Mathematical Techniques for Water Waves

Advances in Fluid Mechanics Volume 8 Series Editor: M. Rahman

Mathematical Techniques for Water Waves Editor:

B.N. Mandal Indian Statistical Institute, Calcutta, India

Computational Mechanics Publications Southampton Boston)

CONTENTS

Series Editor M.Rahman Editor B.N. Manda! Indian Statistical Institute 203 Barrackpore Trunk Road Calcutta 700 035 India

Preface

Xl

Chapter 1 Complementary methods for scattering by thin barriers D. V. Evans, R. Porter

1

Published by

Chapter2 Computational Mechanics Publications Ashurst Lodge, Ashurst, Southampton, S040 7AA, UK Tel: 44 (0)1703 293223; Fax: 44 (0)1703 292853 Email: [email protected] http://www.cmp.co.uk For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 508 667 5841; Fax: 508 667 7582 Email: [email protected] British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN l 85312 413 3 Computational Mechanics Publications ISSN 1353 808X Series Library of Congress Catalog Card Number 96-083306 No responsibility is assumed by the Publisher the Editors and Authors for any injury and/or damage to persons or property as amatterofproducts liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

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The use of multipoles in channel problems C.M Linton

45

Chapter3 Analytical dynamics of wave-body interactions G.A. Athanassoulis

79

Chapter4 The use of Green's theorem in water wave problems P.F. Rhodes-Robinson

155

Chapter 5 Interaction of water waves with thin plates P.A. Martin, NF. Parsons, L. Farina

197

Chapter 6 A survey on two mathematical methods used in scattering of surface water waves A. Chakrabarti

231

Chapter7 On a singular integral equation and its use to some barrier problems B.N Manda/, Sudeshna Banerjea

255

Chapters Hydrodynamic loading on an elliptic cylinder in waves M Rahman, Geeta Agrawal

285

Chapter9 On fourth-order nonlinear evolution equations in water wave theory KP. Das Chapter 10 Propagation of solitary waves in a binary fluid A.S. Gupta

Preface 327

339

The topic of water waves is an important branch of fluid mechanics. The mathematical techniques utilized to handle various problems of water waves are varied and fascinating. In this book some of these techniques are highlighted in connection with investigating some classes of problems in water wave theory by a number of leading researchers in the field. The book contains ten chapters. The first eight chapters involve the linearised theory of water waves, while the last two chapters involve nonlinear theory. In the first chapter entitled 'Complementary methods for scattering by thin barriers', D.V. Evans and R. Porter consider an important technique for handling water wave scattering problems involving thin barriers. The problems are formulated in two ways using either a velocity or a pressure difference integral equation representation for the reflection and transmission coefficients. It is shown that these representations provide upper and lower bounds for quantities related to these coefficients. With a judicious choice of expansion functions, a Galerkin expansion provides extremely accurate results with a minimum of computational effort. Results are presented for a wide range of problems including both vertical barriers and a periodic array of barriers with gaps, extending throughout the depth. In the second chapter entitled 'The use of multipoles in cha,nnel problems', C.M. Linton demonstrates the use of multi po IP expansions in tlte solution of water wave radiation and scattering problems involving bodies in channels of constant finite depth and width. The essence of the method lies in the construction of singular potential functions which satisfy all the boundary conditions of the problem, except that on the boundary. The general radiation problem for an arbitrary positioned vertical circular cylinder extending throughout the depth is considered in some detail. The implication of the existence of trapped modes for such problems is also addressed. Applications of the multipole to truncated cylinders and to arrays of cylinders and their extension are also considered. The third chapter is on 'Analytical dynamics of wave-body interactions' written by G.A. Athanassoulis, wherein the problem of wave-body interaction is studied from the point of view of analytical dynamics. It describes the extension of concepts and techniques of analytical dynamics to hydromechanical systems containing liquid with a free surface, as well as rigid bodies. Some numerical results obtained by discretized methods of Hamiltonian functional equations are also presented, showing that this approach is useful not only from the theoretical, but also from the numerical point of view. In the fourth chapter on 'The use of Green's theorem in water wave problems', P.F. Rhodes- Robinson discusses the use of Green's function for water

