Acoustic and Evolutionary Aspects of Vocalization

Fitch, W T, 2007: Linguistics: An Invisible Hand. Nature, 449:665-667.

Quantitative relationships between how frequently a word is used and how rapidly it changes over time raise intriguing questions about the way individual behaviours determine large-scale linguistic and cultural change.

Fitch, W T, & Hauser, M D, 2004: Computational constraints on syntactic processing in a nonhuman primate. Science, 303:377-380.

The capacity to generate a limitless range of meaningful expressions from a ?nite set of elements differentiates human language from other animal communication systems. Rule systems capable of generating an in?nite set of outputs ("grammars") vary in generative power. The weakest possess only local organizational principles, with regularities limited to neighboring units.Weused a familiarization /discrimination paradigm to demonstrate that monkeys can spontaneously master such grammars. However, human language entails more sophisticated grammars, incorporating hierarchical structure. Monkeys tested with the same methods, syllables, and sequence lengths were unable to master a grammar at this higher, "phrase structure grammar" level.

Hewitt, G, MacLarnon, A, & Jones, K E, 2002: The Functions of Laryngeal Air Sacs in Primates: A New Hypothesis. Folia Primatol., 73(2-3):70-94.

A possible function of laryngeal air sacs in apes and gibbons was investigated by examining the relationships between air sac distribution, call rate, call duration and body weight in a phylogenetic context. The results suggest that lack of sacs in the smaller gibbons and in humans is a derived feature. Call parameters in primates, such as rate and duration, scaled to resting breathing rate (and therefore to body weight) only in species without air sacs, which appear to modify these relationships. Apes and larger gibbons may be able to produce fast extended call sequences without the risk of hyperventilating because they can re-breathe exhaled air from their air sacs. Humans may have lost air sacs during their evolutionary history because they are able to modify their speech breathing patterns and so reduce any tendency to hyperventilate.

Fitch, W T, & Hauser, M D, 2002: Unpacking "Honesty": Vertebrate Vocal Production and the Evolution of Acoustic Signals. ed: Simmons, A M, Fay, R R, & Popper, A N, Acoustic Communication, 65-137, Springer.

When autumn arrives in northern Europe, female red deer (Cervus elaphus L.) begin to congregate. The mating season has begun. They are soon joined by males, who have spent the previous ten months in preparation, feeding and sparring. Some of the males herd females into groups, or "harems," which they vigorously defend against other males. A prominent component of this defense is roaring, a powerful, low-pitched groaning sound made only by males and primarily by harem holders. Why do males poduce these sounds, and what effect do they have on listeners? Early observers (Darling 1937) suggested that the roars intimidated rivals and repelled intruders without the need for a dangerous fight. However, selection should favor opponents who are not so easily intimidated and base their behavior solely on a balanced assessment of their chances of winning a fight and inheriting the herd (Maynard, Smith and Price 1973). In a classic paper, Clutton-Brock and Albon (1979) showed that roaring provides a source of information relevant to this decision: roaring rates of individual males are highly correlated with their fighting ability and thus provide an accurate indication of the males' ability to repel intruders. They also demonstrated, in a series of playback experiments, that rival males attend to this information, responding preferentially to high roar rates and ignoring the roars of young, small males. Red dear roaring has since become a classic example of "truth in advertising" in an animal vocalization. "Honest signalling" in animal communication refers to signals that provide accurate information to perceivers either about the quality or properties of the signaller itself (e.g. advertisement calls) or about something in the environment (e.g. alarm calls). The degree to which animal signals are honest in this sense has been a perennially provocative problem and has generated significant theoretical advances, along with some empirical work, in the last few decades. In this chapter, we will adress the production of acoustic signals from a dynamic evolutionary perspective, paying close attention to the role of physical and phylogenetic constraints on the evolution of acoustic signals and the mechanisms that produce them.

Fitch, W T, & Hauser, M D, 1995: Vocal Production in Nonhuman Primates: Acoustics, Physiology and Functional Constraints on 'Honest' Advertising. Am. J. Primatol. 37(3):191-219.

The physiological mechanisms and acoustic principles underlying sound production in primates are important for analyzing and synthesizing primate vocalizations, for determining the range of calls that are physically producible, and for understanding primate communication in the broader comparative context of what is known about communication in other vertebrates. In this paper we discuss what is known about vocal production in nonhuman primates, relying heavily on models from speech and musical acoustics. We first describe the role of the lungs and larynx in generating the sound source, and then discuss the effects of the supralaryngeal vocal tract in modifying this source. We conclude that more research is needed to resolve several important questions about the acoustics of primate calls, including the nature of the vocal tract's contribution to call production. Nonetheless, enough is known to explore the implications of call acoustics for the evolution of primate communication. In particular, we discuss how anatomy and physiology may provide constraints resulting in honest acoustic indicators of body size.

Hauser, M, Chomsky, N & Fitch, W T, 2002: The Language Faculty: What is it, who has it, and how did it evolve? Science, 298:1569-1579.

We argue that an understanding of the faculty of language requires substantial interdisciplinary cooperation. We suggest how current developments in linguistics can be pro?tably wedded to work in evolutionary biology, anthropology, psychology, and neuroscience. We submit that a distinction should be made between the faculty of language in the broad sense (FLB)and in the narrow sense (FLN). FLB includes a sensory-motor system, a conceptual-intentional system, and the computational mechanisms for recursion, providing the capacity to generate an in?nite range of expressions from a ?nite set of elements. We hypothesize that FLN only includes recursion and is the only uniquely human component of the faculty of language. We further argue that FLN may have evolved for reasons other than language, hence comparative studies might look for evidence of such computations outside of the domain of communication (for example, number, navigation, and social relations).

Fitch, W T, Neubauer, J, & Herzel, H, 2002: Calls out of chaos: The adaptive significance of nonlinear phenomena in mammalian vocal production. Animal Behaviour, 63:407-418.

Recent work on human vocal production demonstrates that certain irregular phenomena seen in human pathological voices and baby crying result from nonlinearities in the vocal production system. Equivalent phenomena are quite common in nonhuman mammal vocal repertoires. In particular, bifurcations and chaos are ubiquitous aspects of the normal adult repertoire in many primate species. Here we argue that these phenomena result from properties inherent in the peripheral production mechanism, which allows individuals to generate highly complex and unpredictable vocalizations without requiring equivalently complex neural control mechanisms. We provide examples from the vocal repertoire of rhesus macaques, Macaca mulatta, and other species illustrating the different classes of nonlinear phenomena, and review the concepts from nonlinear dynamics that explicate these calls. Finally, we discuss the evolutionary significance of nonlinear vocal phenomena. We suggest that nonlinear phenomena may subserve individual recognition and the estimation of size or fluctuating asymmetry from vocalizations. Furthermore, neurally 'cheap' unpredictability may serve the valuable adaptive function of making chaotic calls difficult to predict and ignore. While noting that nonlinear phenomena are in some cases probably nonadaptive by-products of the physics of the sound-generating mechanism, we suggest that these functional hypotheses provide at least a partial explanation for the ubiquity of nonlinear calls in nonhuman vocal repertoires.

Fitch, W T, & Reby, D, 2001: The descended larynx is not uniquely human. Proc. Roy. Soc. B, 268(1477):1669-1675.

Morphological modifications of vocal anatomy are widespread among vertebrates, and the investigation of the physiological mechanisms and adaptive functions of such variants is an important focus of research into the evolution of communication. The `descended larynx' of adult humans has traditionally been considered unique to our species, representing an adaptation for articulate speech, and debate concerning the position of the larynx in extinct hominids assumes that a lowered larynx is diagnostic of speech and language. Here, we use bioacoustic analyses of vocalizing animals, together with anatomical analyses of functional morphology, to document descended larynges in red and fallow deer. The resting position of the larynx in males of these species is similar to that in humans, and, during roaring, red -deer stags lower the larynx even further, to the sternum. These ¢ndings indicate that laryngeal descent is not uniquely human and has evolved at least twice in independent lineages.We suggest that laryngeal descent serves to elongate the vocal tract, allowing callers to exaggerate their perceived body size by decreasing vocal-tract resonant frequencies. Vocal-tract elongation is common in birds and is probably present in additional mammals. Size exaggeration provides a non-linguistic alternative hypothesis for the descent of the larynx in human evolution.

Fitch, W T, 2000: The evolution of speech: a comparative review. Trends Cogn. Sci., 4(7):258-267.

The evolution of speech can be studied independently of the evolution of language, with the advantage that most aspects of speech acoustics, physiology and neural control are shared with animals, and thus open to empirical investigation. At least two changes were necessary prerequisites for modern human speech abilities: (1) modification of vocal tract morphology, and (2) development of vocal imitative ability. Despite an extensive literature, attempts to pinpoint the timing of these changes using fossil data have proven inconclusive. However, recent comparative data from nonhuman primates have shed light on the ancestral use of formants (a crucial cue in human speech) to identify individuals and gauge body size. Second, comparative analysis of the diverse vertebrates that have evolved vocal imitation (humans, cetaceans, seals and birds) provides several distinct, testable hypotheses about the adaptive function of vocal mimicry. These developments suggest that, for understanding the evolution of speech, comparative analysis of living species provides a viable alternative to fossil data. However, the neural basis for vocal mimicry and for mimesis in general remains unknown.

Fitch, W T, 1997: Vocal tract length and formant frequency dispersion correlate with body size in rhesus macaques. J. Acoust. Soc. Am., 102:1213-1222.

Body weight, length, and vocal tract length were measured for 23 rhesus macaques (Macaca mulatta) of various sizes using radiographs and computer graphic techniques. Linear predictive coding analysis of tape-recorded threat vocalizations was used to determine vocal tract resonance frequencies (''formants'') for the same animals. A new acoustic variable is proposed, ''formant dispersion,'' which should theoretically depend upon vocal tract length. Formant dispersion is the averaged difference between successive formant frequencies, and was found to be closely tied to both vocal tract length and body size. Despite the common claim that voice fundamental frequency (F0) provides an acoustic indication of body size, repeated investigations have failed to support such a relationship in many vertebrate species including humans. Formant dispersion, unlike voice pitch, is proposed to be a reliable predictor of body size in macaques, and probably many other species.

Fitch, W T, 1994: Vocal Tract Length Perception and the Evolution of Language, Ph.D. Thesis, Brown University.

The length of the vocal tract is correlated with body size and determines the overall dispersion of formant frequencies in speech. In this thesis I explore the interconnections between vocal tract length, formant dispersion and perceived body size. I used computer-synthesized vowel sounds to show that human subjects use vocal tract length (along with other cues) to gauge the relative body size of a speaker. Vocal tract length assessment may play an important role in "vocal tract normalization", which is crucial for speech perception and language acquisition: listeners must adjust for size differences between speakers' vocal tracts to accurately perceive speech.
A second set of experiments used synthesized vowels to show that the lengthening of the vocal tract accompanying the production of [u] ("boot") vs. [i] ("beet") is perceived by human listeners as an increase in apparent size. This may explain the long-noted phenomenon of "phonetic symbolism" for size in many languages: diminutives and words for small things contain [i], while words for "large" contain [o] or [u], far more often than predicted by chance. A control experiment showed that the phenomenon is not due to other acoustic factors or to conventional linguistic associations.
Non-human primates may also use formant dispersion as a cue to body size. If animals use vocal tract length to gauge the size of a vocalizer, we might expect them to lengthen the vocal tract in situations where it would be advantageous to seem larger (e.g., during aggressive interactions). Video analysis of captive saki monkeys (Pithecia pithecia) showed that they used lip protrusion, which lengthens the vocal tract, during behaviors accompanied by fur erection, which increases their visually -apparent size. These data, along with much of the bioacoustics literature, suggest that nonhumans also use vocal tract length as a cue to body size, and may explain the origin of the primate "fear grimace" and human smile. The framework developed in this thesis provides a unifying theoretical structure within which to empirically investigate important relationships between animal communication systems and human language.

Acoustics of the Vocal Tract

Baer, T, Gore, J C, Gracco, L C, & Nye, R W, 1991: Analysis of vocal tract shape and dimensions using magnetic resonance imaging: vowels. J. Acoust. Soc. Am., 90(2):799-828.

