What does it mean to solve a problem in classical elasticity? The question may appear trivial, but, if we ask scholars working in the field, we receive surprisingly different answers. Let us assume, in order to make the subject more explicit, that the problem concerns the impact between two elastic bodies. For an experimental physicist solving the problem means interpreting those crucial experiments that make it possible to decide which are the important variables in the phenomenon: in this specific case, the densities and elastic moduli of the materials are important, but the temperature and atmospheric pressure, for instance, are not. A theoretical physicist will say instead that the solution consists in formulating the general equations of the problem, having inserted all the significant variables. For a mathematician it will be obvious that solving the problem means finding an existence, uniqueness, and possibly regularity, theorem for the equations of elastic impact. Yet another answer will be given by an engineer, who will require an explicit formula giving the stress components within the two bodies at each point and at each instant.

Confronted with such a variety of answers, a typical student feels disoriented, being immediately aware of the basic ambiguity in the way in which the question itself has been posed. Ludwig Wittgenstein would say in explanation that the confusion arises from the vague use of the verb “to solve.” For the same word has been used in different contexts with different meanings.

Though dissimilar, the four answers have a common characteristic. They represent four attempts at describing the same phenomenon by abstraction, setting aside the unessential details, with the purpose not merely of illustrating but also of predicting. We say that they propose four models for the elastic impact. At this point we immediately ask whether there is a rational criterion for deciding which model is preferable, provided that all satisfy the three necessary requisites of being realistic, logically coherent, and simple. It is evident that a model must not be in obvious conflict with the physical data, nor must it be self-contradictory or too complicated. However, unfortunately, there is no incontrovertible way of establishing that one model is better than another. Setting up a model means creating conceptual conditions suitable for posing a particular question about the problem.

In Chapter 2 we presented some classical ideas in the theory of water waves. One particular concept that we introduced was the phenomenon of a balance between nonlinearity and dispersion, leading to the existence of the solitary wave, for example. Further, under suitable assumptions, this wave can be approximated by the sech2 function, which is an exact solution of the Korteweg–de Vries (KdV) equation; see Section 2.9.1. We shall now use this result as the starting point for a discussion of the equations, and of the properties of corresponding solutions, that arise when we invoke the assumptions of small amplitude and long wave-length. In the modern theories of nonlinear wave propagation – and certainly not restricted only to water waves – this has proved to be an exceptionally fruitful area of study.

The results that have been obtained, and the mathematical techniques that have been developed, have led to altogether novel, important and deep concepts in the theory of wave propagation. Starting from the general method of solution for the initial value problem for the KdV equation, a vast arena of equations, solutions and mathematical ideas has evolved. At the heart of this panoply is the soliton, which has caused much excitement in the mathematical and physical communities over the last 30 years or so. It is our intention to describe some of these results, and their relevance to the theory of water waves, where, indeed, they first arose.

The study of problems in water-wave theory, particularly under the umbrella of the linear approximation, goes back over 150 years. In the intervening time, many different problems – and extensions of standard problems – have been discussed by many authors. In a text such as ours, it is necessary to make a selection from this body of classical work; we cannot hope to describe all the various problems, nor all the subtle variants of standard problems. Our intention is, of course, to include the simplest and most fundamental results (such as, for example, the speed of waves over constant depth and the description of particle paths), but otherwise we choose those topics which contain some interesting and relevant mathematics. However, since we shall not present all that some readers might, perhaps, expect or prefer, we endeavour to remedy this by introducing additional examples through the exercises. The sufficiently dedicated reader is therefore directed to the exercises, particularly if a broader spectrum of water-wave theory is desired.

The material here is presented under two separate headings. The first is linear problems, where, apart from the elementary aspects mentioned above, we single out those topics that are attractive and which will prove relevant to some of our later discussions. Thus we describe waves on sloping beaches, as well as the phenomenon of edge waves.