wave problems in general, and in particular, for two types of incomplete vertical wavemakers in deep water, the presence of surface tension at the free surface also being included. This chapter also illustrates the effectiveness of the technique of Green's integral theorem for dealing with some water wave problems for which an exact Green's function of some sort can be obtained. The fifth chapter, 'Interaction of water waves with thin plates', written by P.A. Martin, N .F. Parsons and L. Farina, emphasizes the use of hypersingular integral equations in water wave interaction problems involving thin plates. Several examples, such as scattering by submerged curved plates and surfacepiercing plates in two dimensions, trapping waves by submerged plates and scattering by flat plates in three dimensions, were considered. In each case, the hypersingular integral equation is solved numerically, using an expansioncollocation method. In the sixth chapter on 'A survey on two mathematical methods used in scattering of surface water waves', A. Chakrabarti discusses two well-known basic problems of water wave scattering by a fully submerged thin vertical barrier and a surface piercing thin vertical barrier by using two methods attributed to Ursell and Williams. He also appended a new method which depends on the exploitation of the singular behaviour of certain Fourier integrals at the turning points. The seventh chapter entitled 'A singular integral equation and its application to some barrier problems' is written by B.N. Manda! and Sudeshna Banerjea. Here the solution of a singular integral equation with a combination of logarithmic and Cauchy type kernels for different ranges was obtained. This singular integral equation arises in a number of water wave problems involving thin vertical barriers. Use of this integral equation in handling some barrier problems in water wave theory is demonstrated. The eighth chapter on 'Hydrodynamic loading on an elliptic cylinder in waves' written by M. Rahman and G. Agrawal, presents an analytical solution to the problem of water wave diffraction by a fixed vertical cylinder of elliptical cross section. The results obtained are compared with existing results. The limiting case of the circular cylinder was also studied. In all the above chapters the various water wave problems were discussed essentially within the framework of the linearised theory of water waves. The last two chapters, however, involve the nonlinear theory. In the ninth chapter entitled 'On fourth order nonlinear evolution equations in water wave theory', written by K.P. Das, a fourth order nonlinear evolution equation for a surface gravity wave is derived by two techniques, namely the method of multiple scaLes and by using Zakharov's integral equation. Fourth order evolution equations are a good starting point for the study of nonlinear sea water waves, as has been pointed out by many researchers. Finally, the tenth chapter written by A.S. Gupta is on 'Propagation of

solitary waves in a binary fluid'. It examines the propagation of solitary wa:es in water stratified with salt. This study has some relevance to wave mot10n at the air-sea interface. When the free surface is subject to uniform heat and mass flux, a long wave length oscillatory instability occurs at a critical value of Rayleigh number. Using the method of reductive perturbation, the nonlinear evolution of this instability is investigated and the analysis reveals the existence of a solitary wave. I am thankful to my research colleagues on water waves from whose association the idea of preparing a book on water waves emphasizing a number of mathematical techniques was mooted. I feel that this volume will be a welcome addition to the water wave literature. I thank my students Dr D.P. Dolai and Dr Mridula Kanoria for their help at various stages in the preparation of this book. I am grateful to Professor M. Rahman, the Series Editor, who induced me to prepa~e this book in his series, and to Mr Lance Sucharov, Publishing Director, Mrs Juliet Booker and Miss Vicky Cuthill, Book Production Editors, at Computational Mechanics Publications for their cooperation and help in publishing this volume. I also express my sincere thanks to the contributors of the various chapters of the book. Finally I express my gratitude to my wife Alpana and my daughter Anindita for their encouragement, patience and understanding while I remained busy away in my office for long hours during the preparation of this volume.

B.N. MANDAL Physics and Applied Mathematics Unit Indian Statistical Institute Calcutta, India May 1996

78

Mathematical Techniques For Water Waves

Parker, R. & Stoneman, S.A.T. (1989) The excitation and consequences of acoustic resonances in enclosed fluid flow around solid bodies, Proc. lnstn Mech. Engrs, 203, 9-19. Spring, B.H. & Monkmeyer, P.L. (1975) Interaction of plane waves with a row of cylinders. Proceedings of the 3rd Conference on Civil Engineering in Oceans, ASCE, Newark, Delaware, pp. 979-998. Srokosz, M.A. (1980) Some relations for bodies in a canal, with an application to wave-power absorption, J. Fluid Mech., 99, 145-162. Thomas, G.P. (1991) The diffraction of water waves by a circular cylinder in a channel, Ocean Engng., 18, 17-44.

Chapter 3 Analytical dynamics of wave-body interaction G.A. Athanassoulis

Department of Naval Architecture and Marine Engineering, National Technical University of Athens, Heroon Polytechniou 9, 157 73 Athens, GREECE

Thorne, R.C. (1953) Multipole expansions in the theory of surface waves, Proc. Camb. Phil. Soc., 49, 707-716.

Abstract

Twersky, V. (1952) Multiple scattering of radiation by an arbitrary configuration of parallel cylinders, J. Acoust. Soc. Am., 24, 42-46.

Williams, A.N. & Vazquez, J.H. (1993) Mean drift loads on array of vertical cylinders in narrow tank, J. Wtrway., Port, Coast., and Oc. Engrg., 119(4), 398-416.