Magnetic resonance imaging (MRI) techniques were used to gather basic data to apply in computational models of speech articulation. Two experiments were performed. In experiment 1, voice recordings from two male subjects were obtained simultaneously with axial, coronal, or midsagittal MR images of their vocal tracts while they produced the four point vowels. Area functions describing the individual tract shapes were obtained by measurements performed on the MR images. Digital filters derived from these functions were then used to resynthesize the vowel sounds which were compared, both perceptually and acoustically, with the subjects' original recordings. In experiment 2, axial images of the pharyngeal cavity were collected during the production of an ensemble of nine vowels. Plots of cross-sectional area versus the midsagittal width of the tract at different locations within the pharynx and for different vowel productions were used to derive a functional relationship between the two variables. Data from experiment 1 relating midsagittal width to cross-sectional area within the oral cavity were also examined.

Ludwig, G D, 1950: The Velocity of Sound through Tissues and the Acoustic Impedance of Tissues. J. Acoust. Soc. Am. 22(6):862-866.

The velocity of sound through various animal organ tissues and through living human tissues is measured, using an ultrasonic pulse method, at 1.25 and 2.5 Mc. The effect of anisotropy (fiber direction) on velocity is determined with beef muscle. Values obtained with the beam traversing the tissue perpendicularly to the long axis of the muscle bundles do not differ significantly from those found with the energy directed parallel with the muscle fibers. Velocity through living human tissues, consisting mostly of muscle, is measured by transmitting the ultrasound through various thicknesses of the arm, leg, and thigh. Specific gravities of the tissues are measured. The characteristic acoustic impedances (rho c values), calculated from the density and velocity data, vary between 1.5X10^6 and 1.7X10^6 g/cm2/sec. The imaginary component of tissue impedance is calculated and found to be negligible at the frequencies at which these measurements are made.

Harper, P, Kraman, S S, Pasterkamp, H, & Wodicka, G R, 2001: An acoustic model of the respiratory tract. IEEE Trans. Biomed. Eng., 48(5):543-550.

With the emerging use of tracheal sound analysis to detect and monitor respiratory tract changes such as those found in asthma and obstructive sleep apnea, there is a need to link the attributes of these easily measured sounds first to the underlying anatomy, and then to specific pathophysiology. To begin this process, we have developed a model of the acoustic properties of the entire respiratory tract (supraglottal plus subglottal airways) over the frequency range of tracheal sound measurements, 100 to 3000 Hz. The respiratory tract is represented by a transmission line acoustical analogy with varying cross sectional area, yielding walls, and dichotomous branching in the subglottal component. The model predicts the location in frequency of the natural acoustic resonances of components or the entire tract. Individually, the supra and subglottal portions of the model predict well the distinct locations of the spectral peaks (formants) from speech sounds such as /a/ as measured at the mouth and the trachea, respectively, in healthy subjects. When combining the supraglottic and subglottic portions to form a complete tract model, the predicted peak locations compare favorably with those of tracheal sounds measured during normal breathing. This modeling effort provides the first insights into the complex relationships between the spectral peaks of tracheal sounds and the underlying anatomy of the respiratory tract.

Alkhairy, A, 2003: Mathematical models of vocal tract with distributed sources. IEEE Conf. Acoustics, Speech & Signal Processing (ICASSP 2003), I:148-151.

Frequecy dornain models for the vocal tract with distributed sources are introduced. Pressure and volume velocity distributed sources are modeled within incremental lossy cylindrical pipes in a manner similar to that of modeling electrical components. Equations for transmission matrices of uniforn-area vocal tract sections with distributed sources are derived. Transfer functions for pressure and volume velocity at various segments of the vocal tract are computed. Comparison of distributed source models with point sorirce models reveal several improvements on the traditional lumped source modeling method and shows that the effects of the finite impedance constriction and back cavity can not be adequately modeled using point sources. The distributed source vocal tract framework is important for building articulatory and acoustic models for fricative sounds and other phoneme categories. This paper provides such a mathematical framework for tlie first time.

Schroeter, J, & Sondhi, M M, 1994: Techniques for estimating vocal-tract shapes from the speech signal. IEEE Trans. Speech & Audio Proc., 2(1 Pt. 2):133-150.

This paper reviews methods for mapping from the acoustical properties of a speech signal to the geometry of the vocal tract that generated the signal. Such mapping techniques are studied for their potential application in speech synthesis, coding, and recognition. Mathematically, the estimation of the vocal tract shape from its output speech is a so-called inverse problem, where the direct problem is the synthesis of speech from a given time-varying geometry of the vocal tract and glottis. Different mappings are discussed: mapping via articulatory codebooks, mapping by nonlinear regression, mapping by basis functions, and mapping by neural networks. Besides being nonlinear, the acoustic-to -geometry mapping is also nonunique, i.e., more than one tract geometry might produce the same speech spectrum. The authors show how this nonuniqueness can be alleviated by imposing continuity constraints.

Fujimura, O, & Lindquist J, 1971: Sweep-tone measurements of vocal-tract characteristics. J. Acoust. Soc. Am., 49:541-558.

The vocal tract was excited transcutaneously at a point just above the glottis by an external sweep-tone signal, in order to measure its transfer characteristics acoustically as continuous frequency functions. An analysis-by-synthesis procedure derived reliable data of vowels, in particular of the formant bandwidths, for three male and three female normal subjects. It has been shown for the closed glottis condition that the first formant bandwidths are higher for close vowels (typically 70 Hz for male subjects) than for semi-open vowels (typically 35 Hz for male subjects). Stationary consonantal articulations including stops, nasals, and nasalized vowels also have been studied, as well as the effect of opening the glottis on the vocal-tract transfer characteristics. The stop articulations give rise to a first-formant frequency slightly below 200 Hz. This fact and the high dissipation of the first formant is explained by assuming nonrigidness of the surrounding wall. Characteristics of nasalized vowels and nasal murmurs are also discussed based on the data obtained in this experiment.

Articulatory Synthesis

Cataldo, E, Leta, F R, Lucero, J, & Nicolato, L, 2006: Synthesis of voiced sounds using low-dimensional models of the vocal cords and time-varying subglottal pressure. Mechanics Res. Comm., 33(2):250-260.

The vocal cords play an important role on voice production. Air coming from the lungs is forced through the narrow space between the two vocal cords that are set in motion in a frequency that is governed by the tension of the attached muscles. The motion of the vocal cords changes the type of flow, that comes from the lungs, to pulses of air, and as the flow passes through the oral and nasal cavities, it is amplified and further modified until it is radiated from the mouth. This complex process can be modeled by a system of integral-differential equations. This paper considers two mechanical models previously used for explaining the dynamics of the vocal cords. It shows that the level of naturalness of the sound generated by these models is rather poor, and it proposes temporal variations of the parameters of the models to increase such level. Examples of synthetic vowels and diphthongs are given to assess the models. In general, the results show that, although the system of voice production is complex, we can achieve satisfactory results with relatively simple low-dimensional models, by suitable temporal variations of the aerodynamic parameters.

Cataldo, E, Lucero, J C, Sampaio, R, & Nicolato, L, 2006: Comparison of some mechanical models of larynx in the synthesis of voiced sounds. J. Braz. Soc. Mech. Sci. & Eng., 28(4):461-466.

The process of voiced sounds production can be described as follows: air coming from the lungs is forced through the narrow space between the two vocal folds, which are set in motion in a frequency governed by the tension of their tissues. The vocal folds change the continuous flow that comes from the lungs into a series of pulses. Then, as the flow passes through the oral and nasal cavities it is amplified and changed until it is finally radiated from the mouth. This complex process can be modeled by a system of integral-differential equations. In spite of such complexity, this paper shows that it is possible to obtain synthetic voice sounds of satisfactory realism using simple mathematical models. The perception of a synthetic sound as natural is increased by choosing suitable waveforms for the time-varying subglottal pressure, rather than by augmenting the number of degrees of freedom of the mechanical model. This paper also shows possible ways to adapt the models to voices of men, women and children.

Badin, P, & Fant, G, 1984: Notes on vocal tract computation. STL-QPSR, 25(2-3):53-108.

In the general frame of vocal tract computation in the frequency domain, we describe experiments aiminq at the evaluation of different methods to handle the boundary conditions and the losses, that is, the radiation load, the viscous and thermal losses. the wall vibrations, the glottal and subglottal impedances, and the constriction resistances. We also discuss a metbd to determine poles and zeros for any transfer function in the tract. We give data for the Russian vowels (FANT, 1960) in different conditions, including diver's speech, and show an example of application to the study of a constrictive consonant.

Fant, G, 1972: Vocal tract wall effects, losses and resonance bandwidths. STL-QPSR, 13(2-3):28-52.

The finite impedance of the vocal tract cavity walls accounts for a shift in formant frequencies compared to the idealized hard-wall conditions. These effects are treated by: 1) low frequency approximation, 2) single-tube tract and lumped element representation of the wall shunt, 3) generalized distributed -element treatment of mass loading and losses. The superposition of sound radiated from the walls and from the mouth is calculated for case 2) above. Data on resonance bandwidths are reviewed and simplified formulas are derived for estimating formant bandwidth contributions from a) radiation, b) classical friction and heat conduction losses, c) dissipation within the cavity walls. The relative importance of these factors in various vocal tract configurations and with respect to different resonances is discussed. Formulas have also been derived for predicting resonance bandwidth from the set of resonance frequencies.

Stevens, K N, Kasowski, S, & Fant, G, 1953: An electrical analog of the vocal tract. J. Acoust. Soc. Am., 25(4):1070-1075.

The design and construction of an electrical analog of the human vocal tract is described. The vocal tract is viewed as an acoustic tube of varying cross-sectional area, terminated by the vocal cords at one end and by the lip opening at the other. The analog is a lumped-constant electrical transmission line consisting of thirty-five pi-sections. Each section represents a 1/2-cm length of the vocal tract and is adjustable to simulate a range of cross-sectional areas from 0.17 to 17 cm^2. The electrical line can be excited by a periodic current source representing the vocal cord output or by a random voltage representing the noise from turbulent air flow at a constriction. The electrical analog can synthesize with good quality all English vowels and some consonants. The physical characteristics of the output of the analog for each sound are shown in terms of measured formant frequencies or by conventional spectrograms. For each sound, the vocal tract dimensions that the electrical network simulates are shown in graphical form. Applications of the speech synthesizer to linguistic and engineering research are discussed briefly.

Richards, M A, & Schafer, R W, 1984: Acoustic tube analysis of formant bandwidths and frequencies in helium speech. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing (ICASSP'84), 64-67.

We use the lossy electric transmission-line analog model of the vocal tract to study the acoustics of speech produced in a hyperbaric helium-oxygen atmosphere. The analysis extends previous work by including more completely the effects of the wall vibration, glottal, and radiation impedances, and by analyzing the formant bandwidths and amplitudes in addition to the formant frequencies. It shows that (1) the classic Fant and Lindquist formula somewhat overstates the formant frequency shift when glottal and radiation effects are included; (2) the lower formant bandwidths increase by much more than commonly assumed; and (3) the upper formant amplitudes are higher relative to the lower formants in helium speech than in normal speech. These results are useful in developing advanced helium speech enhancement algorithms.

Lunde, P, 1985: Acoustic transmission-line analysis of formants in hyperbaric helium speech. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing (ICASSP'85), 1141-1144.

Acoustic transmission-line vocal tract models are used to study formant frequencies, bandwidths and amplitudes of hyperbaric heliox speech versus those of speech in air at 1 ATA. The models account for energy losses due to glottal impedance, lip/nostrils radiation, wall vibration, viscous friction and thermal conduction. New wall impedance values are presented, matching measurements of the closed tract resonance. On basis of, a uniform tube model, an extended version of the classic Fant-Lindquist formula [1] describing formant frequency shift is developed, and formulas for bandwidth and amplitude shifts are given. A multitube vocal tract model is applied for analysis of the effects of nonuniform vocal tract cross-sectional area on the formant shift.

Helmholtz Resonators

Starecki, T, 2007: Loss-improved electroacoustical modeling of small Helmholtz resonators. J. Acoust. Soc. Am., 122(4):2118-2123.

Modeling of small Helmholtz resonators based on electroacoustical analogies often results in significant disagreement with measurements, as existing models do not take into account some losses that are observed in practical implementations of such acoustical circuits, e.g., in photoacoustic Helmholtz cells. The paper presents a method which introduces loss corrections to the transmission line model, resulting in substantial improvement of simulations. Values of the loss corrections obtained from comparison of frequency responses of practically implemented resonators with computer simulations are presented in tabular and graphical form. A simple analytical function that can be used for interpolation or extrapolation of the loss corrections for other dimensions of the Helmholtz resonators is also given. Verification of such a modeling method against an open two-cavity Helmholtz structure shows very good agreement between measurements and simulations.

Photiadis, D M, 1991: The effect of wall elasticity on the properties of a Helmholtz resonator. J. Acoust. Soc. Am., 90(2):1188-1190.