In the earlier chapters we have described the mathematical background – and the mathematical details – of many classical linear and nonlinear water-wave phenomena. In addition, in the later chapters, we have presented many of the important and modern ideas that connect various aspects of soliton theory with the mathematical theory of water waves. However, much that is significant in the practical application of theories to real water waves – turbulence, random depth variations, wind shear, and much else – has been omitted. There are two reasons for this: first, most of these features are quite beyond the scope of an introductory text, and, second, the modelling of these types of phenomena follows a less systematic and well-understood path. Of course, that is not meant to imply that these approaches are unimportant; such studies have received much attention, and with good reason since they are essential in the design of man-made structures and in our endeavours to control nature.

What we have attempted here, in a manner that we hope makes the mathematical ideas transparent, is a description of some of the current approaches to the theory of water waves. To this end we have moved from the simplest models of wave propagation over stationary water of constant depth (sometimes including the effects of surface tension), to more involved problems (for example, with ‘shear’ or variable depth), but then only for gravity waves.

Geometrical Properties of Rods

From the beginnings of human civilization, rods have been used in making rudimentary tools and weapons. In the theory of the elasticity, rods have been used to model a range of physical structures such as beams, columns, and shafts, the characteristics of which is slenderness that is of having a longitudinal dimension much greater than either of the transverse dimensions. The term “bar” is sometimes used synonymously with rod, although in general speech it need not imply that one dimension is very much greater than others. Occasionally, when modeling a physical object, it is possible to identify it immediately as a rod; for example, the mast of a boat. On other occasions a physical structure can be regarded as an assembly of rods; for example, certain buildings or bridges. Rods are important because they offer the simplest but nontrivial model to which the theory of elasticity can be applied and verified experimentally. The earliest theories of rods formulated at the beginning of the seventeenth century arose as a consequence of the research of Beeckman and Mersenne (Truesdell 1960, Prologue, Sec. 3.4), but the most significant progress made in the theory was due to James Bernoulli, Euler, Saint-Venant, Kirchhoff, and Michell. Implicit in using rod models is the facility of being able to work with a single independent variable representing the arc measured along a certain curve. In slender bodies this avoids the mathematical obstacles that arise from having the three independent variables of general continuum theory.

This then requires the construction of a rational scheme for approximating the system of field equations of general continua by a system containing one spatial independent variable and, ultimately, the time. However, in order to be acceptable, the approximation procedure must explain at each stage the relationship between the approximate solutions and those of the original three-dimensional problem; in addition, it must contain, at least in principle, the method of estimating the approximation error; finally, the approximate quantities must have a significant mechanical interpretation in the form of stress averages over certain sections of the body.

We provide brief historical notes on some of the prominent mathematicians, scientists and engineers who have made significant contributions to the ideas that are described in this text. In some cases this contribution is a general mathematical technique, and in others it is a development in fluid mechanics or a specific idea in the theory of water waves. The selection that has been made is, of course, altogether the responsibility of the author, and it includes only those researchers who died at least 20 years ago.

Airy, Sir George Biddell (1801–92) British mathematician and physicist, who was Astronomer Royal for 46 years; he made contributions to theories of light and, of course, to astronomy, but also to gravitation, magnetism and sound, as well as to wave propagation in general and to the theory of tides in particular.

Bernoulli, Daniel (1700–82) Dutch-born member of the famous Swiss family of about 10 mathematicians (fathers, sons, uncles, nephews), best known for his work on fluid flow and the kinetic theory of gases; his equation for fluid flow first appeared in 1738; he also worked in astronomy and magnetism, and was the first to solve the Riccati equation.

Bessel, Friedrich Wilhelm (1784–1846) German mathematician who was, for many years, the director of the astronomical observatory in Königsberg; he was the first to study the equation that bears his name (which arose in some work on the motion of planets); he carried out a lengthy correspondence with Gauss on many mathematical topics.