In this work the problem of wave-body interaction is studied from the point of view of analytical dynamics. The liquid is considered inviscid, incompressible, unbounded, with an unbounded free surface, moving in interaction with freely-moving rigid bodies. Other fixed or moving, in a prescribed way, rigid boundaries may also be present. It is shown that the degrees of freedom of the rigid-body subsystem and the free-surface elevation can be considered as the generalised co-ordinates of the system, although the latter is in fact nonholonomic. It is also shown that the first form of Hamilton's variational principle is applied as if tJ.:te system were holonomic. Then, the principle is used to derive Lagrange equations in independent co-ordinates, which are functional differential equations modelling the nonlocal character of wave-body interaction. Generalised momenta of the system are calculated and the second form of Hamilton's principle, based on generalised co-ordinates and momenta, is formulated and proved. Hamilton's equations of motion are obtained for a simplified version of the system under study. A distinctive feature of this work is that both the first and the second form of Hamilton's variational principle are formulated and proved not only within the context of analytical dynamics, but also as variational theorems on their own, based on general fluid dynamic considerations.

Yeung, R.W. & Sphaier, S.H. (1989a) Wave-interference effects on a truncated cylinder in a channel, J. Engng. Math., 23, 95-117.

1 Introduction

Twersky, V. (1956) On the scattering of waves by an infinite grating, IRE Trans. on Antennas and Propagation, 4, 330-345. Twersky, V. (1962) On scattering of waves by the infinite grating of circular cylinders, IRE Trans. on Antennas and Propagation, 10, 737-765. Ursell, F. (1950) Surface waves on deep water in the presence of a submerged circular cylinder I, Proc. Camb. Phil. Soc., 46, 141-152. Ursell, F. (1951) Trapping modes in the theory of surface waves, Proc. Camb. Phil. Soc., 47, 347-358. Ursell, F. (1991) Trapped modes in a circular cylindrical acoustic waveguide, Proc. Roy. Soc. Land., A, 435, 575-589.

Yeung, R.W. & Sphaier, S.H. (1989b) Wave-interference effects on a floating body in a towing tank, Proceedings of PRADs '89, Varna, Bulgaria. Zaviska, F. (1913) Uber die Beugung elektromagnetischer Wellen an parallelen, unendlich langen Kreiszylindern, Ann. Physik, 40, 1023-1056.

It is well known that the hydrodynamic forces Fa, a= 1,2,. ..,A 5 6, exerted on a rigid body moving through a totally unbounded 1, inviscid and incompressible liquid, may be expressed with the aid of Lagrange's equations of the second kind, provided that the liquid motion is irrotational and acyclic:

F =-!!_()TL+ ()TL a

dt ac,a

()qa'

a=l,2, ... ,A,

(la)

where qa =qa(t) and ifa= if a(t) are the generalised co-ordinates and generalised velocities of the rigid body, respectively, and TL is the kinetic energy of the liquid, expressed as a function of qa(t) and q)t). Here, and sub1

The tenn totally unbounded liquid means that the liquid domain is unbounded in every direction.

80



sequently, a dot over a time-dependent quantity denotes time differentiation. The assumptions that the liquid is inviscid and incompressible and the flow is irrotational and acyclic will be retained throughout the present work. These assumptions together will be indicated by the term ideal liquid. Using eqns (la) and the Lagrangian equations of motion of the rigid body, we obtain d (}(TL +T8

dt

)

(}(TL +T8

)

Qa,

a=l,2, ... ,A,

(lb)

where T8 is the kinetic energy of the body and Qa =Qa(t) are external forces acting on it. Equations (I) were first derived in 1871 by Lord Kelvin with the aid of Hamilton's principle (see Thomson & Tait (1879)), and they seem to be the first application of the methods of analytical dynamics to hydromechanics. The subject has been treated subsequently by many authors, and various proofs, discussions and applications can be found in the literature. See, for example, Lamb (1932), Birkhoff (1960), Milne-Thomson (1968). For the free-motion case, (Qa =0), some deep mathematical results have been obtained, in the context of Hamiltonian dynamics, by Novikov and his associates. For example, Novikov & Shmeltser (1981) showed the existence of periodic solutions, and Perelomov (1981) found new quadratic integrals for the body motion. Equations ( 1) are often referred to as the Kelvin-Kirchhoff hydromechanical equations. The analytical basis on which the validity of eqns (1) relies is the fact that the kinetic energy of the liquid can be expressed as a single-valued function of qa(t) and qa(t). This is possible because the kinematical constraints of irrotationality and incompressibility permit the calculation of the liquid velocity field throughout the whole (totally unbounded) liquid domain, in terms of the boundary geometry and kinematics, the latter being completely determined at every time instant t by qa(t) and qa(t). This situation does not essentially change when more than one freely-moving rigid bodies are present. In this case the number of degrees of freedom is, of course, larger, and all generalised co-ordinates (for all bodies) are dynamically coupled through the liquid, a fact that manifests itself in the form that the kinetic energy of the system has; it is a full quadratic form of the generalised velocities. See, for example, Voinov et al. (1973), Abdel Moneim (1976). Furthermore, eqns (1) for a single- or a multiple-body system remain valid in the presence of other,