The effect of compliant walls on the properties of a Helmholtz resonator is examined. The effective stiffness of the resonator is decreased by the wall compliance, while the effective mass is unchanged to leading order. The radiation resistance is also decreased due to a cancellation between the radiation from the cavity opening and the radiation from the cavity walls. This leads to a reduction in the cross section at resonance by the factor [Zs/(Zs+Z)]^-4, where Zs is the wall impedance and Z is the total fluid loading on the breathing mode of the cavity.

Acoustic Transmission Line Theory

Bansevicius, R, & Kargaudas, V, 2005: Wave propagation in rigid cylindrical tubes with viscous and heat-conducting fluid. Ultragarsas, 56(3):7-10.

The propagation of sound waves in cylindrical tubes with compressible fluid is a fundamental and classical problem. Famous names - Helmholtz, Kirchhoff, Rayleigh - are connected with the first studies. The full Kirchhoff solution of a viscous and heat-conducting fluid in rigid circular tubes was later developed in two directions: analytical approximations of a very complicated transcendental equations for various regions of "wide", "narrow", "wide-narrow", "very wide" tubes and an extension of the theory for non-circular tubes or higher modes in circular tubes. Another development of investigations considered deformations of the pipe. Interaction of the compressible ideal fluid and elastic shell is the principle aim of these investigations.

Tijdeman, H, 1975: On the propagation of sound waves in cylindrical tubes. J. Sound Vib., 39(1):1-33.

It is shown that the two main parameters governing the propagation of sound waves in gases contained in rigid cylindrical tubes, are the shear wave number, s = Rv(??/µ), and the reduced frequency, k = ?R/a. It appears possible to rewrite the most significant analytical solutions for the propagation constant, G, as given in the literature, as simple expressions in terms of these two parameters. With the aid of these expressions the various solutions are put in perspective and their ranges of applicability are indicated.
It is demonstrated that most of the analytical solutions are dependent only on the shear wave number, s, and that they are covered completely by the solution obtained for the first time by Zwikker and Kosten.
The full solution of the problem has been obtained by Kirchhoff in the form of a complicated, complex transcendental equation. In the present paper this equation is rewritten in terms of the mentioned basic parameters and brought in the attractive form F(G,s,k) = 0, which is solved numerically by using the Newton-Raphson procedure. As first estimate in this procedure the value of G according to the solution of Zwikker and Kosten is taken. Results are presented for a wide range of s and k values.

Weston, D E, 1953: The theory of the propagation of plane sound waves in tubes. Proc. Phys. Soc. B, 66(8):695-709.

The propagation of plane sound waves in gases in tubes can be divided into three main types, depending on the radius and frequency involved. These types are described as 'narrow' tube, 'wide' tube and 'very wide' tube propagation. The phase velocity, attenuation and cross section profile of particle velocity etc. are investigated theoretically, and their inter-relation pointed out. The factors affecting the validity of Kirchhoff's formulae are considered, and the theory is applied to some recent work.

Page, N W, & Mee, D J, 1984: Wall effects on sound propagation in tubes. J. Sound Vib., 93(4):473-480.

Numerical solutions have been obtained for the exact equations describing the propagation of periodic axisymmetric waves in a rigid cylindrical tube. Results were obtained for air over a range of conditions corresponding to shear wave numbers, s = R v(??/µ), from 0.2 to 5000 and reduced frequencies, k = ?R/a, from 0.01 to 6. For conciseness and convenient application, the results for the attenuation and phase shift coefficients are given in the form of simple polynomials for the ranges 5 = s = 5000 and 0 .01 = k = 6. This range covers virtually all values of tube diameter and sound frequency likely to be met in practical situations that are consistent with a continuum gas model.

Shields, F D, Lee, K P, & Wiley, W J, 1965: Numerical solution for sound velocity and absorption in cylindrical tubes. J. Acoust. Soc. Am., 37(4):724-729.

A numerical solution of the Kirchhoff equation for the propagation constant of longitudinal sound waves in infinitely long cylindrical tubes has been obtained. The solution, which avoids the wide-tube approximations, shows that the percentage errors in the von Helmholtz-Kirchhoff tube velocity correction and tube absorption are both roughly equal to the percentage the velocity correction is of the free-space velocity. The error in the von Helmholtz-Kirchhoff equations can be plotted as a function of fD/a, pD/?a, and ?. (f is the sound frequency, D the tube diameter, a the free-space velocity, p the gas pressure, ? the viscosity, and ? the ratio of specific heats.) Recent absorption measurements in Ar are in agreement with values calculated numerically, but measured velocities indicate the need for considering molecular slip at the tube wall. Thermal relaxation is introduced into Kirchhoff's basic equation by using the Eucken relation k/c_o? - (9? - 5)/4 and considering ? to be the ratio of complex relaxing specific heats. Viscothermal and relaxation effects are found to be additive only if the frequency is near the cutoff frequency for the first unsymmetric mode and the f/p values do not extend to the megacycle / (second atmospheres) range.

Bruneau, A M, Bruneau, M, Herzog, P, Kergomard, J, 1987: Boundary layer attenuation of higher order modes in waveguides. J. Sound Vib., 119(1):15-27.

The attenuation of higher order modes in rectangular and circular tubes is treated here by using results for the boundary layer admittance for the respective normal modes. Comparison with results available in the literature for propagating modes is given. Results for evanescent modes and at the cut-off frequencies are discussed. Finally, the well-known Kirchhoff theory is extended to obtain a test of validity for the proposed calculations.

Rodarte, E, Singh, G, Miller, N R, & Hrnjak, P, 2000: Sound attenuation in tubes due to visco-thermal effects. J. Sound Vib., 231(5):1221-1242.

The propagation of periodic axial sound waves in gases contained in circular cylindrical structures is a function of four parameters: s = R v(?·?/µ), the shear wave number or Stokes number, k = ?·R/c, known as the reduced frequency, s = v(µC_p/?), the square root of the Prandtl number, and ? = C_p/ C_v, the ratio of specific heats. The complete Kirchhoff solution of the sound propagation in tubes problem obtained in 1868 was expressed in terms of these parameters by Tijdeman [1]. In previous works [1, 2] the complex propagation constant was obtained by solving this expression. The results were presented for a limited range in reference [1] and for a broader range in reference [2] but in both cases only for a single fluid, air. In this work the results of a computer code to solve for this propagation constant are presented. The code was used to find the propagation constants (attenuation and phase -shift coefficients) in the range 5

Cummings, A, 1993: Sound propagation in narrow tubes of arbitrary cross section. J. Sound Vib., 162(1):27-42.

A variational treatment of the problem of sound transmission in narrow tubes is described as an alternative to the more usual analytical procedures, which are limited to mathematically tractable geometries. This method can be used for any cross-sectional geometry, as long as the cross-section of the tube is uniform along its length. Very simple low frequency and high frequency approximate solutions are obtained by the use of idealized trial functions, and these compare tolerably well to existing analytical solutions for a variety of geometries.

Lin, T C, & Morgan, G W, 1956: Wave propagation through fluid contained in cylindrical elastic shell. J. Acoust. Soc. Am., 28(6):1165-1176.

A study is presented of the propagation of axisymmetric waves through compressible, inviscid fluid contained in a cylindrical, elastic shell. The dependence of the phase velocity as a function of frequency on four dimensionless parameters of the system is discussed and illustrated graphically.

Bruneau, M, Herzog, P, Kergomard, J, & Polack, J D, 1989: General formulation of the dispersion equation in bounded viscothermal ?uid. Wave Motion, 11(5):441-451.

The purpose of this paper is to give a new set of equations derived from the basic classical theory of the acoustic propagation in visco-thermal fluid and valid in the time domain, and to provide us with a general dispersion equation for harmonic waves in several boundary problems of interest. It is shown that this dispersion equation generalizes some known results as the equivalent specific impedance of plane boundaries and resonance frequencies of spherical resonators, and that it provides us with a new general equation giving the propagation constant of waves for all kind of modes in rigid walled cylindrical tubes.

Christensen, R, Juhl, P, & Henriquez, V C, 2008: Practical modeling of acoustic losses in air due to heat conduction and viscosity. Proc. Acoustics 08 Conference.

Accurate acoustic models of small devices with cavities and narrow slits and ducts should include the so-called boundary layer attenuation caused by thermal conduction and viscosity. The purpose of this paper is to present and compare different methods for including these loss mechanisms in analytical and numerical models. Two test cases with circular geometry have been used as references and are investigated both through measurements and the different models. Four simulation methods are compared. The transmission line model is an analytical model which can be modi?ed to include loss. Additionally, three numerical models have been tested. Two different implementations of the so-called Full Navier-Stokes model, one using the commercial package COMSOL Multiphysics, the other using Boundary Element code speci?cally aimed at the test case, are considered. The third numerical method, the so-called Low reduced frequency model, is evaluated using the commercial package ACTRAN.

Beltman, W, 1999: Viscothermal wave propagation including acousto-elastic interaction, part I: Theory. J. Sound Vib., 227(3):555-586.

This research deals with pressure waves in a gas trapped in thin layers or narrow tubes. In these cases viscous and thermal effects can have a significant effect on the propagation of waves. This so-called viscothermal wave propagation is governed by a number of dimensionless parameters. The two most important parameters are the shear wave number and the reduced frequency. These parameters were used to put into perspective the models that were presented in the literature. The analysis shows that the complete parameter range is covered by three classes of models: the standard wave equation model, the low reduced frequency model and the full linearized Navier-Stokes model. For the majority of practical situations, the low reduced frequency model is sufficient and the most efficient to describe viscothermal wave propagation. The full linearized Navier-Stokes model should only be used under extreme conditions.

Beltman, W, 1999: Viscothermal wave propagation including acousto-elastic interaction, part II: Applications. J. Sound Vib., 227(3):587-609.

In Part I of the present paper a parameter analysis showed that the most efficient model to describe viscothermal wave propagation is the low reduced frequency model. In order to demonstrate the wide range of applicability of the low reduced frequency model, a number of examples from the literature are discussed in Part II. An overview of fundamental solutions and general applications is given. Because the models are all written in terms of dimensionless parameters and solutions for various co-ordinate systems are given, this paper also serves as a solution overview.

Beltman, W M, Wijnant, Y H, & Nijhof, M J J, 2008: An overview of models for viscothermal wave propagation, including fluid-structure interaction. Proc. Acoustics 08, 3275-3280.

In acoustics, the standard wave propagation models neglect the effects of viscosity and thermal conductivity. When waves propagate in narrow tubes or thin layers, these simplifications may not be accurate. This paper presents an overview of models that take into account the effects of inertia, viscosity, thermal conductivity and compressibility. Based on the use of dimensionless parameters, three classes of models are outlined. The most important dimensionless parameter is the shear wave number, an unsteady Reynolds number that indicates the ratio between inertial and viscous effects. These viscothermal wave propagation models can be coupled to structural models to capture the fluid-structure interaction. Analytical solutions can be found for these coupled scenarios for simple geometries and boundary conditions. For more complex geometries, numerical models were developed. Examples of applications of these models are also presented.

Zuercher, J C, Carlson, E V, & Killion, M C, 1988: Small acoustic tubes: New approximations including isothermal and viscous effects. J. Acoust. Soc. Am., 83(4):1653-1660.

Algebraic expressions are given (in fortran format for convenient use) that accurately approximate the value of the per unit-length acoustic impedance of small tubes (radii less than 1/16 acoustic wavelength) including isothermal and viscous effects. Their accuracy of typically 3% or better was obtained as a result of adding a shunt conductance term to the traditional approximations and including frequency-dependent multiplicative factors. The multiplicative factors are simple expressions for the real or imaginary terms, providing a factor of 40 increase in calculational speed over the full Bessel-function solution. Experimental verification is presented for various combinations of small tubes and cavities relevant to hearing aid applications, and the differences between the present approximations and previous approximations are illustrated using a 1-mm-diam tube as an example. Normalized complex series impedance and shunt admittance, characteristic impedance, time delay, and attenuation constant plots are provided as a reference utilizing the more nearly exact Bessel solution.

Fuller, C R, & Fahy, F J, 1982: Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. J. Sound Vib., 81(4):501-518.

The dispersion behaviour and energy distributions of free waves in thin walled cylindrical elastic shells filled with fluid are investigated. Dispersion curves are presented for a range of parameters and the behaviour of individual branches is explained. A non-dimensional equation which determines the distribution of vibrational energy between the shell wall and the contained fluid is derived and its variation with frequency and material parameters is studied.

