Physics

New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity by Matthew Wright and Richard Weaver (Cambridge University Press) The field of acoustics is of immense industrial and scientific importance. The subject is built on the foundations of linear acoustics, which is widely regarded as so mature that it is fully encapsulated in the physics texts of the 1950s. This view was changed by developments in physics such as the study of quantum chaos. Developments in physics throughout the last four decades, often equally applicable to both quantum and linear acoustic problems but overwhelmingly more often expressed in the language of the former, have explored this. There is a significant new amount of theory that can be used to address problems in linear acoustics and vibration, but only a small amount of reported work does so. This book is an attempt to bridge the gap between theoreticians and practitioners, as well as the gap between quantum and acoustic. Tutorial chapters provide introductions to each of the major aspects of the physical theory and are written using the appropriate terminology of the acoustical community. The book will act as a quick-start guide to the new methods while providing a wide-ranging introduction to the physical concepts.

Matthew Wright is a senior lecturer in Acoustics at the Institute of Sound and Vibration Research (ISVR). His B.Eng. was in engineering acoustics and vibration, and his Ph.D. was in Volterra series characterization and identification of nonlinear bioacoustic systems, both from the University of Southampton. Since then he has worked on flow control for drag and noise reduction, turbofan inlet design, aeroacoustic theory, violin acoustics, and quantum chaos in acoustics, for the study of which he was awarded an EPSRC Advanced Research Fellowship. His current interests include wind farm noise and the neuroscience of hearing. He is a Fellow of the Institute of Acoustics, a Fellow of the Institute of Mathematics and Its Applications, a Senior Member of the American Institute of Aeronautics and Astronautics, a Member of the Acoustical Society of America, and the book reviews editor of the Journal of Sound and Vibration. He teaches musical instrument acoustics and acoustical design.

Richard Weaver received an A.B. degree in physics from Washington University in St. Louis in 1971 and a Ph.D. in astrophysics from Cornell University in 1977. He has been at the University of Illinois since 1981, after a research associateship in theoretical elastic wave propagation and ultrasonics at Cornell. He was elected a Fellow of the Acoustical Society of America in 1996 and received the Hetényi Award from the Society for Experimental Mechanics in 2004. He is associate editor of the Journal of the Acoustical Society of America.

In the early 1970s, Martin Gutzwiller and Roger Balian and Claude Bloch described quantum spectra in terms of classical periodic orbits, and in the mid 1970s it became clear that the random matrix theory devised for nuclear physics would also describe the statistics of quantum energy levels in classically chaotic systems. It seemed obvious even then that these two great ideas would find application in acoustics, but it has taken more than three decades for this insight to be fully implemented. The chapters in this fine collection provide abundant demonstration of the continuing fertility, in the understanding of acoustic spectra, of periodic orbit theory and the statistical approach.

First, here is a simple argument for periodic orbit theory being the uniquely appropriate tool for describing the acoustics of rooms. The reason for confining music and speech within auditoriums — at least in climates where there is no need to protect listeners from the weather — is to prevent sound from being attentuated by radiating into the open air. But if the confinement were perfect, that is, if the walls of the room were completely reflecting, sounds would reverberate forever and get confused. To avoid these extremes, the walls in a real room must be partially absorbing. This has the effect of converting the discrete eigenvalues with perfectly reflecting walls into resonances. I will argue that for real rooms the width of resonances usually exceeds their spacing. This is important because it casts doubt on the usefulness of the concept of an individual mode in assessing the acoustic response of rooms; a smoothed description of the spectrum seems preferable. But smoothing is precisely what periodic orbit theory naturally describes. When there is no absorption, the contributions from the long periodic orbits make the convergence of the sum problematic, frustrating the direct calculation of individual eigenvalues, for example, in quantum chaology. Absorption attentuates the long orbits, and the oscillatory contributions from few shortest orbits are sufficient to describe the acoustic response. But these few orbits are important: the crudest smoothing, based simply on the average spectral density, obliterates all the spectral oscillations and fails to capture the characteristics of most real rooms.

To assess the significance of absorption, start from the Weyl counting formula for the number N of modes with frequencies less than f, for a room of volume L3: if the speed of sound is c = 330 ms-1.

In the presence of absorption, modeled approximately by an exponential amplitude decay time T, that is, intensity ~ exp(-2t/ T), the resonance width corresponds to a frequency broadening.

Thus, incorporating the reverberation time T60, corresponding to 60-dB intensity reduction, that is, T = T60/3 loge 10, the number AN of modes smoothed over by the broadening.

For estimates, we can choose the frequency middle A (f = 440 Hz). Then, for a small auditorium with L = 6 m, and a reverberation time T60 = 0.7s, AN ~ 23, which is unexpectedly large for such a small room. For the Albert Hall in London, where the effective L- 60 m, and taking T60 = 2 s, A N ^ 8,200. These estimates strongly suggest that there is little sense in studying individual modes.