81

fixed-in-space, rigid boundaries, confining internally and/or externally the liquid domain. Various applications of eqns (1) in the presence of fixed boundaries, either rigid or equipotential, have been presented by Chou et al. (1976), Miloh & Landweber (1981), and Miloh & Hauptan (1980). When a deformable boundary is present, such as a free surface, a liquidliquid interface or a liquid-elastic body interface, new features come into play and new difficulties of analytical character arise. Of course, the presence of a deformable boundary does not essentially alter the physics of the problem; the liquid remains an ideal mechanical system the kinematics of which is completely defined through its boundary kinematics. However, the boundary geometry cannot now be defined in terms of a finite number of generalised co-ordinates. An infinity of generalised co-ordinates is now needed, and the kinetic energy of the liquid becomes a nonlinear, non-local functional on the deformable boundary geometry and kinematics. In the present work, we shall study the dynamics of a hydromechanical system, denoted by Y, made up of two interacting (through their interface) subsystems: • A liquid subsystem, denoted by YI,, consisting of an ideal liquid that occupies an unbounded, topologically connected domain D of Euclidean space R..3, upper-limited by an unbounded free surface. • A rigid-body subsystem, denoted by :f's , consisting of one or more non-deformable rigid bodies, possibly subjected to rheonomic holonomic constraints, which are moving within the liquid subsystem. To avoid specific complications related to water-entry or water-emergence problems2, we assume that the moving rigid bodies do not intersect the free surface of the liquid at any time t . A more detailed description of the system Y is given in Section 4, where an illustration (Fig. 1) is also presented. Both subsystems YI, and :f's lie within a gravity field of constant intensity (acceleration due to gravity) g. Clearly, the dynamics of the two subsystems are coupled, and the problem may be called the wave-body interaction problem. We can distinguish two classes of wave-body interaction problems: the class of problems in which all body boundaries are moving in a prescribed manner, and the class of problems in which freely-moving rigid bodies are also present. Although the second class is more general than the first one and, in fact, includes it as a special case, this distinction is of some importance from the methodological point of view. 2

See, for example, Dobrovolskaia (1969), Greenhow (1987), Cointe (1991).

82


Before proceeding to the study of the wave-body interaction problem in the context of analytical mechanics, it is helpful to discuss the partial problem of the dynamics of water waves, which is characterised by the absence of any rigid boundary (moving or fixed) other than a plane horizontal bottom. Our discussion will focus on the case of a laterally unbounded free surface. The corresponding problem when the free surface is bounded (which is not dealt with in this work) has also been studied in the context of analytical dynamics. (In fact, it was the first free-surface problem studied in this way, in the early fifties). The interested reader is referred to the extended papers by Moiseev (1964), Moiseev & Petrov (1966), Miles (1976), Limarchenko (1980), and the book by Moiseev & Rumyantsev (1968). It is worth noticing here that, although the underlying physical assumptions are the same for the two problems (bounded and unbounded free surface), there are profound differences concerning the dynamic behaviour of the two systems. For example, in the case of a bounded free surface, there are isolated eigenfrequencies and finite energy eigenstates, while in the case of an unbounded surface, the spectrum is, in general, continuous and the corresponding eigenstates are of infinite energy. It should be noted, however, that there are circumstances under which discrete eigenvalues and eigenstates of finite energy (trapped modes) are possible, even when the free surface extends to infinity. This phenomenon was discovered by Ursell (1951), and studied in depth by Jones (1953). See also Evans et al. (1994). It seems that the foundations of the analytical dynamics of nonlinear water waves have been laid by Petrov (1964) and Zakharov (1968). Petrov (1964) seems to be the first author who proved the validity of Hamilton's principle for nonlinear water waves, treating the sloshing problem in a container. However, his proof applies equally well to exterior flows, provided that the free-surface elevation tends sufficiently fast to zero at infinity. Zakharov (1968) recognized that the free-surface elevations and the free-surface values of the velocity potential constitute a pair of canonical variables, obtained the corresponding canonical equations, and used them to study the stability of nonlinear water waves. In the Western world the analytical dynamics of water waves has been rediscovered, independently, at least twice (see Miles (1981)), by Broer (1974) and Miles (1977). Since then a large number of relevant works have appeared, developing further and extending the Lagrangian and Hamiltonian treatment of water waves to various, more complicated situations. Among