Pavic, G, 1992: Vibroacoustical energy flow through straight pipes. J. Sound Vib., 154(3):411-429.

At lower relative (i.e., non-dimensional) frequencies, four propagating waves exist in fluid-filled pipes. Each of these waves carries energy in the pipe wall, while three waves carry energy in the fluid as well. The otherwise fairly complex dispersion laws for waves in pipes simplify in the frequency region considered to simple rod- and beam-type laws. It is shown that these laws can be determined by approximate formulae fairly accurately, the accuracy decreasing with increase in frequency. Due to fluid -wall coupling, expressed again by simplifications, the energy flow in both the wall and the fluid can be evaluated in principle from knowledge of surface vibrations only. The portions of the flow in the solid and the fluid fluctuate along the pipe axis, and consequently spatial averaging has to be done in order to obtain useful results. In this way, the pipe becomes a homogeneous one-dimensional waveguide, suitable for measurements of energy flow by detection of surface vibrations only. Specific transducer patterns for this purpose are described. At higher frequencies however, where additional propagating waves take place, simplifications are no longer possible. The exact expression for the unit-length energy flow can be then employed in conjunction with averaging around the circumference to evaluate flow in the wall at a particular axial position.

Feng, L, 1994: Acoustic properties of fluid-filled elastic pipes. J. Sound Vib., 176(3):399-413.

The acoustic properties of an infinite, fluid-filled pipe have been investigated. The relation between radiated sound power and the system power distribution for a single mode is analyzed. Energy flow is examined. Coupling effects on acoustic properties are compared. Whether coupling with fluid will increase or decrease pipe response, and hence noise power, is largely dependent on the frequency range and on the method of excitation.

Kuhn, G F, & Morfey, C L, 1976: Transmission of low-frequency internal sound through pipe walls. J. Sound Vib., 47(2):147-161.

Transmission loss measurements are reported for long steel pipes of circular crosssection, with air inside and out, excited by internal sound. At low frequencies (wavelength greater than the pipe diameter), most of the radiated sound is accounted for by pipe bending waves. In order to approach the much higher transmission loss predicted for pure breathing motion of the pipe, bending waves must be suppressed; this has been achieved for a straight pipe by careful isolation. A sharp 90 bend in the pipe is shown to cause significant bending-wave excitation when plane waves are incident on the bend.

Smith Jr, P W, 1957: Sound transmission through thin cylindrical shells. J. Acoust. Soc. Am., 29(6):721-729.

An analysis presented of the impedance presented by a thin cylindrical elastic shell to a pressure or normal stress as a function of the axial wavelength and the angular dependence of the forces. Results of computation are presented graphically. This information is then used to compute a measure of the sound transmitted through the shell immersed in air for various particular cases. The theory of the scattering and absorption of waves incident upon a cylinder at angles other than normal is developed for this purpose. The results and their implications are discussed in detail.

Cummings, A, 1978: Low frequency acoustic transmission through the walls of rectangular ducts. J. Sound Vib., 61(3):327-345.

A simple theory is described for the transmission of low frequency sound through the walls of rectangular ducts, particularly those in air conditioning systems. The model is based on a coupled acoustic/structural wave system, and it is assumed that the duct radiates in the same way as a finite -length line source incorporating a single travelling wave. Measurements of wall transmission loss on two types of duct system are compared to theoretical predictions, and good agreement is obtained within the frequency range of validity of the theory. It is concluded that the present approach should give reliable estimates of noise transmission in practical situations.

Cummings, A, 2001: Sound transmission through duct walls. J. Sound Vib., 239(4):731-765.

Acoustic "breakout" and "breakin" through duct walls had, until the late 1970s, been a rather neglected topic of research, particularly in the field of heating, ventilating and air-conditioning ducts. Since then, interest has grown and many publications have appeared in which predictive methods have been reported. Research in this area, especially that which has been conducted over the past two decades, is reviewed in this paper. Efforts are made to identify the main physical processes involved and to present some relevant published data, rather than to give a finely detailed, comprehensive, account of this research. Some comments are made concerning the possible direction of future research.

Stinson, M R, 1991: The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape. J. Acoust. Soc. Am., 89(2):550-558.

The general Kirchhoff theory of sound propagation in a circular tube is shown to take a considerably simpler form in a regime that includes both narrow and wide tubes. For tube radii greater than r_w = 10^-3 cm and sound frequencies f such that r_w f^3/2 < 10^6 cm s^-3/2, the Kirchhoff solution reduces to the approximate solution suggested by Zwikker and Kosten. In this regime, viscosity and thermal conductivity effects are treated separately, within complex density and complex compressibility functions. The sound pressure is essentially constant through each cross section, and the excess density and sound pressure (when scaled by the equilibrium density and pressure of air, respectively) are comparable in magnitude. These last two observations are assumed to apply to uniform tubes having arbitrary cross-sectional shape, and a generalized theory of sound propagation in narrow and wide tubes is derived. The two-dimensional wave equation that results can be used to describe the variation of either particle velocity or excess temperature over a cross section. Complex density and compressibility functions, propagation constants, and characteristic impedances may then be calculated. As an example, this procedure has been used to determine the propagation characteristics for a tube of rectangular cross section.

Moloney, M J, & Hatten, D L, 2001: Acoustic quality factor and energy losses in cylindrical pipes. Am. J. Phys., 69(3):311-314.

The quality factor Q of a damped oscillator equals 2pi times the ratio of stored energy to the energy dissipated per cycle. This makes Q a sensitive probe of energy losses. Using modest equipment, we measured the acoustical Q for a set of cylindrical pipes having the same resonant frequency, but different diameters D. The graph of Q vs D could be well fitted with two parameters, one of which corresponds to energy loss via radiation from the ends of the pipe, and the other to thermal and viscous losses very close to the pipe wall. The wall loss parameter was quite constant no matter where the pipes were located, but the radiative loss parameter varied significantly with location inside a room, suggesting that room reflections affected the sound radiated from the pipe. This study offers valuable insights at no great expense, and could be the basis of an upper-division undergraduate laboratory experiment.

White, J E, 1950: A Method for Measuring Source Impedance and Tube Attenuation. J. Acoust. Soc. Am., 22(5):565-567.

If the active face, or acoustic output terminal, of a sinusoidal sound source moves as a plane piston, then the source can be characterized by a blocked pressure and an acoustic output impedance. If this piston is coupled to a microphone by means of a closed air column, the pressure at the microphone depends on the acoustic impedance of the microphone, on the impedance of the source, and on the air column. An expression for this pressure as a function of the length of the air column is developed, and data are presented which show how source impedance, tube attenuation and other quantities may be obtained.

Miles, J W, 1947: The Equivalent Circuit for a Bifurcated Cylindrical Tube. J. Acoust. Soc. Am., 19(4):579-584.

Using the impedance concepts developed in earlier papers, the problem of the diffraction of sound, caused by a bifurcation of a cylindrical tube, is solved. The equivalent circuit elements are shown to be related to the analogous changes of cross section. It is shown that the transmitted power is divided between the two tubes resulting from the bifurcation in proportion to their areas and without frequency distortion or reflection, and that the bifurcation may, therefore, be regarded as taking place in a virtual plane parallel to, but somewhat removed frown, the geometrical plane. A formula is given to locate this plane.The results are applied to a concentric-circular bifurcation and to a rectangular-tube bifurcated parallel to one of its walls. Numerical results are given which may be applied both to the circular change of cross section and to the concentric bifurcation of a circular tube.

Wente, E C, 1922: The thermophone. Phys. Rev., 19(4):333-345.

Acoustic Efficiency of Thermophones of the Heated Foil or Wire Type.-Theoretical formulę are derived for the maximum value of the alternating pressure produced within the enclosure of any thermophone when a given alternating current, superposed on a direct current, is passed through the central foil or wire. The effect of certain simplifying assumptions which are made is shown to be small in practical cases. As an experimental verification of the formulę, an electrostatic transmitter was calibrated for a wide range of frequencies with four thermophones which differed greatly in their physical constants, the formulę being used to compute the pressures produced. The four calibrations thus obtained agree with each other closely and also with an independent calibration made with a pistonphone.
Methods of Calibrating Acoustical Transmitters.-In addition to the pistonphone, the thermophone is now available. The circuits used in calibrating an electrostatic transmitter are given.
Method of Measuring the Thermal Conductivity of a Gas by Using a Thermophone is Suggested.-It would have the advantage of avoiding difficulties due to convection.

Mawardi, O K, 1949: On the Propagation of Sound Waves in Narrow Conduits. J. Acoust. Soc. Am., 21(5):482-486.

The analysis of the propagation of sound waves in narrow tubes has usually been restricted to shapes yielding tractable mathematical expressions. A great number of practical applications do not fall within these categories and await a solution. An approximate solution of sufficient accuracy for narrow tubes of arbitrary shapes developed in this paper has been applied to a wire-filled tube. The theoretical predictions check satisfactorily with the experimental results. It is believed that this study will be useful in other similar applications.

Daniels, F B, 1947: Acoustical Impedance of Enclosures. J. Acoust. Soc. Am., 19(4):569-571.

Formulas are derived for the acoustical impedance of three types of enclosures, a sphere, a cylinder, and a narrow rectangular box. The solutions are valid throughout the entire range from adiabatic to isothermal conditions.

Daniels, F B, 1950: On the Propagation of Sound Waves in a Cylindrical Conduit. J. Acoust. Soc. Am., 22(5):563-564.

The characteristic impedance and propagation constant of a cylindrical conduit are calculated on the basis of an equivalent electrical T-section. Numerical values of the results are plotted for air at 20°C, for a range of values of the independent variable which includes the region of transition from isothermal to adiabatic conditions.

Iberall, A S, 1950: Attenuation of oscillatory pressures in instrument lines. J. Res. Nat'l Bureau of Standards., 45:85-108.

A theoretical investigation has been made of the attenuation and lag of an oscillatory pressure variation applied to one end of a tube, when the other end is connected to a pressure sensitive element. An elementary theory based on incompressible viscous-fluid flow is first developed. The elementary solution is then modified to take into account compressibility; finite pressure amplitudes; appreciable fluid acceleration; and finite length of tubing (end effects). Account is taken of heat transfer into the tube. The complete theory is derived in an appendix. The results are summarized in eight graphs in a form convenient for use in computing the lag and attenuation of a sinusoidal oscillation in a transmission tube.

Harris, J, Peev, G, & Wilkinson, W L, 1969: Velocity profiles in laminar oscillatory flow in tubes. J. Phys. E: Sci. Instrum., 2(11):913-916.

The oscillatory laminar flow of a Newtonian fluid in a circular tube has been investigated when the fluid is subjected to a periodic pressure gradient. A technique has been developed which allows the velocity amplitude to be determined as a function of radius and the results agree closely with theoretical predictions. The work is being continued with a view to employing the technique for the determination of the rheological properties of viscoelastic fluids.

Benade, A H, 1968: On the propagation of sound waves in a cylindrical conduit. J. Acoust. Soc. Am., 44(2):616-623.

The series impedance and shunt admittance of an acoustic line is calculated from the linearized acoustic equations. Exact and limiting formulas for small and large tubes are provided for R, L, G, C, the real and imaginary parts of the characteristic impedance Z0, as well as the phase velocity v and attenuation constant . All results are presented in convenient form for quick computation on the basis of tables and graphs. A self-consistent set of molecular data is presented. Accuracies of formulas and of the data are discussed in detail.

Keefe, D H, 1984: Acoustical wave propagation in cylindrical ducts: Transmission line approximation for isothermal and nonisothermal boundary conditions. J. Acoust. Soc. Am., 75(1):58-62.

Approximate expressions are given for the characteristic impedance and propagation wavenumber for linear acoustic transmission through a gas enclosed in a rigid cylindrical duct. These expressions are most complicated in the transition zone where the thermoviscous boundary layers are on the order of the tube radius. The approximations are accurate to within 1% for all frequencies and tube diameters except within the transition zone where the approximations are accurate to within 10%. A simple modification of the transmission line parameters is presented for the case where the tube walls are nonisothermal.

Maisano, J, 1994: An improved transmission line model for visco-thermal lossy sound propagation. Proc. 96th AES Convention, 8.6-8.22.

A transmission line model for lossy sound propagation has been obtained by solving the state law of air and the Navier-Stokes, mass conservation. Fourier heat equations. The series impedance and shunt admittance of general sound propagation has been established in order to obtain the acoustic equivalent elements representing the visco-thermal effects. Acoustic equivalent elements are given for sound propagation in various structures such as holes, cavities, or ducts present in every miniaturized transducer and in particular in integrated microphones and earphones.

