One of the fascinating aspects of the field known colloquially as quantum chaos is the immense variety of physical contexts in which it appears. In the late 1980s it was recognized that ocean acoustics was one such context. It was discovered that the internal state of the ocean leads to multiple scattering of sound as it propagates and leads to an underlying ray dynamics which is predominantly unstable, that is, chaotic. This development helped motivate a resurgence of interest in extending dynamical systems theory suitably for applying ray theory in its full form to a “chaotic” wave mechanical propagation problem. A number of theoretical tools are indispensable, including semiclassical methods, action-angle variables, canonical perturbation theory, ray stability analysis and Lyapunov exponents, mode approximations, and various statistical methods. In the current work, we focus on these tools and how they enter into an analysis of the propagating sound.

Introduction

Acoustic wave propagation through the ocean became a topic of immense physical interest in the latter half of the twentieth century. Beyond the evident sonar applications, acoustic waves offer a means with which to probe the ocean itself. It is possible to monitor bulk mean ocean temperatures over time, which gives important information for studying global warming, and to obtain other information about the internal state of the ocean, that is, currents, eddies, internal waves, seafloor properties, and the like (Flatté et al. 1979, Munk et al. 1995).

The modeling of field distributions by Gaussian random wavefields is reviewed. Basic properties of Gaussian random functions are discussed and applied to the specific examples of randomly scattered fields (speckle patterns) and random eigenfunctions in chaotic enclosures, according to Berry's ergodic mode hypothesis. Sabine's law for reverberation time is derived for an ergodic random mode.

Introduction

The fact that deterministic wavefields in complex geometries can be represented and analyzed by statistical methods is an important assumption in many physical systems. In acoustics, such random-seeming deterministic fields can arise in scattering from a rough surface or in the spatial structure of the eigenfunctions of a perfectly resonant ergodic cavity. The random fields in the former case originate from approximating the surface roughness with randomly placed secondary sources, whose superposition gives rise to a random “speckle pattern,” typical for a wide range of wave scattering systems, especially laser light (Goodman 1985, 2007), and long studied in acoustics (Morse & Bolt 1944, Ebeling 1984). More surprising is Berry's hypothesis (Berry 1977), originally proposed in the context of quantum chaos, that a typical mode of an irregular cavity itself strongly resembles a typical Gaussian random function for high frequencies (i.e., in the semiclassical limit); again, this notion has existed in some sense for a long time in acoustics (Morse & Bolt 1944).

If Mathematics is the language of Physics, then the case for the use of Differential Geometry in Mechanics needs hardly any advocacy. The very arena of mechanical phenomena is the space-time continuum, and a continuum is another word for a differentiable manifold. Roughly speaking, this foundational notion of Differential Geometry entails an entity that can support smooth fields, the physical nature of which is a matter of context. In Continuum Mechanics, as opposed to Classical Particle Mechanics, there is another continuum at play, namely, the material body. This continuous collection of particles, known also as the body manifold, supports fields such as temperature, velocity and stress, which interact with each other according to the physical laws governing the various phenomena of interest. Thus, we can appreciate how Differential Geometry provides us with the proper mathematical framework to describe the two fundamental entities of our discourse: the space-time manifold and the body manifold. But there is much more.

When Lagrange published his treatise on analytical mechanics, he was in fact creating, or at least laying the foundations of, a Geometrical Mechanics. A classical mechanical system, such as the plane double pendulum shown in Figure 1.4, has a finite number of degrees of freedom. In this example, because of the constraints imposed by the constancy of the lengths of the links, this number is 2.

The Norwegian mathematician Sophus Lie (1842–1899) is rightly credited with the creation of one of the most fertile paradigms in mathematical physics. Some of the material discussed in the previous chapter, in particular the relation between brackets of vector fields and commutativity of flows, is directly traceable to Lie's doctoral dissertation. Twentieth-century Physics owes a great deal to Lie's ideas, and so does Differential Geometry.

Introduction

We will revisit some of the ideas introduced in Example 4.6 from a more general point of view. Just as the velocity field of a fluid presupposes an underlying flow of matter, so can any vector field be regarded as the velocity field of the steady motion of a fluid, thereby leading to the mathematical notion of the flow of a vector field. Moreover, were one to attach a marker to each of two neighbouring fluid particles, representing the tail and the tip of a vector, as time goes on the flow would carry them along, thus yielding a rate of change of the vector they define. The rigorous mathematical counterpart of this idea is the Lie derivative.

Let V : M → TM be a (smooth) vector field. A (parametrized) curve γ : I → M is called an integral curve of the vector field if its tangent at each point coincides with the vector field at that point.