83

them we mention Milder (1977), who first introduced the analog of eqn (1) for the case of a free surface; Crawford et al. (1981), West (1981,1983) and Zakharov himself ( 1991 ), who applied Zakharov' s equations to studying the nonlinear dynamics of ocean waves; Goncharov (1980, 1984) and Romanova (1989), who developed the Hamiltonian formalism for the two layer, oceanatmosphere system; Benjamin & Olver (1982), who derived the full symmetry group and the complete set of conservation laws for the nonlinear water wave problem; Henyey et al. (1988), who extended Hamiltonian formulation to study the dynamics of small waves riding on larger waves; Creamer et al. (1989), Wright and Creamer (1994), who exploit Hamiltonian theory and coordinate transformation to obtained improved linear representations of surface waves; Marchenko & Shrira (1991), who extended the Hamiltonian treatment to the study of waves of an ice covered liquid surface. Let us now return to the wave-body interaction problem. Obviously, to develop the analytical dynamics for this problem, one has to take into account all aspects of the analytical dynamics of the two partial problems, namely, the problem of rigid bodies moving through an ideal liquid, and the water-wave problem. Although no new fundamental difficulties in physics arise, the mathematical formalism presents new features that deserve careful analysis. For example, in this case, spatially periodic flows are not, in general, possible and thus a series-expansion representation of spatial quantities in terms of simple free-surface modes is not applicable. Moreover, the simultaneous appearance of discrete and continuously distributed degrees of freedom (in the case of freely-moving rigid bodies), is a new feature of this problem which affects the form of the equations of motion of system ::r. Contrary to the case of the water-wave problem, only very few works have appeared dealing with the analytical dynamics of wave-body interaction problem. Perhaps the first application of analytical dynamic equations to this problem was tried by Wang (1976). He used the Lagrangian expressions (1) of the hydrodynamic forces, valid in the absence of a free surface, in the study of a special free-surface problem; namely, the steady-state timeharmonic oscillations of a two-dimensional body floating on an unbounded liquid. To obtain results, he applied the Lagrangian differential operator, not on the actual kinetic energy of the liquid, which is infinite, but on an "energy function" which is bounded. Wang, despite some controversies, treated the ship motion (radiation) problem in the context of strip theory, ending in interesting and correct results with the proviso that, in the intermediate stages,

84


some physically unacceptable terms of hydrodynamic reactions are discarded on an intuitive basis. Wang's method has been subsequently extended by Loukakis & Sclavounos (1978) and Pawlowski (1982) to cover the diffraction problem as well. Despite its (conditional) success, the approach initiated by Wang did not win wide recognition, probably because of its controversial character. Motivated by the above works, Athanassoulis (1982) developed a consistent Langrangian and Hamiltonian formalism for the transient wave-body interaction problem. He formulated and proved Hamilton's principle in its first fundamental form and he used it as a basis for the consistent development of the analytical dynamics of wave-body interactions. He then derived Lagrangian and Hamiltonian equations, as well as a version of the HamiltonJacobi equation. Also, Athanassoulis (1982) extended Luke's variational principle (Luke ( 1967)) to the case of the wave-body interaction problem. In a subsequent paper (Athanassoulis & Loukakis (1985)), the Lagrangian equations of motion of the system Y were rederived by means of the principle of virtual velocities (virtual power), which is valid for both holonomic and linearly nonholonomic systems (as the system Y is). The extension of these ideas to steady-state wave-body interaction problems presents additional difficulties, since, in this case, the total energy of the liquid is infinite and the flow is not spatially periodic. This subject was addressed by Athanassoulis et al. (1991) and Theodoulidis (1995). A noticeable characteristic of the analytical dynamic formulation of the wave-body interaction problem is that the complete classical differential formulation (equations and boundary conditions) is not a prerequisite for this. As always in analytical dynamics, only the kinematical constraints satisfied by the system Y are taken as a priori conditions, the dynamical equations being obtained by means of the analytical dynamic principle used (Hamilton's principle here). To emphasize this fact, the complete classical differential formulation of the problem is not contained in the main part of this work. However, for reference purposes, it is presented and briefly discussed in Appendix 1. It is also noted that the classical formulation, given in Appendix 1, although very common in fluid mechanics, exhibits some awkward characteristics (e.g. unknown boundaries, nonlinear boundary conditions) that make it inappropriate to be used directly for the mathematical treatment and/or numerical solution of the problem. In fact, most attempts to obtain mathematical or numerical results start by devising another, more convenient, formulation of the problem. The analytical dynamic treatment, presented herewith,


