews www.AnnualReviews.org. Photograph of Geoffrey Ingram Taylor at age 69 in his laboratory reprinted with permission from the AIP Emilio Segre` Visual Archieves. Copyright, American Institute of Physics, 2000. Fluid Mechanics Fourth Edition Pijush K. Kundu Oceanographic Center Nova Southeastern University Dania, Florida Ira M. Cohen Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, Pennsylvania with contributions by P. S. Ayyaswamy and H. H. Hu AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400 Burlington, MA 01803, USA Elsevier, The Boulevard, Langford Lane Kidlington, Oxford, OX5 1GB, UK © 2008 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).
    [Show full text]
  • Marine Forecasting at TAFB [email protected]
    Marine Forecasting at TAFB [email protected] 1 Waves 101 Concepts and basic equations 2 Have an overall understanding of the wave forecasting challenge • Wave growth • Wave spectra • Swell propagation • Swell decay • Deep water waves • Shallow water waves 3 Wave Concepts • Waves form by the stress induced on the ocean surface by physical wind contact with water • Begin with capillary waves with gradual growth dependent on conditions • Wave decay process begins immediately as waves exit wind generation area…a.k.a. “fetch” area 4 5 Wave Growth There are three basic components to wave growth: • Wind speed • Fetch length • Duration Wave growth is limited by either fetch length or duration 6 Fully Developed Sea • When wave growth has reached a maximum height for a given wind speed, fetch and duration of wind. • A sea for which the input of energy to the waves from the local wind is in balance with the transfer of energy among the different wave components, and with the dissipation of energy by wave breaking - AMS. 7 Fetches 8 Dynamic Fetch 9 Wave Growth Nomogram 10 Calculate Wave H and T • What can we determine for wave characteristics from the following scenario? • 40 kt wind blows for 24 hours across a 150 nm fetch area? • Using the wave nomogram – start on left vertical axis at 40 kt • Move forward in time to the right until you reach either 24 hours or 150 nm of fetch • What is limiting factor? Fetch length or time? • Nomogram yields 18.7 ft @ 9.6 sec 11 Wave Growth Nomogram 12 Wave Dimensions • C=Wave Celerity • L=Wave Length •
    [Show full text]
  • Waves and Structures
    WAVES AND STRUCTURES By Dr M C Deo Professor of Civil Engineering Indian Institute of Technology Bombay Powai, Mumbai 400 076 Contact: [email protected]; (+91) 22 2572 2377 (Please refer as follows, if you use any part of this book: Deo M C (2013): Waves and Structures, http://www.civil.iitb.ac.in/~mcdeo/waves.html) (Suggestions to improve/modify contents are welcome) 1 Content Chapter 1: Introduction 4 Chapter 2: Wave Theories 18 Chapter 3: Random Waves 47 Chapter 4: Wave Propagation 80 Chapter 5: Numerical Modeling of Waves 110 Chapter 6: Design Water Depth 115 Chapter 7: Wave Forces on Shore-Based Structures 132 Chapter 8: Wave Force On Small Diameter Members 150 Chapter 9: Maximum Wave Force on the Entire Structure 173 Chapter 10: Wave Forces on Large Diameter Members 187 Chapter 11: Spectral and Statistical Analysis of Wave Forces 209 Chapter 12: Wave Run Up 221 Chapter 13: Pipeline Hydrodynamics 234 Chapter 14: Statics of Floating Bodies 241 Chapter 15: Vibrations 268 Chapter 16: Motions of Freely Floating Bodies 283 Chapter 17: Motion Response of Compliant Structures 315 2 Notations 338 References 342 3 CHAPTER 1 INTRODUCTION 1.1 Introduction The knowledge of magnitude and behavior of ocean waves at site is an essential prerequisite for almost all activities in the ocean including planning, design, construction and operation related to harbor, coastal and structures. The waves of major concern to a harbor engineer are generated by the action of wind. The wind creates a disturbance in the sea which is restored to its calm equilibrium position by the action of gravity and hence resulting waves are called wind generated gravity waves.
