Water Wave Mechanics for Engineers and Scientists

Robert G Dean, Robert A Dalrymple

• Publisher: WSPC
• Year: 1991
• ISBN:
  9789814365697
• Topic: Scienze fisiche
• Subtopic: Oceanografia

This book is intended as an introduction to classical water wave theory for the college senior or first year graduate student. The material is self-contained; almost all mathematical and engineering concepts are presented or derived in the text, thus making the book accessible to practicing engineers as well.

The book commences with a review of fluid mechanics and basic vector concepts. The formulation and solution of the governing boundary value problem for small amplitude waves are developed and the kinematic and pressure fields for short and long waves are explored. The transformation of waves due to variations in depth and their interactions with structures are derived. Wavemaker theories and the statistics of ocean waves are reviewed. The application of the water particle motions and pressure fields are applied to the calculation of wave forces on small and large objects. Extension of the linear theory results to several nonlinear wave properties is presented. Each chapter concludes with a set of homework problems exercising and sometimes extending the material presented in the chapter. An appendix provides a description of nine experiments which can be performed, with little additional equipment, in most wave tank facilities.

Contents:

• Introduction to Wave Mechanics
• A Review of Hydrodynamics and Vector Analysis
• Small-Amplitude Water Wave Theory Formulation and Solution
• Engineering Wave Properties
• Long Waves
• Wavemaker Theory
• Wave Statistics and Spectra
• Wave Forces
• Waves Over Real Seabeds
• Nonlinear Properties Derivable from Small-Amplitude Waves
• Nonlinear Waves
• A Series of Experiments for a Laboratory Course Component in Water Waves

Readership: Coastal and ocean engineers.

1

Introduction to Wave
Mechanics

Dedication
SIR HORACE LAMB

Sir Horace Lamb (1849–1934) is best known for his extremely thorough and well-written book, Hydrodynamics, which first appeared in 1879 and has been reprinted numerous times. It still serves as a compendium of useful information as well as the source for a great number of papers and books. If this present book has but a small fraction of the appeal of Hydrodynamics, the authors would be well satisfied.

Sir Horace Lamb was born in Stockport, England in 1849, educated at Owens College, Manchester, and then Trinity College, Cambridge University, where he studied with professors such as J. Clerk Maxwell and G. G. Stokes. After his graduation, he lectured at Trinity (1822–1825) and then moved to Adelaide, Australia, to become Professor of Mathematics.

After ten years, he returned to Owens College (part of Victoria University of Manchester) as Professor of Pure Mathematics; he remained until 1920.

Professor Lamb was noted for his excellent teaching and writing abilities. In response to a student tribute on the occasion of his eightieth birthday, he replied: “I did try to make things clear, first to myself…and then to my students, and somehow make these dry bones live.”

His research areas encompassed tides, waves, and earthquake properties as well as mathematics.

1.1 INTRODUCTION

Rarely can one find a body of water open to the atmosphere that does not have waves on its surface. These waves are a manifestation of forces acting on the fluid tending to deform it against the action of gravity and surface tension, which together act to maintain a level fluid surface. Thus it requires a force of some kind, such as would be caused by a gust of wind or a falling stone impacting on the water, to create waves. Once these are created, gravitational and surface tension forces are activated that allow the waves to propagate, in the same manner as tension on a string causes the string to vibrate, much to our listening enjoyment.

Waves occur in all sizes and forms, depending on the magnitude of the forces acting on the water. A simple illustration is that a small stone and a large rock create different-size waves after impacting on water. Further, different speeds of impact create different-size waves, which indicates that the pressure forces acting on the fluid surface are important, as well as the magnitude of the displaced fluid. The gravitational attraction of the moon, sun, and other astronomical bodies creates the longest known water waves, the tides. These waves circle halfway around the earth from end to end and travel with tremendous speeds. The shortest waves can be less than a centimeter in length. The length of the wave gives one an idea of the magnitude of the forces acting on the waves. For example, the longer the wave, the more important gravity (comprised of the contributions from the earth, the moon, and the sun) is in relation to surface tension.

The importance of waves cannot be overestimated. Anything that is near or in a body of water is subject to wave action. At the coast, this can result in the movement of sand along the shore, causing erosion or damage to structures during storms. In the water, offshore oil platforms must be able to withstand severe storms without destruction. At present drilling depths exceeding 300 m, this requires enormous and expensive structures. On the water, all ships are subjected to wave attack, and countless ships have foundered due to waves which have been observed to be as large as 34 m in height. Further, any ship moving through water creates a pressure field and, hence, waves. These waves create a significant portion of the resistance to motion enountered by the ships.