85

can be considered as a systematic approach to obtain an alternative formulation, free of the peculiarities of the classical one mentioned above. Indeed, the Lagrange and Hamilton equations obtained do not contain such features as unknown boundaries or nonlinear boundary conditions. Instead they are functional differential equations3 comprising the coupled dynamics of freelymoving rigid bodies and the free surface. These equations generalise the classical Lagrange and Hamilton's equations to a higher level of mathematical sophistication, retaining most of their structural characteristics (Poisson brackets, the Hamilton-Jacobi equation, etc.). It is interesting to note that functional differential equations of exactly the same type as Hamilton's equations of the wave-body interaction problem, obtained in Section 8 of this work, were introduced and studied, in an abstract context, by Volterra as early as 1914. See, e.g., Volterra (1929/1959), pp. 163-164, and the references cited therein. Of course, these equations are by no means easy to solve. However, on the basis of their analogy to the classical equations of analytical dynamics, one may expect that they would be expedient in various directions as, for example, for stability analysis, and for obtaining a consistent stochastic mathematical model describing the evolution of the system under stochastic excitation (e.g. stochastic surface pressure due to overflowing wind). These two features of the Hamiltonian description have been very lively discussed by Bruce West (1981). See also the papers by West and Zakharov mentioned earlier. Another legitimate question about the value of the analytical dynamic formulation for the wave-body interaction problem presented here, is whether this formulation is beneficial for obtaining numerical solutions of the problem. The answer to this question should be positive, since there have been very successful numerical techniques such as, for example, those introduced by Dommermuth & Yue (1987), that essentially solve numerically Hamilton's equations of motion. See also Liu et al. (1992) and the references cited therein. In this work a unified introduction to the analytical dynamics of both the water-wave and the wave-body interaction problems is developed. Our intention is to present the subject in a systematic and consistent way, showing as clearly as possible its hidden beauty, which lies in the underlying reasons 3

That is, equations containing functional derivatives. For the reader's convenience the basic definitions and results concerning functional derivatives (in the sense of Volterra), along with extensive references to original and modern treatments of the subject, are presented in Appendix 2.

86



and physical interpretations rather than in the final results. The content of this work is organised as follows. Although the rudiments of analytical dynamics are assumed to be known, a brief survey of basic concepts is presented in Section 2. In the same section, an abstract account of the formalism which is subsequently used is also presented, containing the generalised forms of both the first and the second form of Hamilton's principle, as well as the forms of Lagrange and Hamilton's equations of motion in the required level of mathematical sophistication (i.e. as equations in functional derivatives). As we have already noted, only systems with an unbounded free surface are considered here. For such systems motions with infinite energy are possible, a fact that seriously complicates the analytical dynamic considerations. These aspects are qualitatively, yet systematically, discussed in Section 3, where the unbounded mechanical systems are classified either as a priori closed-at-infinity or as conditionally closed-at-infinity. The first class contains systems having finite total energy, while the second class contains systems of which the total energy may become infinite. System studied in this work, is a conditionally closed-at-infinity system provided that additional constraints are imposed on it in order to exclude infinite-energy states. In Section 4 it is shown that the boundary geometry and kinematics completely define, by means of a well-defined auxiliary boundary-value problem for the Laplace equation, the velocity of each element (and thus the kinetic energy) of the system .7. The auxiliary boundary-value problem is free of all peculiarities appearing in the classical differential formulation of the wave-body interaction problem. The only complexity now is the presence of infinite boundaries, which does not cause any trouble in the finite energy case, where all disturbances die out sufficiently fast at infinity. Also, in Section 4 it is shown that system .'f' is a linearly non-holonomic, ideal, mechanical system, which can be studied as if it were holonomic, since all non-holonomic constraints refer to ignorable generalised co-ordinates. In Section 5 the kinetic and potential energies of the system .'f' are calculated in terms of qa(t} and qa(t) (the rigid-body generalised co-ordinates and velocities), and q;(t) and II;;'= II;';' II XpX~ =IIx~xp.

(39)

n..

can be symmetrically extended by defining flaa = fl 00 nxpa =naxp, etc. These symmetry relations are entirely justified on the basis of the definition of coefficients n... For example, eqn (38d) expresses a reciprocity theorem justifying that II a; =II ;a .

Moreover,

5.3 Potential energy of the liquid subsystems The potential energy of a liquid element pdx in a homogenous gravitational field with intensity g is pg(x3 -x30 )dx, where x 30 represents a reference level. Accordingly, the total potential energy of the liquid is

vl =pg

f

(x3 -x3o)d\'.