Introduction

The propagation of waves through heterogeneous media occurs in many forms, including acoustic, electromagnetic, and elastic. As these waves propagate, the wave front is altered because of spatial variations in properties. The result of the interaction with the medium is that the incident energy is dispersed in many directions – the input energy is said to be scattered. If the scattering is strong and one waits long enough, the signal received will become complex because of multiple scattering effects. Understanding this process is necessary for locating an object within a scattering medium and/or for quantifying the properties of the medium itself. The focus here is on the use of diagrams than can aid in analysis of the multiple scattering process.

Multiple scattering has been discussed by theorists since the time of Rayleigh (1892, 1945). Systems with distributions of discrete inclusions (scatterers) in a homogeneous background were studied by Foldy (1945), Lax (1951, 1952), Waterman and Truell (1961), and Twersky (1977) in terms of assumed exact descriptions of scattering by isolated inclusions. This approach may be contrasted with a model of the heterogeneous medium as having continuously varying properties. This approach entails stochastic operator theory and includes the work of Karal and Keller (1964), Frisch (1968), McCoy (1981), Stanke and Kino (1984), and Hirsekorn (1988). Both approaches seek the wave speed and attenuation of an ensemble average field, although the connection to measurements in a single sample is not always obvious.

We give an overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. In the high-frequency regime, a number of universal (geometry-independent) properties that are described by random matrix theory emerge. Asymptotic methods based on the underlaying rays explain this universality and are able to go beyond it to account for geometry-specific effects. We discuss in this context statistics of the scattering matrix, scattering states, the fractal Weyl law for resonances, and fractal resonance wavefunctions.

Introduction

Our purpose here is to give an introductory overview of wave scattering in complex geometries, where the corresponding rays are typically chaotic. For simplicity, we focus our discussion on domains with lossless walls, inside which the wave speed is constant. The rays then are straight, with specular reflections at the boundaries. This situation is often encountered in experiments (Stöckmann 1999, Kuhl et al. 2005, Tanner & Søndergaard 2007) and in acoustic applications. However, many of the features we shall identify occur much more generally. Indeed, most recent developments in the subject have taken place in the context of quantum wave scattering, where the underlying rays are the classical trajectories of Newtonian mechanics. Much of this review will be devoted to translating quantum results into the language of classical wave scattering.

One of the main observations we wish to make is that many of the essential mathematical features of wave scattering in complex geometries can be found in certain very simple discrete models, which we here call wave maps.

Continuum mechanics

Matter is composed of discrete molecules, which in turn are made up of atoms. An atom consists of electrons, positively charged protons, and neutrons. Electrons form chemical bonds. An example of mechanical (i.e., has no living cells) matter is a carbon nanotube (CNT), which consists of carbon molecules in a certain geometric pattern in equilibrium with each other, as shown in Figure 1.1.1.

Another example of matter is a biological cell, which is a fundamental unit of any living organism. There are two types of cells: prokaryotic and eukaryotic cells. Eukaryotic cells are generally found in multicellular organs and have a true nucleus, distinct from a prokaryotic cell. Structurally, cells are composed of a large number of macromolecules, or large molecules. These macromolecules consist of large numbers of atoms and form specific structures, like chromosomes and plasma membranes in a cell. Macromolecules occur as four major types: carbohydrates, proteins, lipids, and nucleic acids. To highlight the hierarchical nature of the structures formed by the macromolecule in a cell, let us analyze a chromosome.

Chromosomes, which are carriers of hereditary traits in an individual, are found inside the nucleus of all eukaryotes.

Introduction

Virtually every phenomenon in nature, whether mechanical, biological, chemical, geological, or geophysical, can be described in terms of mathematical relations among various quantities of interest. Such relationships are called mathematical models and are based on fundamental scientific laws of physics that are extracted from centuries of research on the behavior of mechanical systems subjected to the action of external stimuli. What is most exciting is that the laws of physics also govern biological systems because of mass and energy transports. However, biological systems may require additional laws, yet to be discovered, from biology and chemistry to complete their description.

This chapter is devoted to the study of the fundamental laws of physics as applied to mechanical systems. The laws of physics are expressed in analytical form with the aid of the concepts and quantities introduced in previous chapters. The laws or principles of physics that we study here are the principle of conservation of mass, the principle of conservation of linear momentum, the principle of conservation of angular momentum, and the principle of conservation of energy. These laws allow us to write mathematical relationships – algebraic, differential, or integral – of physical quantities such as displacements, velocities, temperatures, stresses, and strains in mechanical systems.