Guianvarc'h, C, Durochera, J-N, Bruneau, M, & Bruneau, A-M, 2006: Acoustic transfer admittance of cylindrical cavities. J. Sound Vib., 292(3-5):595-603.

The reciprocity calibration method uses two microphones acoustically connected by a coupler, a cylindrical cavity closed at each end by the diaphragms of the transmitting and receiving microphones. The acoustic transfer admittance of the coupler, including the thermal conductivity effect of the fluid, must be modelled precisely to obtain the accurate sensitivity of the microphones from the electrical transfer impedance measurement. It appears that the analytical model quoted in the current standard [International Electrotechnical Commission IEC 61064-2, Measurement Microphones, Part 2: Primary Method for Pressure Calibration of Laboratory Standard Microphones by the Reciprocity Technique, 1992] is not the appropriate one and that it should be revised, as also suggested by a recent EUROMET project report [K. Rasmussen, Datafiles simulating a pressure reciprocity calibration of microphones, EUROMET Project 294 Report PL-13, 2001]. Thus, it is the aim of the paper to investigate analytically the acoustic field inside the coupler, revisiting the assumptions of the earlier work, leading to a coherent description and therefore providing clarity which should facilitate discussion of a possible revised standard.

Ballagh, K O, 1987: Acoustical admittance of cylindrical cavities. J. Sound Vib., 112(3):567-569.

The acoustical admittance of cylindrical cavities is required to be known very accurately for precise calibration of laboratory standard microphones.

Jarvis, D R, 1987: Acoustical admittance of cylindrical cavities. J. Sound Vib., 117(2):390-392.

A recent letter to this journal from Ballagh explained how the present method of correcting the acoustical admittance of a cavity for heat conduction to the cavity wall can be replaced by one which allows for the interaction between heat conduction and wave motion in the cavity. It was shown that in the pressure calibration of one-inch microphones by the reciprocity method, heat conduction has a significant effect over the whole of the audible frequency range and not just at low frequencies as previously assumed. At NPL we have been using a theory similar to this in work to extend the reciprocity method to the calibration of half-inch microphones, and would like to present some evidence in support of Ballagh. We would also like to explain a further modification to the method proposed by Ballagh and that recommended in IEC 327, which takes into account the impedance of the driver.

Scheichl, S, 2004: On the calculation of the transmission line parameters for long tubes using the method of multiple scales. J. Acoust. Soc. Am., 115(2):534-555.

The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Anderson, M J, & Vaidya, P G, 1991: Thermo-viscous effects on finite amplitude sound propagation in a rectangular waveguide. J. Acoust. Soc. Am., 90(2):1188-1190.

The role that thermo-viscous effects play in the propagation of finite level sound in a waveguide has been reexamined from a fundamental perspective. In the past, nonlinear acoustic interactions have been described by energy conserving modulation of spectral amplitudes as wave packets travel axially down the waveguide. To account for thermo-viscous effects in this modulation, investigators have included without formal justification into the modulation equations dissipative terms with a magnitude corresponding to the Kirchhoff rate of attenuation encountered in linear theory. In this investigation, the problem of the propagation of finite magnitude plane waves is analyzed in a different manner. As opposed to previous investigations, all three modes (acoustic, vorticity, and entropy) are considered from the outset. The boundary conditions are extended to include vanishing normal and tangential fluid velocity, as well as vanishing fluid temperature perturbations. A new solution at second order is presented (second order being the first correction due to nonlinearity), which is uniform in the spatial variables. As a consequence, it is shown that the thermo-viscous effects are incorporated into the spectral amplitude modulation equations through one of the boundary conditions. These modulation equations apply to both plane and higher-order modes, including the region arbitrarily near the cutoff frequency for the higher-order modes. It is shown that the small parameter 1/(N)^1/2, where N = rho _0 Dc / µ (the acoustic Reynolds number), is a special scale for analysis of nonlinear interactions in a waveguide. In particular, the relative magnitude of the sound source and 1/(N)^1/2 is a determining factor that predicts whether nonlinear interactions will be significant.

Stolyarov, E P, 2005: Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium. Fluid Dynamics, 40(3):403-412.

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn << 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M a = v/a => 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.

Trochidis, A, & Radcliffe, C J, 1991: Sound transmission in a duct with an array of lined resonators. J. Vib. Acoust., 113(2):.245-249.

A simple method is presented for describing the sound transmission in a duct containing an array of lined resonators. The impedance of a resonator element is calculated, including the effect of the lining and the duct is modeled as a one-dimensional waveguide with lumped impedances. An expression for the TL is derived from considerations of pressure and mass flow continuity along the duct. Experimental data compared to numerical computations show that the method developed here describes satisfactorily the performance of the duct and allows useful parametric analyses which can lead to improved design.

Hull, A J, & Radcliffe, C J, 1991: An eigenvalue based acoustic impedance measurement technique. J. Vib. Acoust., 113(2):250-254.

A method is developed for measuring acoustic impedance. The method employs a one-dimensional tube or duct with excitation at one end and an unknown acoustic impedance at the termination end. Microphones placed in the tube are then employed to measure the frequency response of the system from which acoustic impedance of the end is calculated. This method uses fixed instrumentation and takes advantage of modern Fast Fourier Transform analyzers. Conventional impedance tube methods have errors resulting from movement of microphones to locate the maxima and minima of the wave pattern in the impedance tube or require phase matched microphones with specific microphone spacing. This technique avoids these problems by calculating the acoustic impedance from measured duct eigenvalues. Laboratory tests of the method are presented to demonstrate its accuracy.

Figueroa, H M, & Juan, E J, 2006: Estimation of Tube Wall Compliance Using Pulse-Echo Acoustic Reflectometry. 28th Ann. Int'l Conf. IEEE Eng. in Med. & Biol. Soc. (EMBS'06), 2852-2855.

This work was primarily supported by the NIH/MBRS-SCORE program. Abstract-This paper focuses on the estimation of tube wall compliance using reflection analysis of acoustic pulses. The wall compliance of a rubber latex tube was found theoretically using an acoustical transmission line model. Wall compliance was also obtained experimentally from acoustical and mechanical measurements. The acoustically estimated, mechanically estimated, and simulated wall compliances were CwExp=6.55·10-7 cm5/dyne, CwMech=6.89·10-7 cm5/dyne and CwSim= 5.18·10-7 cm5/dyne, respectively. The methods developed and the preliminary results obtained from this research could serve as the groundwork for the development of a device that determines the pathological condition of compliant biological conduits such as the airways.

Albors, G O, Kyle, A M, Wodicka, G R, & Juan, E J, 2007: Computer simulation tool for predicting sound propagation in air-filled tubes with acoustic impedance discontinuities. Proc. 29th Ann. Int'l Conf. IEEE Eng. Med. Biol. Soc., 2203-2206.

A computer tool, based on an acoustic transmission line model, was developed for modeling and predicting sound propagation and reflections in cascaded tube segments. This subroutine considered the number of interconnected tubes, their dimensions and wall properties, as well as medium properties to create a network of cascaded transmission line model segments, from which the impulse response of the network was estimated. Acoustic propagation was examined in air-filled cascaded tube networks and model predictions were compared to measured acoustic pulse responses. The model was able to accurately predict the location and morphology of reflections. The developed code proved to be a useful design tool for applications such as the guidance of catheters through compliant air-filled biological conduits.

Coltman, J W, 1976: Jet drive mechanisms in edge tones and organ pipes. J. Acoust. Soc. Am., Vol. 60(3):725-733.

Measurements of the phases of free jet waves relative to an acoustic excitation, and of the pattern and time phase of the sound pressure produced by the same jet impinging on an edge, provide a consistent model for Stage I frequencies of edge tones and of an organ pipe with identical geometry. Both systems are explained entirely in terms of volume displacement of air by the jet. During edge-tone oscillation, 180° of phase delay occur on the jet. Peak positive acoustic pressure on a given side of the edge occurs at the instant the jet profile crosses the edge and starts into that side. For the pipe, additional phase shifts occur that depend on the driving points for the jet current, the Q of the pipe, and the frequency of oscillation. Introduction of this additional phase shift yields an accurate prediction of the frequencies of a blown pipe and the blowing pressure at which mode jumps will occur.

Horne, M P, & Hansen, R J, 1982: Sound propagation in a pipe containing a liquid of comparable acoustic impedance. J. Acoust. Soc. Am., 71(6):1400-1405.

A detailed experimental study of sound propagation in liquids contained by pipes constructed of polymeric materials is discussed. Experiments were conducted with vertically aligned cylinders containing water ensonified at one end by a piston-driven sound source. Significant sound attenuation (as much as 60 dB) was observed in pipes made of flexible polymeric materials, the effect increasing with frequency and loss tangent. Sound propagation in more rigid polymeric pipes exhibited similar characteristics to that in metallic pipe in that negligible attenuation was observed. In this latter case, a comparison was made with recent analytical work for which excellent agreement was obtained.

Morse, P M, 1939: Some Aspects of the Theory of Room Acoustics. J. Acoust. Soc. Am., 10(1):55-66.

An exact solution for the decay of sound in a rectangular room is obtained; assuming that each wall is uniformly covered with absorbing material, which may differ from wall to wall. It is concluded, from recent experimental measurements, that the boundary conditions for the sound field are correctly expressed in terms of the effective normal impedance of the wall material. The sound is analyzed into its component normal modes of vibration, and the reverberation times and frequencies of the different normal modes are calculated as functions of the wall impedances and their phase angles. Curves are given for these quantities for a wide range of the parameters involved. The effect of the absorbing material in distorting the sound field is shown, and several other interesting points are brought out in the discussion: that waves which travel "parallel" to a wall are absorbed by the wall, but are not absorbed as much as are waves striking at more oblique angles; that it is sometimes possible to increase the reverberation time for a standing wave by decreasing a wall's effective acoustic resistance; etc.

Rossetti, S, Gardonio, P, & Brennan, M J, 2005: A wave model for rigid-frame porous materials using lumped parameter concepts. J. Sound Vib., 286(1-2):81-96.

The work presented in this paper concerns the behaviour of porous media when exposed to a normal incidence sound field. A propagating wave model based on lumped parameter concepts of acoustic mass, stiffness and damping is used to investigate the absorption phenomena due to the wave propagation in the layer(s) and interference effects due to the wave reflection-transmission at the interfaces of the layer(s). Results from the theoretical model have been validated by measurements on samples of consolidated rubber granulate material. Two typical installations where a layer of porous material is placed next to a rigid wall, and where it is placed at a distance from a rigid wall are used as reference cases. The geometrical and physical properties of porous materials can be described by such parameters as the non-dimensional shape factor and the porosity. The propagating model introduced is used to investigate the effect of these two parameters on acoustic absorption and thus relate the physical properties to the acoustic behaviour.

Schneider, J B, & Ramahi, O M, 1998: The complementary operators method applied to acoustic finite-difference time-domain simulations. J. Acoust. Soc. Am., 104(2):686-693.

The complementary operators method (COM) has recently been introduced as a mesh-truncation technique for open-domain radiation problems in electromagnetics. The COM entails the construction of two solutions that employ absorbing boundary conditions (ABCs) with complementary behavior, i.e., the reflection coefficients associated with the two ABCs are exactly opposite each other. The average of these solutions then yields a new solution in which the errors caused by artificial reflections from the termination of grid are nearly eliminated. In this work, COM is introduced for the finite-difference time -domain (FDTD) solution of acoustics problems. The development of COM is presented in terms of Higdon's absorbing boundary operators, but generalization to non-Higdon operators is straightforward. The effectiveness of COM in comparison to other absorbing boundary conditions is demonstrated with numerical experiments in two and three dimensions.

Electromagnetic / Acoustic / Mechanical Analogy and Duality

Pueo, B, Escolano, J, & Romį, M, 2002: Advanced electric lumped model of a transmision line loudspeaker system. 33rd Congresos Nacionales de Acśstica, Seville, Spain.

Many models based on physical properties of damped pipes have been proposed to characterize the Transmission Line Loudspeaker Systems. Unfortunately, Thiele/Small parameters, usually employed in enclosure designs, are not useful as parameters of design for these loudspeakers. Derived from Locanthi horn model, an advanced circuit with lumped elements is presented. It can accommodate arbitrary flare shapes and damping. The influence of the main empirical parameters is tested on a prototype to validate the model. The main contribution of this paper consists on the model of the acoustic radiation impedance at the end of a long tube.