(40)

D(t)

This integral, however, is not well defined in the present case, where the

domain D(t) is unbounded. This complication may easily be remedied if we replace definition (40) by

VL=pg~~=l Jx DR(t)

3 dx-

Jx,),j

(41)

DR(O)

where DR(t) is the volume of the liquid lying inside a sphere with radius R, and D (0) is the initial volume of the liquid lying inside the same sphere; see Fig. 3. ;.he physical meaning of eqn (41) is that the initial state is taken as the reference state for the liquid potential energy. Definition (41) holds both for a bounded and an unbounded liquid. Using eqn (41) and Green's theorem we obtain

116


1 +2pg

Mathematical Techniques For Water Waves 11 7 Og(t)

[I2x 3n3ds8 - J2x 3n3ds8 1 ), oD8 (t)

6 Direct proof of the first form of Hamilton's principle

~here aDF'.0), aDp(O) and aD8 (0) are the initial positions of the correspond~ng boundanes. Recalling that dDp(f) dDp8 (t)u JDP the latter being fixed m space, we see that the two integrals on dDp cancel each other and thus ,

00

fx:n3dsp fx:n3dsp = fx~n3 dsp- fx~n3 dsp. JDp(t)

JDp8 (f)

JDP101

JDp (0) 8

Now, neglecting the constant terms (i.e. the integrals on aD (0) aD (0) and aD (0)) . ' 2 2 F ' PB s , , usmg the equation x 3 n3 dsF =q~(t)~, and applying once again Green s theorem to transform the boundary integrals on aD (t) and aDPB(t)t~ volume integrals o~er the interior domains DB(t)=int(aD:(t)) and Dp8 (t)=mt(aDp8 (t)), we obtam Vr

f

= Vr(q{a}(t),qw(t),t) =~pg qi(t)d~

I

pg

X3dx-pg

Ds(t)

I

X3dx.

Dps(t)

(43) Recall that the convergence of all integrals over the free surface is a consequence ~f the assu~ption that q~(t) belongs to the finite-energy class and, thus, satisfies the fimte-energy constraints (22a).

5.4 Generalised forces associated with the free-surface degrees of freedom It is clear that the generalised forces associated with the free-surface degrees of freedom should be expressed in terms of the pressure applied externally to t~e free su~ace. Sine~ the virtual work of a system of applied pressure p(~,t), ~ E..:: , to any Virtual displacement, oq~(t), is given by

owr

it follows that

=-ff(~,t)oqg(t)d~,

(45)

(42)

oD8 (0)

00

= - ,D(~,t).

(44)

In Section 4 we described the configuration space of our hydromechanical system and in Section 5 we calculated its kinetic and potential energy, as well as the virtual work of any admissible system of external forces. Also, it became clear that the structure of our system perfectly fits the abstract structure developed in Section 2. Thus, at first sight at least, it seems that all that remains to be done is to come back in Section 2 and specify everything to the concrete case of the hydromechanical system ,a(xF,t)qa(t)+ Pfi;5(XF,t)D5(f) +P

I

eqn(28)

I

N(xF Xp,t)Dn(xp,t)dsp

=

p
1

1

1 ) l, F 1 ) 5 1 ::::51 K 1 (t) ::::51 ( q{g}(t),@(t) ( 1 1 1 0

0

0

~pf {[+~,;,;+gx3 )w

0

Zakharov, V.E. (1968) Stability of periodic waves of finite amplitude on a surface of a deep fluid, J Appl. Mech. Tech. Phys., 9(2), 190-194. Zakharov, V.E. (1992) Inverse and direct cascade in the wind-driven surface turbulence and wave breaking, Breaking Waves, ed. M.L. Banner & R.H.J. Grimshaw, IUTAM Symposium, Sydney, Australia, 1991, Springer-Verlag, Berlin, pp. 69-91.

Appendix 1:

The classical formulation of the transient wavebody interaction problem

The classical (differential) formulation of the wave-body interaction problem corresponding to our hydromechanical system !/' (unbounded, ideal liquid with an unbounded free surface, containing freely-moving rigid bodies, bodies/boundaries moving in a prescribed manner, and fixed boundaries through which a prescribed inflow/outflow takes place) can be stated as follows: Find the free-surface elevation q~(t), ~EE, the body motions qa(t) a=l,2, ... ,A, and the velocity potential (x,t), xED(t) =int(dD(t)) 18 satisfying the following equations: ,;;(x,t)=O,

q~(t)

(Al.l)

xED(t),

+ ,1(Xp,t)q~,1(f) + ,z(Xp,t)q~, 2 (t) Xp

.