Second, here is an unusual application of spectral statistics from 1993, inspired by a visit to Loughborough University, where I talked about quantum chaos and mentioned that the ideas could be usefully applied in acoustics. Afterward, Robert Perrin showed me his measurements (Perrin et al. 1983) of eigenfrequencies of one English church bell, ranging from 292.72 Hz - the lowest mode, called the hum, through the first few harmonics, with their traditional names Fundamental, Tierce, Quint, Nominal, Twister, Superquint - up to the 134th frequency of 9,285 Hz. This provided sufficient data to make a first attempt to understand the frequency spacings distribution.

I did this in two ways. First, taking the whole set of 134 frequencies, unfolding them by fitting the counting function (spectral staircase) to a cubic function, and then calculating the 133 spacings, normalized to unit mean. The resulting cumulative spacings distribution C(S) = fraction of spacings less than S, fits the Poisson distribution 1 - exp(-S) reasonably well (the thin and dashed curves in the figure). This is not surprising because the bell has approximate rotation symmetry, and the whole set of frequencies conflates subsets with different numbers l of nodal meridians ("angular momentum quantum number"). Fortunately the value of / for each frequency was given; / ranged from 0 to 28, but only the subsets with 0 < / < 10 included sufficient frequencies to generate sensible statistics. In the second procedure, I unfolded these subsets separately and conflated the spacings afterwards, thereby generating the heavy curve in the figure. This is better fitted to the Wigner cumulative distribution 1 - exp(-S2/4) (the dotted curve in the figure), indicating strong repulsion of neighboring frequencies in each /-subset. The precise fit is not important because the Wigner distribution should apply when the ray geodesics on the bell - "classical paths" - are chaotic, whereas the vibrations of the bell, regarded as a thin elastic sheet, are probably integrable, with frequencies given by the modes of a one-dimensional "radial" equation, albeit of fourth order.

This book has some of its genesis in the, possibly apocryphal, story that at an acoustics conference in the late 1980s a certain distinguished professor, tiring of the proceedings, turned to the assembled researchers and announced

Listen! If what you're doing isn't nonlinear or transonic, then don't bother! It's all been done!

Certainly it has become easy to think of linear acoustics as essentially completed. After all, classic texts such as Morse and Feshbach (1953) give admirably thorough expositions of very general techniques, particularly those based on Green's functions. Cases described by coordinate systems in which the governing equations are separable are extensively tabulated and admit analytic solutions. The alternative is to employ numerical methods, many of them also based on Green's functions, which work in arbitrarily complex geometries. There is perhaps a perception that notwithstanding a host of important applied problems, there are no fundamental issues remaining in linear acoustics. Increased understanding of the richness and complexity of nonlinear problems with the explosion of interest in chaos only serves to make linear systems seem "done and dusted" in comparison.

And yet this picture is overly dismissive. A solution of a linear differential equation depends nonlinearly on its coefficients and the shape of the boundary. The dependence is all the richer if those coefficients are random or if boundary reflections are defocusing. Developments in physics throughout the last four decades, often equally applicable to both quantum and linear acoustic problems, but overwhelmingly more often expressed in the language of the former, have explored this. More than that they have provided a new way of thinking about such things. We have been impressed at the significant new body of theory that can be used to address problems in linear acoustics and vibration, although also disappointed at the small amount of reported work that does so. This book is an attempt to bridge the gap between theoreticians and practitioners, as well as the gap between quantum and acoustic, a gap that is mostly terminological but should nevertheless not be underestimated. Our hope is that acousticians and vibration engineers who wish to see what can be done with these new tools will find in this book a comprehensible introduction and that physicists may also learn what problems might usefully be addressed.

So what is on offer? We begin with what is known as the semiclassical trace formula (Chapter 1), which expresses the modal density of a closed, lossless enclosure (membrane or cavity) in terms of its periodic orbits, closed internal ray paths that repeat indefinitely. As a way to determine eigenvalues (let alone response to arbitrary excitations) it cannot compete with the numerical techniques that have been refined for use in engineering (such as finite elements) or physics (such as plane-wave decomposition); its significance lies in the fact that it provides an explicit link between the shape of an enclosure and its acoustic characteristics, both in an average sense (via the Weyl series) and at the level of individual eigenvalues, and in a way that doesn't depend on separability.

This connection is important because for many shapes the periodic orbits are unstable and the ray paths are chaotic, the implications of which are explored in Chapter 2. It can be disconcerting to find chaos having such a profound influence on linear systems. This is due to the nonlinearity of ray motion in the high-frequency limit, and the study of the effects of this on the finite-frequency wave motion has come to be known as quantum chaology or (despite linguistic objections) quantum chaos. It used to be easy to imagine that almost all ordinary differential equations had well-behaved, predictable solutions because almost all the ones in books did. That misapprehension was shattered by the explosion of awareness about chaos. In the same way it is easy to fall into the trap of thinking that modeshapes and natural frequencies are as simple and regular in arbitrary shapes as those of the simple textbook examples used to teach the subject. They are not, and for very similar reasons.