Few words are used with so many different meanings as the term “model.” In everyday language the word “model” can be applied in a moral, fashion, economic, linguistic, or scientific context; in each case it means something completely different. Even if we restrict ourselves to the category of scientific models, the notion is ambiguous, because it could signify the reproduction in miniature of a certain physical phenomenon, and at the same time present a theoretical description of its nature that preserves the broad outline of its behavior. It is the theoretical aspect of models that we wish to consider; in order to emphasize this, we describe this type of model as “mathematical” (Tarski 1953). Formulating a mathematical model is a logical operation consisting in: (i) making a selection of variables relevant to the problem; (ii) postulating statements of a general law in precise mathematical form, establishing relations between some variables said to be data and others unknown; and (iii) carrying out the treatment of the mathematical problem to make the connections between these variables explicit.

The motivations underlying the use of mathematical models are of different types. Sometimes a model is the passage from a lesser known theoretical domain to another for which the theory is well established, as, for example, when we describe neurological processes by means of network theory. In other cases a model is simply a bridge between theory and observation (Aris 1978). The word “model” must be distinguished from “simulation.” The simulation of a phenomenon increases in usefulness with the quantity of specific details incorporated, as, for example, in trying to predict the circumstances under which an epidemic propagates. The mathematical model should instead include as few details as possible, but preserve the essential outline of the problem. The “simulation’ is concretely descriptive, but applies to only one case; the “mathematical model” is abstract and universal. Another special property of a good mathematical model is that it can isolate only some aspects of the physical fact, but not all. The merit of such a model is not of finding what is common to two groups of observed facts, but rather of indicating their diversities. A long-debated and important question is that of how to formulate a model in its most useful form.

The inclusion of viscosity in the modelling of the fluid requires that, at the free surface, the stresses there must be known (given) and, at the bottom, that there is no slip between the fluid and the bottom boundary. The surface stresses are resolved to produce the normal stress and any two (independent) tangential stresses. The normal stress is prescribed, predominantly, by the ambient pressure above the surface, but it may also contain a contribution from the surface tension (see Section 1.2.2). The tangential stresses describe the shearing action of the air at the surface, and therefore may be significant in the analysis of the motion of the surface which interacts with a surface wind. The bottom condition is the far simpler (and familiar) one which states that, for a viscous fluid, the fluid in contact with a solid boundary must move with that boundary.

The appropriate stress conditions are derived by considering the equilibrium of an element of the surface under the action of the forces generated by the stresses. The normal and shear stresses in the fluid (see Appendix A) produce forces that are resolved normal and tangential to the free surface, although the details of this calculation will not be reproduced here. It is sufficient for our purposes (and for general reference) to quote the results – in both rectangular Cartesian and cylindrical coordinates – for the three surface stresses.

Geometric Definitions and Mechanical Assumptions

A curved plate or shell may be described by means of its middle surface, its edge line, and its thickness 2h. We shall take the thickness to be constant, and consider the two surfaces of constant normal distance h from the middle surface and placed on opposite sides of it. These two surfaces constitute the faces of the shell. Let s denote a closed curve drawn on the strained middle surface, and consider the outer normal v to this curve at a point P1? drawn in the tangent plane to the surface at P1 and let s be the unit vector tangent to s, directed counterclockwise. Then let n denote the unit normal to the middle surface at Pi oriented positively so that the triad (v,s,n) is righthanded (Love 1927, Art. 328). We consider a point P1 on the curve s, at a small distance 8s from Pl5 and take the two segments constituted by those parts of the normals to the middle surface at Px and P1 which are included between the faces of the shell. These two pieces of normal at Pi and P1 mark out an element of area δA (Figure 54.1) belonging to a developable surface the generators of which are perpendicular to the middle surface. The contact tractions exerted across the area δA are made statically equivalent to a force and to a couple applied at Pj. We define the averaged components of this force and couple per unit length of s by dividing their components by δs. The limits of these averages for δs → 0 are the stress-resultants and the stress couples transmitted across the curve s at P1. We assume that all these limits exist, and that the component of the stress couple along n vanishes. On taking a Cartesian system x', y’ z along v, s, n, with its origin at P1, the stress-resultants are denoted by T, S, and N, respectively, and the nonzero couple resultants by H and G. These stress- and couple-resultants may be written in terms of the stress components referred to the axes x', y’ and z.