(Xp,f)

- , 3 (xp,f)=0,

=(;,q~(tl) E aDp(f)'

1

,o{;,t)

+ z.,;(Xp,f),;(Xp,f) + gq~{t)=--p-, xp

,n(x3,f)

=

w 0 (x8 ,t)q0 (t)

=(;,q~(tl)EaDp(t),

(AI.2b)

+ Wc;(x8 ,t)iic;(x8 ,t), x8

18

(AI.2a)

E

dD8 (t), (Al.3a)

Recall here that itD(t)=itD(q{a}(t),qgi(t),t). For a detailed description of the liquid domain con-

figuration see Section 4.2 of the main part of this work.

(x8 ,t)ds8

145

=Q0 (t),

ifD 8 (t)

a=l,2, ... ,A, (Al.3b) (AI.4) (Al.5) In the above equations, as in the main part of this work, p(~,t) is the external pressure applied to the free surface, LB =LB lq{aJ (t),q{aJ (t),t) is the Lagrangian of the rigid-body subsystem in the absence of the liquid, w)xB,t) and wa(xB,t) are generalised direction-cosines, Q0 (t) are the generalised forces corresponding to the generalised co-ordinates of the rigid bodies, and iia(xB,t) and iin(xP,t) are given velocities. The above equations should be supplemented by appropriate initial conditions, in the case of a transient problem, or by an appropriate description of the behaviour at infinity, in the case of a steady-state problem. The former are the standard initial conditions of Newtonian mechanics (initial positions and velocities of all particles of the system), while the latter are problem-dependent and difficult to find. (See the discussion at the end of Section 3). In any case, we will not explicitly describe these supplementary conditions here. Standard sources for the classical formulation of the wave-body interaction problem are the monographs by Stoker (1957) and Wehausen & Laitone (1960). See also Wehausen (1971, 1973), Mei (1983). A few comments are in order here. The classical formulation presented above appears to define a very unusual initial-boundary value problem: Time differentiation is not applied to the "differential equation of the problem", the Laplace equation (Al.l), but it is only applied to the boundary conditions (Al.2a,b), (Al.3a) and to the ordinary differential eqns (AI.3b). Also, the potential oo

F(q@ +q;n)-F(q@)

•

Figure 5 illustrates the idea of a Volterra shrinking sequence for the case where E is the real-number axis. Note, however, that the positivity, ~ unimodality and n -monotonicity (qin+ 1 (~) < q;)~)) of the sequence shown in the figure are not required (and are not contained in Definition A2.2). Now we are in a position to state the following:

f

lfln@ct;

•

provided that this limit exists.

By direct application of the above definition we can ea~ily find the. following formulae for the first Volterra derivatives of the lmear functional f:(q ) eqn (A2 1) and the quadratic functionals F22 (q1;il, Fz1(qgi}, eqns 1

Definition A2.3: Let qi" be a Volterra shrinking sequence at the point ~ 1 EE. The first functional derivative in the sense of Volterra (or first Volterra derivative, for short) of the functional F=F(q1 ~l) with respect to the function q ~ at the point ~ 1 , denoted by 8F / 8q g , is defined by means of the • • 1 hmtt

(A2.7)

{{)

'

.

'

(A2.2a,b): (A2.8)

(A2.9a)

8F21=2A21(;1)q~.

8q

(A2.9b)

1

~1

In calculating the derivative (A2.9a) it is assumed that A22(~1,~2) =A 22 (~ 2 .~ 1 ), i.e. A 22 (•,•) is symmetric. Using the notion of the Volterra derivative, it can be shown that the first variation of F corresponding to an arbitrary (admissible) variation oq; of qg, extending over the whole domain E, is given by the formula:

8F=f l {;) ~l. • respect to q~ at the point ~ 2 , we obtained the functional

152


F(2

l( .;: 1 ;: 2 )-0F< l(q@;~ 1 )_0 F(q@) -

8

- 8

q~2

8

q~2 q~l

,

(A2. I I) where the residual

which is also an ordinary function of the two variables ; 1 , ; 2 . This functional is naturally defined as the second Volterra derivative of F =F(q ) with {/;) ' respect to the function qi; at the points ; 1 and ; 2 . Higher order Volterra derivatives are easily defined by induction. If we denote the nth order Volterra derivative of the functional F (qg ), with re1 spect to the function qi; at the points ; 1 , ; 2 ,. • • ,;n, by

F(n)(

.): ;:

;: )-

q{~},'::>l,'::>2"'"'='n

= s:

OnF(q{~}) s:

s:

uq~1 vq~2. "Vq~n

(A2.12)

Rn+i

Rn+1(q@,8gi,£) = -

is given by the formula

1

-ff . .f

p