Nicolas, L, Furstoss, M, & Galland, M-A, 1998: Analogy electromagnetism-acoustics: Validation and application to local impedance active control for sound absorption. Eur. Phys. J. Appl. Phys., 4(1):95-100.

An analogy between electromagnetism and acoustics is presented in 2D. The propagation of sound in presence of absorbing material is modeled using an open boundary microwave package. Validation is performed through analytical and experimental results. Application to local impedance active control for free field sound absorption is finally described.

Le Corbeiller, P, & Yeung, Y-W, 1952: Duality in Mechanics. J. Acoust. Soc. Am., 24(6):643-648.

Given an electric network Ev which has M meshes and P node-pairs, its electric dual Ei will have P meshes and M node-pairs and its classical mechanical analog Mf will have M+1 nodes, M independent node-pairs, and P independent node cycles. A second mechanical system Mv, the classical analog of Ei, will have M cycles and P node-pairs. If, for example, M = 2, P = 3, the systems Ev and Mv, analogs in the Firestone or "mobility" method, will be governed by two mesh equations, expressing that the algebraic sum of the voltages or velocities around any loop is zero; the systems Ei and Mf, also Firestone analogs, will satisfy two node equations, expressing that the algebraic sum of the currents or forces leaving any node is zero. These four sets of equations are identical, interchanging symbols suitably. The consideration of the four systems, Ev, Ei, Mv, Mf, forming a complete set, shows the advantages of the Firestone over the classical system of analogies and suggests a systematic use of duality in mechanical as well as in electrical systems.

Elecromagnetic Transmission Line Theory

Jordan, P M, & Puri, A, 1999: Digital signal propagation in dispersive media. J. Appl. Phys., 85(3):1273-1282.

In this article, the propagation of digital and analog signals through media which, in general, are both dissipative and dispersive is modeled using the one-dimensional telegraph equation. Input signals are represented using impulsive, Heaviside unit step, Gaussian, rectangular pulse, and both unmodulated and modulated sinusoidal pulse type boundary data. Applications to coaxial transmission lines and freshwater signal propagation, for both digital and analog signals, are included. The analysis presented here supports the finding that digital transmission in dispersive media is generally superior to that of analog. The boundary data (input signals) give rise to solutions of the telegraph equation which contain propagating discontinuities. It is shown that the magnitudes of these discontinuities, as a function of distance, can be found without the need of solving the governing equation. Thus, for digital signals in particular, signal strength at a given distance from the input source can be easily determined. Furthermore, the magnitudes of these discontinuities are found to be independent of both the dispersion coefficient k and the elastic coefficient b. In addition, it is shown that, depending on the algebraic sign of k, one of two distinct forms of dispersion is possible and that for small-time intervals, solutions are approximately independent of k.

Djordjevic, A R, Zajic, A G, Tosic, D V, & Hoang, T, 2003: A note on the modeling of transmission-line losses. IEEE Trans. Microwave Theory & Techniques, 51(2):483-486.

We consider uniform lossy transmission lines characterized by their primary parameters. Exact and approximate formulas for the characteristic impedance and propagation coefficient are reviewed and discussed for low-loss lines. Approximating the characteristic impedance by its real part can lead to erroneous results for the input impedance of short- and open-circuited stubs. This problem is analytically demonstrated on electrically short stubs. Results obtained using the exact and approximate expressions are compared with numerical solutions that are generated by various circuit simulation software.

Dieckmann, A, 2001: Pulse Propagation along close conductors. <pi.physik.uni-bonn.de/~dieckman/Pulse.pdf>.

The propagation and reflection of arbitrarily shaped pulses on non-dispersive parallel conductors of finite length with user defined cross section is simulated employing the discretized telegraph equation. The geometry of the system of conductors and the presence of dielectric material determine the capacities and inductances that enter the calculation. The values of these parameters are found using an iterative Laplace equation solving procedure and confirmed for certain calculable geometries including the line charge inside a box. As an example a pair of microstrips as used in the ATLAS vertex detector is analysed.

Weeks, W T, 1972: Multiconductor Transmission-line Theory in the TEM Approximation. IBM J. Res. Develop. 1972(Nov):604-611.

Starting with Maxwell's equations, the transmission line equations are derived for a system consisting of an arbitrary number of conductors. The derivation is rigorous for long lossless conductors embedded in a uniform perfect dielectric. The presentation is essentially tutorial, most of the results being well known, at least for two- and three-conductor systems. The novelty lies in the point of view adopted in obtaining a systematic generalization to the case of an arbitrary number of conductors. Explicit expressions are obtained for the electric and magnetic fields in the dielectric surrounding the conductors, and a rigorous formulation is given for the problem of calculating the coefficients of capacitance and inductance.

Riederer, A J, & Weinberg, L, 1984: Synthesis of lumped-distributed cascades with lossy transmission lines. IEEE Trans. Circuits & Systems, 31(5):485-500.

In the analysis of large systems such as high-speed digital computing networks and circuits on an LSI or VLSI silicon chip, lossy lumped-distributed networks have been used to model their interconnections. A solution of the synthesis problem for these networks will aid in the design of these circuits. This paper establishes single-variable realizability conditions and synthesis procedures for the class of lossy lumped-distributed cascade networks which contain lossy transmission lines and are described by a driving point impedance expression of a particular form. The cascade networks consist of commensurate, uniform and/or tapered, lossy (except distortionless [3], [4]) transmission lines interconnected by passive, lumped (lossless and/or lossy) two-ports and terminated in a passive load. This class includes general lines, leakage-free lines, RC-lines and acoustic filters. The results also apply to cascades with noncommensurate lines and to cascades of mixed transmission-line types.

Wave Propagation in Dissipative Media

Beyer, R T, 1978: Radiation pressure - the history of a mislabeled tensor. J. Acoust. Soc. Am., 63(4):1025-1030.

The acoustic radiation pressure has found practical application in recent years in instruments measuring sound intensity and in experiments on acoustic levitation. The concept of radiation pressure has, however, long fascinated both optical and acoustical physicists. The history of light radiation pressure goes back more than 200 years to Leonhard Euler, while the concept of acoustic radiation pressure dates from the time and work of Rayleigh. It was pointed out by Brillouin that what we call radiation pressure is not a pressure at all, but a diagonal tensor, all the diagonal terms of which are not identical. The size of the effect is small, and the values obtained for the radiation pressure are very sensitive to boundary conditions and to the approximations that must necessarily be employed. In addition, although the phenomenon is primarily one of nonlinear acoustics, it can be observed down to the lowest sound intensities under certain conditions. Thus, the Rayleigh radiation pressure vanishes for the linear case, but the usually measured Langevin pressure does not. It might be said that radiation pressure is a phenomenon that the observer thinks he understands-for short intervals, and only every now and then.

Ruggeri, T, 1983: Symmetric-hyperbolic system of conservative equations for a viscous heat conducting fluid. Acta Mechanica, 47(3-4):167-183.

Some critical considerations on the models of extended irreversible thermodynamics are given. By developing a methodology (invariance of the generators) based both on the ideas of the extended irreversible thermodynamics and the entropy principle in its general formulation of rational thermodynamics, a theory for a newtonian thermoviscous fluid is proposed. The theory has the following properties, new when compared with previous ones: a) the system of equations is hyperbolic for any value of the field variables, provided that the usual thermodynamic stability condition (maximum entropy at equilibrium) holds; all wave-propagation speeds are then real and finite; b) the system is conservative and it is possible to seek for weak solutions and, in particular, for shockwaves; moreover, the system is symmetric-hyperbolic in the sense of Friedrichs; special properties hold therefore for weak solutions and shocks; c) the only thermodynamic variables at non-equilibrium, modified with respect to the corresponding ones at equilibrium, are the entropy density and chemical potential; consequentely, there exists only a single absolute temperature, playing an important role in relaxations; d) the entropy principle is automatically satisfied.

Joseph, D D, 2003: Viscous potential flow. J. Fluid Mech., 479:191-197.

Potential flows ${\bm u} = {\bm\nabla} \phi$ are solutions of the Navier-Stokes equations for viscous incompressible fluids for which the vorticity is identically zero. The viscous term $\mu \nabla^2 {\bm u} = \mu{\bm \nabla}\nabla^2\phi$ vanishes, but the viscous contribution to the stress in an incompressible fluid (Stokes 1850) does not vanish in general. Here, we show how the viscosity of a viscous fluid in potential flow away from the boundary layers enters Prandtl's boundary layer equations. Potential flow equations for viscous compressible fluids are derived for sound waves which perturb the Navier-Stokes equations linearized on a state of rest. These linearized equations support a potential flow with the novel features that the Bernoulli equation and the potential as well as the stress depend on the viscosity. The effect of viscosity is to produce decay in time of spatially periodic waves or decay and growth in space of time-periodic waves. In all cases in which potential flows satisfy the Navier-Stokes equations, which includes all potential flows of incompressible fluids as well as potential flows in the acoustic approximation derived here, it is neither necessary nor useful to put the viscosity to zero.

Cox, R H, 1968: Wave Propagation through a Newtonian Fluid Contained within a Thick-Walled, Viscoelastic Tube. Biophys. J., 8(6):691-709.

The propagation of harmonic pressure waves through a Newtonian fluid contained within a thick-walled, viscoelastic tube is considered as a model of arterial blood flow. The fluid is assumed to be homogeneous and Newtonian, and its motion to be laminar and axisymmetric. The wall is assumed to be isotropic, incompressible, linear, and viscoelastic. It is also assumed that the motion is such that the convective acceleration is negligible. The motion of the fluid is described by the linearized form of the Navier-Stokes equations and the motion of the wall by classical elasticity theory. The frequency dependence of the wall mechanical properties are represented by a three parameter, relaxation-type model. Using boundary conditions describing the continuity of stress and velocity components in the fluid and the wall, explicit solutions for the system of equations of the model have been obtained. The longitudinal fluid impedance has been expressed in terms of frequency and the system parameters. The frequency equation has been solved and the propagation constant also expressed in terms of frequency and system parameters. The results indicate that the fluid impedance is smaller than predicted by the rigid tube model or by Womersley's constrained elastic tube model. Also, the velocity of propagation is generally slower and the transmission per wavelength less than predicted by Womersley's elastic tube model. The propagation constant is very sensitive to changes in the degree of wall viscoelasticity.

Tsaturyan, A K, 1975: Electrical simulation of the circulatory system. Mech. Comp. Mater., 11(4):657-660.

An electrical model of the arterial part of the human vascular system is proposed. This model is used to investigate the impedance, the shape of the pressure and flow waves, and the static characteristics of the vascular system under normal conditions and in the case of artificial circulation. In order to simulate the ACA pump, the problem of the quasi-one-dimensional flow of a viscous fluid in a tube with wave -type variation of the radius is considered. A comparison of the results with the results of direct measurements in man shows that they are in qualitative and quantitative agreement. Simulation also reveals certain characteristics of pulse-wave propagation in artificial circulation. An explanation of the effects observed is proposed, and their possible influence on the activity of the organism is discussed.

Walther, K, 1960: Reflection factor of gradual-transition absorbers for electromagnetic and acoustic waves. IRE Trans. Antennas & Propagation, 8(6):608-621.

Absorbers for electromagnetic or acoustic waves are described, for which a good impedance match and low reflection factor can be achieved by providing a gradual transition of material constants into the lossy medium. The reflection factor can be calculated by means of a Riccati-differential equation. General conclusions from the WKB-perturbation method can be drawn for absorbers, the layer thickness of which is either very small or very large in comparison to the wavelength. For "thin" layers, wave energy penetrates the whole thickness of the absorber. Suitable average values of the material constants are derived to describe the performance of the panel in this case. For "thick" layers only the initial part of the panel is energized. The asymptotic expressions contain only the material constants of this part. The results are interpreted physically. Numerical solutions of the reflection factor for highly refractive panels with exponentially varying material constants are reported.

Anderson, W, & Moore, R, 1960: Frequency spectra of transient electromagnetic pulses in a conducting medium. IRE Trans. Antennas & Propagation, 8(6):603-607.

The energy density spectra of transient electromagnetic fields generated by a pulsed ideal dipole source in an infinite conducting medium have been investigated for various distances from the source. A characteristic frequencyomega_{c}, corresponding either to the peak of the spectrum or to its haft -width, is defined and shown to vary inversely as the square of distance at large distances. The behavior ofomega_{c}with distance is a measure of the behavior of the pulse energy. Thus, at large distances it appears that the attenuation factor associated withomega_{c}, exp {-rsqrt{omega_{c}musigma/2}}, is independent ofr, due to the constancy of the productrsqrt{omega_{c}}. From this point of view, the transient fields do not decrease exponentially asr, but as inverse powers ofr. This should not be construed as meaning that the transient possesses an advantage over CW. The attenuation for monochromatic components of the pulse is the same as for continuous waves of the same frequency and at large distances the energy put into the high frequency components is wasted. The phenomenon is illustrated by calculations that have been carried out for the case of pulses in sea water.