Constitutive Properties of Membranes

A curved membrane can be described geometrically by means of its middle surface, its contour, and its thickness. We shall take, for the moment, the thickness to be constant and equal to 2h, so that the upper and lower faces of the membrane are two surfaces each at a distance h from the middle surface and situated on opposite sides of it. We draw a closed curve s on the middle surface and consider the surface described by the normals to the middle surface drawn from the points of s and bounded by the two outer surfaces of the membrane with spacing 2h. We call this surface . The edge of the membrane is a surface like. If j1 is smooth, we can define the outer normal v at any point on s, so that v lies in the tangent plane to the middle surface, while s is the tangent to the line s, and n the normal to the middle surface, so that v, s, and n form a right-handed triad.

Let δs be a short arc of the curve s and take two generating lines of drawn through the extremities of 8s so as to include an area δA of. The tractions on the area δA are statically equivalent to a force at the centroid of δA together with a couple. The components of the forces in the v, s, and n directions are denoted by δT, δS, and δN, and those of the couple are denoted by δH, δG, and δK. So far we have introduced no specific hypothesis that distinguishes membranes from other continua having the same geometrical definition, called shells. The distinction arises from the constitutive behavior of membranes of being unable to transmit couples or forces perpendicular to the middle surface. In mathematical terms, this property is equivalent to the fact that, when δs is diminished indefinitely, the limits δH/δs, δG/δs, and δK/δs are zero, as is the limit of δN/δs. Only the limits δT/δs and δS/δs can be finite. Expressed in another way, a membrane is the two-dimensional analog of a string, that is to say, it is a two-dimensional continuum such that the only forces interacting between its parts are tangential to it.

Equilibrium Equations

Strings are slender bodies like rods, but are characterized by the properties that they cannot withstand compression or bending and can adapt their shape to any form of loading. They are important because they have wide technical applications, such as those required for the construction of suspension bridges, wire meshes, musical instruments, and nets of textile material. Strings are the vehicle through which some of the methods of mathematical physics have found their simplest applications.

Strings were used by Stevin in 1586 to given an experimental demonstration of the law of the triangle of forces. It seems likely that, in 1615, Beeckman solved the problem of finding the shape of a cable modeled as a string under a load uniformly distributed in a plane, and found that the string hangs in a parabolic arc. The problem of finding the configuration of a chain hanging under its own weight was considered by Galilei (1638), in his Discorsi, who concluded, erroneously, that the chain assumes a parabolic form. Subsequently, Leibniz, Huygens and James Bernoulli, apparently independently, discovered the solution now known as the catenary, taking its name from the Latin for a suspended chain. In investigating the catenary, different approaches were employed: Huygens relied on geometrical considerations, while Leibniz and James Bernoulli relied on the calculus. However, at the same time, Hooke (1675) found that a moment-free arch supports its own weight with a curve which is an inverted catenary. Another long-debated problem was that of velaria, that is, the curve assumed by an inextensible weightless string subjected to a normal force of constant magnitude. This is the form of a cylindrical sail under the action of a uniform wind, the name having its origin in this example. Huygens incorrectly stated that the curve is a parabola, but James Bernoulli aptly proved that the velaria is a circle.

The vibration of taut strings was extensively studied by d'Alembert, Daniel Bernoulli and Lagrange in the early part of the eighteenth century. Daniel Bernoulli (1733) found a solution for the natural frequencies of a chain hanging from one end.