One of the consequences of chaotic ray motion is that eigenfunctions often resemble superpositions of Gaussian random waves, the properties of which are explored in more detail in Chapter 4. Those that do not are referred to as "scarred modes"; Chapter 5 presents an ingenious formulation that allows the eigenfunctions to be represented with impressive efficiency in a basis built out of deliberately constructed scar functions. Of course acousticians rarely encounter truly lossless systems in practice; so some of the implications of opening the enclosure are explored in Chapter 6. And in Chapter 7 the central result of the periodic orbit theory is re-derived in a form suitable for elasticity so as to expand the range of possible applications.

Before that, however, we introduce the second major theme of this book: random matrix theory. The study of the statistics of the eigenvalues of ensembles of matrices whose elements are random variables and exhibit a particular symmetry began in nuclear physics as an exploration of the conjecture that a sufficiently complex system might have properties statistically similar to those of a random Hamiltonian. Modern computational capabilities have made it easier to test conjectures and confirm analytic results. For example, the fact that the normalized spacings of the eigenvalues of a large Gaussian Orthogonal matrix are close to the Rayleigh distribution (obeyed exactly by an ensemble of pairs of eigenvalues of 2 x 2 Gaussian orthogonal matrices) can be shown using less than 10 lines of MATLABt and can be computed in a few seconds. Chapter 3 introduces the theory that allows such predictions and, as its name implies, explores why such an approach should be so effective in describing the behavior of the wave-bearing and vibrating systems we are considering here.

Our third theme, complexity does not get a chapter to itself or even an index entry. Instead it is embedded throughout the book in the richness of the behavior of simple systems and the diversity of applications in the later chapters. Each reader will make their own connections between the various topics here, but one striking example is worth noting here: how in a multitude of contexts "the part contains the whole." Just as each cell of an organism contains the DNA of the whole being, a few short periodic orbits contain information about a large part of the eigenstructure; in seismology and underwater acoustics a short part of a time history reveals information about the whole system.

Subsequent chapters survey several applied topics related in varying degrees to the earlier chapters. Inasmuch as multiple scattering plays such a recurrent and important role in mesoscopics (the subject of Chapter 8), we also include a review of the, often too obscure to the non-initiate, diagrammatic methods for the theory of randomly scattered acoustics in Chapter 9. The surprising and highly applicable results of the theory of time-reversed waves are explored in Chapter 10 with particular reference to the themes of this book, which have led to important applications in ultrasonics.

Chapter 11 shows the relevance of ray chaos for long-range propagation in the ocean, whereas Chapter 12 demonstrates applications in seismology. Chapter 13 shows how random matrix theory can be applied to structural acoustics and vibrations, whereas Chapter 14 explains an alternative random matrix theory approach to the problem of estimating the likely variation in response that results from the inevitable small variations that arise in manufacturing.

It is impossible in a book of practical length to cover all the modern applications of these ideas that we might have, and we apologize to those who have noted holes in our coverage. Perhaps there will be a need for another book.

Dictionary of Acoustics edited by Christopher L. Morfey (Academic Press) Covers all areas of fundamental acoustics excepting those terms specific to music and speech. A resource for anyone in acoustics, from undergraduates to researchers and practitioners. Offers mathematics as a supplement to add depth to definitions.

The science and technology of acoustics embraces an unusually wide range of disciplines, from aircraft noise reduction to ultrasonics in medicine, and from psychoacoustics to signal processing. The student of acoustics has to become familiar with a corresponding range of specialist terms in order to communicate with others and to understand the literature. Here, in one informative dictionary, for the first time, are listed accurate and helpful definitions to provide the student-or the specialist from another discipline-with a point of entry into the world of acoustics. The dictionary's 2600 entries cover most of the essential concepts and terminology that the practicing acoustician needs to understand, outside the subfields of music and speech communication. The author has drawn on experience gained during a long career spent mostly at Southampton University's multidisciplinary Institute of Sound and Vibration Research, supplemented by the expertise and perspective of a team of subject specialists.

The
Dictionary of Acoustics is:

* broad-ranging and comprehensive, covering all areas of fundamental acoustics
(except terms specific to music and speech)

* multilevel in its appeal: a reference source that can be used by
undergraduates as well as Ph.D.-level researchers, practitioners, and
consultants

* informative, with extended definitions of important concepts and a
bibliography pointing to sources for further study

* easy to use: all entries are arranged alphabetically and are thoroughly
cross-referenced

* supplemented with mathematical foundations and equations to add depth to
definitions

* written in consultation with 15 advisors on special topics and reviewed by
acousticians of international standing

About Editor: Professor Christopher Morfey has almost 40 years experience in acoustics, both in research and as a consultant. He is currently at the Institute of Sound and Vibration Research, Southampton University, U.K

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