Direct and Inverse Scattering

Jaulent, M, 1982: The Inverse Scattering Problem for LCRG Transmission Lines. J. Math. Phys., 23(12):2286-2290.

The inverse scattering problem for one-dimensional nonuniform transmission lines with inductance L(z), capacitance C(z), series resistance R(z) and shunt conductance G(z) per unit length (zR) is considered. It is reduced to the inverse scattering problem for the Zakharov-Shabat system. It is found that one can construct from the data the following functions of the travel time x: ±(x)=[(1/4)(d/dx)(ln(L/C))±(1/2 )(R/L-G/C)] ×exp((R/L+G/C)dy).

Kristensson, G, & Krueger, R J, 1986: Direct and inverse scattering in the time domain for a dissipative wave equation. I. Scattering operators. J. Math. Phys., 27(6):1667-1682.

This is the first part of a series of papers devoted to direct and inverse scattering of transient waves in lossy inhomogeneous media. The medium is assumed to be stratified, i.e., it varies only with depth. The wave propagation is modeled in an electromagnetic case with spatially varying permittivity and conductivity. The objective in this first paper is to analyze properties of the scattering operators (impulse responses) for the medium and to introduce the reader to the inverse problem, which is the subject of the second paper in this series. In particular, imbedding equations for the propagation operators are derived and the corresponding equations for the scattering operators are reviewed. The kernel representations of the propagation operators are shown to have compact support in the time variable. This property implies that transmission and reflection data can be extended from one round trip to arbitrary time intervals. The compact support of the propagator kernels also restricts the admissible set of transmission kernels consistent with the model employed in this paper. Special cases of scattering and propagation kernels that can be expressed in closed form are presented.

Kristensson, G, & Krueger, R J, 1986: Direct and inverse scattering in the time domain for a dissipative wave equation. II. Simultaneous reconstruction of dissipation and phase velocity profiles. J. Math. Phys., 27(6):1683-1693.

The one-dimensional inverse scattering problem for inhomogeneous lossy media is considered. The model problem involves electromagnetic wave propagation in a medium of unknown thickness with spatially varying conductivity and permittivity. Two inversion algorithms are developed in the time domain using data obtained from normally incident plane waves. These algorithms utilize reflection data from both sides of the medium, and one of them also uses transmission data. These algorithms are implemented numerically on several examples, one of which includes the effects of noisy data. The possibility of using one-sided reflection data and no transmission data is reviewed and analyzed.

Yagle, A E, & Levy, B C, 1985: The Schur algorithm and its applications. Acta Applicandae Mathematicae, 3(3):255-284.

The Schur algorithm and its time-domain counterpart, the fast Cholesky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Cholesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.

Transmission Line Matrix (TLM) Methods and Huygens' Principle

Enders, P, 2001: Huygens' principle in the transmission line matrix method (TLM). Global theory. Int'l J. Numer. Model., 14(5):451-456.

Huygens' principle (HP) is understood as a universal principle governing not only the propagation of light, but also of acoustic waves, heat and matter diffusion, Schrödinger's matter waves, random walks, and many more. According to Hadamard's rigorous definition, HP comprehends the principle of action- by-proximity (cf. Faraday's field theory, etc.) and the superposition of secondary wavelets (Huygens' construction). This definition is reformulated for discrete spaces. The global aspect concerns the propagation of fields (e.g. wavefronts). Within TLM, the appropriate field propagator (Green's function) is the Johns matrix. The compatibility with HP explains the success of TLM in computing propagation, transport, and other evolution processes from a different point of view. A possible practical application of these results for computing eigenmodes is mentioned.

Enders, P, & Vanneste, C, 2003: Huygens' principle in the transmission line matrix method (TLM). Local theory. Int'l J. Numer. Model., 16(2):175-178.

Huygens' principle (HP) is a well-known fundamental principle of wave propagation. More generally, it can be understood as representing the principle of action-by-proximity (cf. Faraday's field theory etc.) and the superposition of secondary wavelets re-irradiated at each point of the wavefront (Huygens' construction). These wavelets are isotropic in free space and in isotropic materials. We will show, that HP is realized within the transmission line matrix method (TLM) for scalar fields in free space of any dimension, if one considers only the scattered fields to represent the secondary wavelets. This corrects and generalizes the previous result for the total field in 2D. This property of TLM provides another explanation for its wide range of applicability.

Ferrari, R L, 2001: An extended Huygens' principle for modelling scattering from general discontinuities within hollow waveguides. Int'l J. Numer. Model., 14(5):411-422.

The modal fields, generalized scattering matrix (GSM) theory and dyadic Green's functions relating to a general uniform hollow waveguide are briefly reviewed in a mutually consistent normalization. By means of an analysis linking these three concepts, an extended version of the mathematical expression of Huygens' principle is derived, applying to scattering from an arbitrary object within a hollow waveguide. The integral-equation result expresses the total field in terms of the incident waveguide modal fields, the dyadic Green's functions and the tangential electromagnetic field on the surface of the object. It is shown how the extended principle may be applied in turn to perfect conductor, uniform material and inhomogeneous material objects using a quasi method of moments (MM) approach, coupled in the last case with the finite element method. The work reported, which indicates how the GSM of the object may be recovered, is entirely theoretical but displays a close similarity with established MM procedures.

O'Connor, W J, & Clune, F J, 2001: TLM-based solutions of the Klein-Gordon equation, Part I. Int'l J. Numer. Model., 14(5):439-449.

The transmission line matrix (TLM) method has become well established as a numerical solution scheme for wave problems in electromagnetics and, to a lesser extent, in acoustics and mechanics. It has also been applied to diffusion/heat-conduction problems. Here the technique is extended to solving the Klein-Gordon equation that arises in Quantum Mechanics and in the dynamics of an elastically anchored vibrating string. In Part I, two novel, TLM-based algorithms are presented and verified. By considering them as solving a special case of the more general forced wave equation, they illustrate how, with care, the TLM algorithm can be adapted to model a wide range of effects.

O'Connor, W J, & Clune, F J, 2001: TLM-based solutions of the Klein-Gordon equation, Part II. Int'l J. Numer. Model., 15(2):215-220.

In Part I, two TLM-based solutions were presented for the Klein-Gordon Equation in its basic form, with the TLM pulses representing the primary variable. In Part II, two further approaches are presented in which the TLM pulses now represent derivatives of the primary variable, with respect to either space or time. As in Part I, the two solution schemes were verified symbolically and numerically. They illustrate further ways to extend the power of TLM beyond its traditional application areas. Some of these areas are discussed briefly.

Ahmadian, M, 2001: Transmission Line Matrix (TLM) modelling of medical ultrasound, Ph.D. Thesis, University of Edinburgh.

This thesis introduces TLM as a new method for modelling medical ultrasound wave propagation. Basic TLM theory is presented and how TLM is related to Huygens principle is discussed. Two dimensional TLM modelling is explained in detail and one dimensional and three dimensional TLM modelling are explained. Implementing TLM in single CPU computers and parallel computers is discussed and several algorithms are presented together with their advantages and disadvantages. Inverse TLM and modelling non linear wave propagation and different types of mesh are discussed. A new idea for modelling TLM as a digital filter is presented and removing the boundary effect based on digital filter modelling of TLM is discussed. Some modelling experiments such as focusing mirror, circular mirror, array transducers and doppler effect are presented and how to use TLM to model these experiments is explained. A new low sampling rate theory for TLM modelling is proposed and verified. This new theory makes the modelling of a much larger spaces practical on a given hardware platform.

Tai, C, 1960: A concise formulation of Huygens' principle for the electromagnetic field. IRE Trans. Antennas & Propagation, 8(6):634.

This communication is to direct attention to the fact that Huygens' principle for the electromagnetic field can be stated in a concise mathematical form by using the unified electromagnetic field vector, together with the dyadic Green's function.

Digital Waveguide Methods

Abel, J, Smyth, T, & Smith, J O, 2003: A simple, accurate wall loss filter for acoustic tubes. Proc. 6th Int'l Conf. Digital Audio Effects (DAFx-03), 1-5.

This research presents a uniform approximation to the formulas of Benade and Keefe for the propagation constant of a cylindrical tube, valid for all tube radii and frequencies in the audio range. Based on this approximation, a simple expression is presented for a filter which closely matches the thermoviscous loss filter of a tube of specified length and radius at a given sampling rate. The form of this filter and the simplicity of coefficient calculation make it particularly suitable for real-time music applications where it may be desirable to have tube parameters such as length and radius vary during performance.

Smith, J O, 2006: A basic introduction to digital waveguide synthesis for the technically inclined. <ccrma.stanford.edu/~jos/swgt/swgt.pdf>.

Digital waveguide synthesis models are computational physical models for certain classes of musical instruments (string, winds, brasses, etc.) which are made up of delay lines, digital filters, and often nonlinear elements. This paper gives a quick overview of the basics.

Välimäki, V, Hernoux, B, Huopaniemi, J, & Karjalainen, M, 1995: Measurement and Analysis of Acoustic Tubes Using Signal Processing Techniques. Proc. Finnish Sig. Proc. Symp. (FINSIG'95), 1-5.

Physical modeling is a modern approach to musical acoustics and high-quality sound synthesis of musical instruments. In this paper, a time-domain method is used to measure acoustical properties of wind instruments. The instrument bore is modeled by an acoustical tube and a DSP-based measurement system is used to characterize transmission and reflection functions in the duct. Inverse filtering techniques are applied to generate a pulse-like signal in the tube, and a sensor switching method is used for automatic calibration during the impulse response recordings. The results can be used to estimate the parameters of a sound synthesis model.

Smith, J O, & Cook, P R, 1992: The second-order digital waveguide oscillator. Proc. Int'l Computer Music Conf. (ICMC'92), 150-153.

A digital sinusoidal oscillator derived from digital waveguide theory is described which has good properties for VLSI implementation. Its main features are no wavetable and a computational complexity of only one multiply per sample when amplitude and frequency are constant. Three additions are required per sample. A piecewise exponential amplitude envelope is available for the cost of a second multiplication per sample, which need not be as expensive as the tuning multiply. In the presence of frequency modulation (FM), the amplitude coefficient can be varied to exactly cancel amplitude modulation (AM) caused by changing the frequency of oscillation.

Smith, J O, & Rocchesso, D, 1997: Aspects of digital waveguide networks for acoustic modeling applications. <ccrma.stanford.edu/~jos/wgj/wgj.pdf>.

This paper collects together various facts about digital waveguide networks (DWN) used in acoustic modeling, particularly results pertaining to lossless scattering at the junction of intersecting digital waveguides. Applications discussed include music synthesis based on physical models and delay effects such as artificial reverberation. Connections with Wave Digital Filters (WDF), ladder/lattice digital filters, and other related topics are outlined. General conditions for losslessness and passivity are specified. Computational complexity and dynamic range requirements are addressed. Both physical and algebraic analyses are utilized. The physical interpretation leads to many of the desirable properties of DWNs. Using both physical and algebraic approaches, three new normalized ladder filter structures are derived which have only three multiplications per two-port scattering junction instead of the four required in the well known version. A vector scattering formulation is derived which maximizes generality subject to maintaining desirable properties. Scattering junctions are generalized to allow any waveguide to have a complex wave impedance which is equivalent at the junction to a lumped load impedance, thus providing a convenient bridge between lumped and distributed modeling methods. Junctions involving complex wave impedances yield generalized scattering coefficients which are frequency dependent and therefore implemented in practice using digital filters. Scattering filters are typically isolable to one per junction in a manner analogous to the multiply in a one-multiply lattice-filter section.

Rocchesso, D, & Smith, J O, 2003: Generalized Digital Waveguide Networks. IEEE Trans. Speech & Audio Processing, 11(3):242:254.

Digital waveguides are generalized to the multivariable case with the goal of maximizing generality while retaining robust numerical properties and simplicity of realization. Multivariable complex power is defined, and conditions for "medium passivity" are presented. Multivariable complex wave impedances, such as those deriving from multivariable lossy waveguides, are used to construct scattering junctions which yield frequency dependent scattering coefficients which can be implemented in practice using digital filters. The general form for the scattering matrix at a junction of multivariable waveguides is derived. An efficient class of loss-modeling filters is derived, including a rule for checking validity of the small-loss assumption. An example application in musical acoustics is given.