    [Show full text]
  • Answers of the Reviewers' Comments Reviewer #1 —- General
    Answers of the reviewers’ comments Reviewer #1 —- General Comments As a detailed assessment of a coupled high resolution wave-ocean modelling system’s sensitivities in an extreme case, this paper provides a useful addition to existing published evidence regarding coupled systems and is a valid extension of the work in Staneva et al, 2016. I would therefore recommend this paper for publication, but with some additions/corrections related to the points below. —- Specific Comments Section 2.1 Whilst this information may well be published in the authors’ previous papers, it would be useful to those reading this paper in isolation if some extra details on the update frequencies of atmosphere and river forcing data were provided. Authors: More information about the model setup, including a description of the open boundary forcing, atmospheric forcing and river runoff, has been included in Section 2.1. Additional references were also added. Section 2.2 This is an extreme case in shallow water, so please could the source term parameterizations for bottom friction and depth induced breaking dissipation that were used in the wave model be stated? Authors: Additional information about the parameterizations used in our model setup, including more references, was provided in Section 2.2. Section 2.3 I found the statements that "<u> is the sum of the Eulerian current and the Stokes drift" and "Thus the divergence of the radiation stress is the only (to second order) force related to waves in the momentum equations." somewhat contradictory. In the equations, Mellor (2011) has been followed correctly and I see the basic point about radiation stress being the difference between coupled and uncoupled systems, so just wondering if the authors can review the text in this section for clarity.
    [Show full text]
  • On the Transport of Energy in Water Waves
    On The Transport Of Energy in Water Waves Marshall P. Tulin Professor Emeritus, UCSB [email protected] Abstract Theory is developed and utilized for the calculation of the separate transport of kinetic, gravity potential, and surface tension energies within sinusoidal surface waves in water of arbitrary depth. In Section 1 it is shown that each of these three types of energy constituting the wave travel at different speeds, and that the group velocity, cg, is the energy weighted average of these speeds, depth and time averaged in the case of the kinetic energy. It is shown that the time averaged kinetic energy travels at every depth horizontally either with (deep water), or faster than the wave itself, and that the propagation of a sinusoidal wave is made possible by the vertical transport of kinetic energy to the free surface, where it provides the oscillating balance in surface energy just necessary to allow the propagation of the wave. The propagation speed along the surface of the gravity potential energy is null, while the surface tension energy travels forward along the wave surface everywhere at twice the wave velocity, c. The flux of kinetic energy when viewed traveling with a wave provides a pattern of steady flux lines which originate and end on the free surface after making vertical excursions into the wave, even to the bottom, and these are calculated. The pictures produced in this way provide immediate insight into the basic mechanisms of wave motion. In Section 2 the modulated gravity wave is considered in deep water and the balance of terms involved in the propagation of the energy in the wave group is determined; it is shown again that vertical transport of kinetic energy to the surface is fundamental in allowing the propagation of the modulation, and in determining the well known speed of the modulation envelope, cg.
    [Show full text]
  • Detection and Characterization of Meteotsunamis in the Gulf of Genoa
    Journal of Marine Science and Engineering Article Detection and Characterization of Meteotsunamis in the Gulf of Genoa Paola Picco 1,*, Maria Elisabetta Schiano 2, Silvio Incardone 1, Luca Repetti 1, Maurizio Demarte 1, Sara Pensieri 2,* and Roberto Bozzano 2 1 Italian Hydrographic Service, passo dell’Osservatorio, 4, 16135 Genova, Italy 2 Institute for the study of the anthropic impacts and the sustainability of the marine environment, National Research Council of Italy, via De Marini 6, 16149 Genova, Italy * Correspondence: [email protected] (P.P.); [email protected] (S.P.) Received: 30 May 2019; Accepted: 10 August 2019; Published: 15 August 2019 Abstract: A long-term time series of high-frequency sampled sea-level data collected in the port of Genoa were analyzed to detect the occurrence of meteotsunami events and to characterize them. Time-frequency analysis showed well-developed energy peaks on a 26–30 minute band, which are an almost permanent feature in the analyzed signal. The amplitude of these waves is generally few centimeters but, in some cases, they can reach values comparable or even greater than the local tidal elevation. In the perspective of sea-level rise, their assessment can be relevant for sound coastal work planning and port management. Events having the highest energy were selected for detailed analysis and the main features were identified and characterized by means of wavelet transform. The most important one occurred on 14 October 2016, when the oscillations, generated by an abrupt jump in the atmospheric pressure, achieved a maximum wave height of 50 cm and lasted for about three hours.
    [Show full text]