Motivation

In the mathematical description of equations governing a continuous medium, we derive relations between various quantities that describe the response of the continuum by means of the laws of nature, such as Newton's laws. As a means of expressing a natural law, a coordinate system in a chosen frame of reference is often introduced. The mathematical form of the law thus depends upon the chosen coordinate system and may appear different in another coordinate system. However, the laws of nature should be independent of the choice of coordinate system, and we may seek to represent the law in a manner independent of a particular coordinate system. A way of doing this is provided by objects called vectors and tensors. When vector and tensor notation is used, a particular coordinate system need not be introduced. Consequently, the use of vector and tensor notation in formulating natural laws leaves them invariant, and we may express them in any chosen coordinate system. A study of physical phenomena by means of vectors and tensors can lead to a deeper understanding of the problem, in addition to bringing simplicity and versatility to the analysis. This chapter is dedicated to the algebra and calculus of physical vectors and tensors, as needed in the subsequent study.

This book is a simplified version of the author's book, An Introduction to Continuum Mechanics with Applications, published by Cambridge University Press (New York, 2008), intended for use as an undergraduate textbook. As most modern technologies are no longer discipline-specific but involve multidisciplinary approaches, undergraduate engineering students should be educated to think and work in such environments. Therefore, it is necessary to introduce the subject of principles of mechanics (i.e., laws of physics applied to science and engineering systems) to undergraduate students so that they have a strong background in the basic principles common to all disciplines and are able to work at the interface of science and engineering disciplines. A first course on principles of mechanics provides an introduction to the basic concepts of stress and strain and conservation principles and prepares engineers and scientists for advanced courses in traditional as well as emerging fields such as biotechnology, nanotechnology, energy systems, and computational mechanics. Undergraduate students with such a background may seek advanced degrees in traditional (e.g., aerospace, civil, electrical or mechanical engineering; physics; applied mathematics) as well as interdisciplinary (e.g., bioengineering, engineering physics, nanoscience and engineering, biomolecular engineering) degree programs.

There are not many books on principles of mechanics that are written that keep the undergraduate engineering or science student in mind.

Deformation and configuration

The present chapter is devoted to the study of geometric changes in a continuous medium that is in static or dynamic equilibrium under the action of some stimuli, such as mechanical, thermal, or other types of forces. The change of geometry or rate of change of geometry of a continuous medium can be used as a measure of so-called strains or strain rates, which are responsible for inducing stresses in the continuum. In the subsequent chapters, we will study stresses and physical principles that govern the mechanical response of a continuous medium. The study of geometric (or rate of geometric) changes in a continuum without regard to the stimuli (forces) causing the changes is known as kinematics.

Consider a continuous body of known geometry, material constitution, and loading in a three-dimensional space; the body may be viewed as a set of particles, each particle representing a large collection of molecules, having a continuous distribution of matter in space and time. Examples of such a body are provided by a diving board, the artery in a human body, a can of soda, and so on. Suppose that the body is subjected to a set of forces that tend to change the shape of the body.

Introduction

The kinematic relations developed in Chapter 3, and the principles of conservation of mass and momenta and thermodynamic principles discussed in Chapter 5, are applicable to any continuum irrespective of its physical constitution. The kinematic variables such as the strains and temperature gradient, and kinetic variables such as the stresses and heat flux were introduced independently of each other. Constitutive equations are those relations that connect the primary field variables (e.g., ρ, T, x, and u or v) to the secondary field variables (e.g., e, q, and σ). In essence, constitutive equations are mathematical models of the behavior of materials that are validated against experimental results. The differences between theoretical predictions and experimental findings are often attributed to inaccurate representation of the constitutive behavior.

A material body is said to be homogeneous if the material properties are the same throughout the body (i.e., independent of position). In a heterogeneous body, the material properties are a function of position. An anisotropic body is one that has different values of a material property in different directions at a point, that is, material properties are direction-dependent.

Introduction

In the beginning of Chapter 3, we briefly discussed the need to study deformation in materials that we may design for engineering applications. All materials have a certain threshold to withstand forces, beyond which they “fail” to perform their intended function. The force per unit area, called stress, is a measure of the capacity of the material to carry loads, and all designs are based on the criterion that the materials used have the capacity to carry the working loads of the system. Thus, it is necessary to determine the state of stress in materials that are used in a system.

In the present chapter, we study the concept of stress and its various measures. For instance, stress can be measured as a force (that occurs inside a deformed body) per unit deformed area or undeformed area. Stress at a point on the surface and at a point inside a three-dimensional continuum are measured using different entities. The stress at a point on the surface is measured in terms of force per unit area and depends on (magnitude and direction) the force vector as well as the plane on which the force is acting.