1.2 CHARACTERISTICS OF WAVES

The important parameters to describe waves are their length and height, and the water depth over which they are propagating. All other parameters, such as wave-induced water velocities and accelerations, can be determined theoretically from these quantities. In Figure 1.1, a two-dimensional schematic of a wave propagating in the x direction is shown. The length of the wave,

Figure 1.1 Wave characteristics.

L, is the horizontal distance between two successive wave crests, or the high points on a wave, or alternatively the distance between two wave troughs. The wave length will be shown later to be related to the water depth h and wave period T, which is the time required for two successive crests or troughs to pass a particular point. As the wave, then, must move a distance L in time T, the speed of the wave, called the celerity, C, is defined as C = L/T. While the wave form travels with celerity C, the water that comprises the wave does not translate in the direction of the wave.

The coordinate axis that will be used to describe wave motion will be located at the still water line, z = 0. The bottom of the water body will be at z = −h.

Waves in nature rarely appear to look exactly the same from wave to wave, nor do they always propagate in the same direction. If a device to measure the water surface elevation, η, as a function of time was placed on a platform in the middle of the ocean, it might obtain a record such as that shown in Figure 1.2. This sea can be seen to be a superposition of a large number of sinusoids going in different directions. For example, consider the two sine waves shown in Figure 1.3 and their sum. It is this superposition of sinusoids that permits the use of Fourier analysis and spectral techniques to be used in describing the sea. Unfortunately, there is a great amount of randomness in the sea, and statistical techniques need to be brought to bear. Fortunately, very large waves or, alternatively, waves in shallow water appear

Figure 1.2 Example of a possible recorded wave form.

Figure 1.3 Complex wave form resulting as the sum of two sinusoids.

to be more regular than smaller waves or those in deeper water, and not so random. Therefore, in these cases, each wave is more readily described by one sinusoid, which repeats itself periodically. Realistically, due to shallow water nonlinearities, more than one sinusoid, all of the same phase, are necessary; however, using one sinusoid has been shown to be reasonably accurate for some purposes. It is this surprising accuracy and ease of application that have maintained the popularity and the widespread usage of so-called linear, or small-amplitude, wave theory. The advantages are that it is easy to use, as opposed to more complicated nonlinear theories, and lends itself to superposition and other complicated manipulations. Moreover, linear wave theory is an effective stepping-stone to some nonlinear theories. For this reason, this book is directed primarily to linear theory.

1.3 HISTORICAL AND PRESENT LITERATURE

The field of water wave theory is over 150 years old and, of course, during this period of time numerous books and articles have been written about the subject. Perhaps the most outstanding is the seminal work of Sir Horace Lamb. His Hydrodynamics has served as a source book since its original publication in 1879.

Other notable books with which the reader should become acquainted are R. L. Wiegel's Oceanographical Engineering and A. T. Ippen's Estuary and Coastline Hydrodynamics. These two books, appearing in the 1960s, provided the education of many of the practicing coastal and ocean engineers of today.

The authors also recommend for further studies on waves the book by G. B. Witham entitled Linear and Nonlinear Waves, from which a portion of Chapter 11 is derived, and the article “Surface Waves,” by J. V Wehausen and E. V Laitone, in the Handbuch der Physik.

In terms of articles, there are a number of journals and proceedings that will provide the reader with more up-to-date material on waves and wave theory and its applications. These include the American Society of Civil Engineers’ Journal of Waterway, Port, Coastal and Ocean Division, the Journal of Fluid Mechanics, the Proceedings of the International Coastal Engineering Conferences, the Journal of Geophysical Research, Coastal Engineering, Applied Ocean Research, and the Proceedings of the Offshore Technology Conference.

2

A Review of
Hydrodynamics and
Vector Analysis

Dedication
LEONHARD EULER

Leonhard Euler (1707–1783), born in Basel, Switzerland, was one of the earliest practitioners of applied mathematics, developing with others the theory of ordinary and partial differential equations and applying them to the physical world. The most frequent use of his work here is the use of the Euler equations of motion, which describe the flow of an inviscid fluid.

In 1722 he graduated from the University of Basel with a degree in Arts. During this time, however, he attended the lectures of Johan I. Bernoulli (Daniel Bernoulli's father), and turned to the study of mathematics. In 1723 he received a master's level degree in philosophy and began to teach in the philosophy department. In 1727 he moved to St. Petersburg, Russia, and to the St. Petersburg Academy of Science, where he worked in physiology and mathematics and succeeded Daniel Bernoulli as Professor of Physics in 1731.

In 1741 he was invited to work in the Berlin Society of Sciences (founded by...