Välimäki, V, Pakarinen, J, Erkut, C, & Karjalainen, M, 2006: Discrete-time modelling of musical instruments. Rep. Prog. Phys., 69(1):1-78.

This article describes physical modelling techniques that can be used for simulating musical instruments. The methods are closely related to digital signal processing. They discretize the system with respect to time, because the aim is to run the simulation using a computer. The physics-based modelling methods can be classified as mass-spring, modal, wave digital, finite difference, digital waveguide and source-filter models. We present the basic theory and a discussion on possible extensions for each modelling technique. For some methods, a simple model example is chosen from the existing literature demonstrating a typical use of the method. For instance, in the case of the digital waveguide modelling technique a vibrating string model is discussed, and in the case of the wave digital filter technique we present a classical piano hammer model. We tackle some nonlinear and time-varying models and include new results on the digital waveguide modelling of a nonlinear string. Current trends and future directions in physical modelling of musical instruments are discussed.

Helie, T, Mignot, R, & Matignon, D, 2007: Waveguide Modeling of Lossy Flared Acoustic Pipes: Derivation of a Kelly-Lochbaum Structure for Real-Time Simulations. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, 267-270.

This paper deals with the theory and application of waveguide modeling of lossy flared acoustic pipes. The novelty lies in a refined 1D-acoustic model: the Webster-Lokshin equation. This model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross section, visco -thermal losses at the walls, and without assuming plane waves or spherical waves. Solving this model for a section of pipe leads to a quadripole made of four transfer functions which imitate the global acoustic effects. Moreover, defining progressive waves and introducing some "relevant" physical interpretations enable the isolation of elementary transfer functions associated with elementary acoustic effects. From this decomposition, a standard Kelly-Lochbaum structure is recovered and efficient low -cost digital simulations are obtained. Thus, this work improves the realism of the sound synthesis of wind instruments, while it preserves waveguide techniques which only involve delay lines and digital filters.

Karjalainen, M, & Erkut, C, 2004: Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling. EURASIP J. Appl. Sig. Proc., 7:978-989.

Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one -dimensional (1D) band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D) and three-dimensional (3D) mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP )-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented.

Bilbao, S, & Smith, J O, 2003: Finite difference schemes and digital waveguide networks for the wave equation: stability, passivity, and numerical dispersion. IEEE Trans. Speech & Audio Processing, 11(3):255-266.

In this paper, some simple families of explicit two-step finite difference methods for solving the wave equation in two and three spatial dimensions are examined. These schemes depend on several free parameters, and can be associated with so-called interpolated digital waveguide meshes. Special attention is paid to the stability properties of these schemes (in particular the bounds on the space- step/time-step ratio) and their relationship with the passivity condition on the related digital waveguide networks. Boundary conditions are also discussed. An analysis of the directional numerical dispersion properties of these schemes is provided, and minimally directionally-dispersive interpolated digital waveguide meshes are constructed.

Bilbao, S, 2001: Wave and scattering methods for the numerical integration of partial differential equations, Ph.D. Thesis, Stanford University.

Digital filtering structures have recently been applied toward the numerical simulation of distributed physical systems. In particular, they have been used to numerically integrate systems of partial differential equations (PDEs), which are time-dependent, and of hyperbolic type (implying wave-like solutions, with a finite propagation velocity). Two such methods, the multidimensional wave digital filtering and digital waveguide network approaches both rely heavily on the classical theory of electrical networks, and make use of wave variables, which are reflected and transmitted throughout a grid of scattering junctions as a means of simulating the behavior of a given model system. These methods possess many good numerical properties which are carried over from digital filter design; in particular, they are numerically robust in the sense that stability may be maintained even in finite arithmetic. As such, these methods are potentially useful candidates for implementation in special purpose hardware.
In this thesis, the subtext is that such scattering-based methods can and should be treated as finite difference schemes, for purposes of analysis and comparison with standard differencing forms. In many cases, these methods can be shown to be equivalent to well-known differencing approaches--we pay close attention to the relationship between digital waveguide networks and finite difference time domain (FDTD) methods. For this reason, it is probably most useful to think of scattering forms as alternative realizations of these schemes with good numerical properties, in direct analogy with ladder, lattice and orthogonal digital filter realizations of direct form filters. We make use of this correspondence in order to import (from the finite difference setting) two techniques for approaching problems with irregular boundaries, namely coordinate changes, and a means of designing interfaces between grids of different densities and/or geometries. We also make use of the finite difference formulation in order to examine initial and boundary conditions, parasitic modes, and take an extended look at the numerical properties of all the commonly encountered forms of the waveguide network in two and three spatial dimensions.
Another question is of the relationship between wave digital and waveguide network schemes. Although they are quite similar from the standpoint of the programmer, in that the main operation, scattering, is the same in either case, conceptually they are very different. A multidimensional wave digital network is derived from a compact circuit representation of model system of PDEs. The numerical routine is itself a discrete time and space image of the original network. Waveguide meshes, however, are usually formulated as collection of lumped scattering junctions which span the problem domain, connected by bidirectional delay lines. Lacking a multidimensional representation, then, it is not straightforward to design a mesh which numerically solves a given problem. A useful result is that waveguide meshes can be obtained directly from a system by almost exactly the same means as a wave digital network. This unification of the two methods opens the door to a larger class of methods which are of neither type, and yet which consist of the same numerically robust basic building blocks.
On the applied side, special attention is paid to problems in beam, plate and shell dynamics; though these systems are in general much more complex than the transmission line and parallel-plate problems which have been discussed extensively in the literature, they can be dealt with using both wave digital filters and waveguide networks, though several new techniques must be introduced. Several simulations are presented.

Van Duyne, S A, & Smith, J O, 1993: Physical modeling with the 2-d digital waveguide mesh. <ccrma.stanford.edu/~jos/pdf/mesh.pdf>.

An extremely efficient method for modeling wave propagation in a membrane is provided by the multidimensional extension of the digital waveguide. The 2-D digital waveguide mesh is constructed out of bidirectional delay units and scattering junctions. We show that it coincides with the standard finite difference approximation scheme for the 2-D wave equation, and we derive the dispersion error. Applications may be found in physical models of drums, soundboards, cymbals, gongs, small-box reverberators, and other acoustic constructs where a one-dimensional model is less desirable.

Van Duyne, S A, & Smith, J O, 1994: The tetrahedral digital waveguide mesh. <ccrma.stanford.edu/~jos/pdf/tetrmesh.pdf >.

The 2D digital waveguide mesh [7, 8] has proven to be effective and efficient in the modeling of musical membranes and plates, particularly in combination with recent simplifications in modeling stiffness [6], nonlinearities [5], and felt mallet excitations [5]. The rectilinear 3D extension to the mesh had been suggested [8], and has been applied to the case of room acoustics [2]. However, it requires the use of 6-port scattering junctions, which make a multiply-free implementation impossible in the isotropic case. The 4-port scattering junctions of the 2D mesh required only an internal divide by 2, which could be implemented as a right shift in binary arithmetic. However, the 6-port junction requires a divide by 3. The multiply-free cases occur for N-port junctions in which N is a power of two [3]. We propose here a tetrahedral distribution of multiply-free 4-port scattering junctions filling space much like the molecular structure of the diamond crystal, where the placement of the scattering junctions corresponds to the placement of the carbon nuclei, and the bidirectional delay units correspond to the four tetrahedrally spaced single bonds between each pair of nuclei. We show that the tetrahedral mesh is mathematically equivalent to a finite difference scheme (FDS) which approximates the 3D lossless wave equation. We further compute the frequency- and direction-dependent plane wave propagation speed dispersion error.

Numerical Methods for Partial Differential Equations

Geroch, R, 1996: Partial Differential Equations of Physics. arXiv:gr-qc/9602055v1.

Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of differential equations so formulated. Examples of such features include hyperbolicity of the equations, constraints and their roles (e.g., in connection with the initial-value formulation), how diffeomorphism freedom is manifest, and how interactions between systems arise and operate. Second, we give a number of examples that illustrate how the equations for physical systems are cast into this form. These examples suggest that the first-order, quasilinear form for a system is often not only the simplest mathematically, but also the most transparent physically.

Sarlet, W, & Cantrijn, F, 1981: Generalizations of Noether's Theorem in Classical Mechanics. SIAM Rev., 23(4):467-494.

In this paper, a review is presented of various approaches to the generalization of the version of Noether's theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity -dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an "interrelationship" between symmetries and first integrals there. We then compare different proposed generalizations of Noether's theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invariance of the action functional, and that of invariance of d? (? being the Cartan -form). A number of arguments are discussed in favor of this last version of a generalized Noether theorem.Throughout the analysis we pay attention to practical considerations, such as the complexity of the Killing-type partial differential equations in each approach, which must be solved in order to identify "Noether-transformations."

Kastrup, H A, 2008: On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics. Annalen der Physik, 17(9-10):631-690.

The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl's attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture. The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations.

Cannas da Silva, A, 2004: Symplectic Geometry. ed: Dillen, F J E, & Verstraelen, L C A, Handbook of Differential Geometry, vol. 2.

Eshelby, J D, 1951: The Force on an Elastic Singularity. Phil. Trans. Roy. Soc. Lond. A, 244(877):87-112.

The parallel between the classical theory of elasticity and the modern physical theory of the solid state is incomplete; the former has nothing analogous to the concept of the force acting on an imperfection (dislocation, foreign atom, etc.) in a stressed crystal lattice. To remedy this a general theory of the forces on singularities in a Hookean elastic continuum is developed. The singularity is taken to be any state of internal stress satisfying the equilibrium equations but not the compatibility conditions. The force on a singularity can be given as an integral over a surface enclosing it. The integral contains the elastic field quantities which would surround the singularity in an infinite medium, multiplied by the difference between these quantities and those actually present. The expression for the force is thus of essentially the same form whether the force is due to applied surface tractions, other singularities or the presence of the free surface of the body ('image force'). A region of inhomogeneity in the elastic constants modifies the stress field; if it is mobile one can define and calculate the force on it. The total force on the singularities and inhomogeneities inside a surface can be expressed in terms of the integral of a 'Maxwell tensor of elasticity' taken over the surface. Possible extensions to the dynamical case are discussed.

Chien, N, Honein, T, & Herrmann, G, 1996: Dissipative systems, conservation laws and symmetries. Int'l J. Solids & Structures, 33(20-22):2959-2968.

In a recent note "On Conservation Laws for Dissipative Systems", a new method of constructing conservation laws applicable to dissipative systems was proposed. It is the purpose of this present paper to explore how this new method, called the "Neutral Action Method", is related to the concept of symmetry, and how it embodies the classical methods for obtaining conservation laws of Noether and Bessel-Hagen which are applicable only to Lagrangian systems.

Nordbrock, U, & Kienzler, R, 2003: A Betti-Maxwell reciprocal theorem for a rotordynamic system with gyroscopic terms. J. Sound Vib., 259(4):981-985.

Conservation laws have been under consideration for a long time. The classical method in constructing conservation laws, based on Noether's theorem, can only be applied, if a Lagrangian function is available for the system of interest. By using the recently developed neutral action (NA) method, this requirement can be dropped, since a given set of governing partial differential equations is suf?cient to construct conservation laws. But even if a Lagrangian function is available, the NA method delivers the same results as Noether's method, if, in addition, the Bessel-Hagen extension would be applied.

Nordbrock, U, & Kienzler, R, 2007: Conservation laws derived by the Neutral-Action method - A simple application to the Schrödinger equation. Eur. Phys. J. D, 44(3):407-410.

Conservation laws are a recognized tool in physical and engineering sciences. The classical procedure to construct conservation laws is to apply Noether's Theorem. It requires the existence of a Lagrange -function for the system under consideration. Two unknown sets of functions have to be found. A broader class of such laws is obtainable, if Noether's Theorem is used together with the Bessel-Hagen extension, raising the number of sets of unknown functions to three. By using the recently developed Neutral-Action Method, the same conservation laws can be obtained by calculating only one unknown set of functions. Moreover the Neutral Action Method can also be applied in the absence of a Lagrangian, since only the governing differential equations are required for this procedure. In the paper, an application of this method to the Schrödinger equation is presented.

Herrmann, G, & Kienzler, R, 2005: On new relations in dispersive wave motion. Wave Motion, 42(3):274-284.

Based on Whitham's variational approach and employing the 4 × 4 formalism for dispersive wave motion, new balance and conservation laws were established. The general relations are illustrated with a specific example.

R. Müller, R, & Maugin, G A, 2002: On material forces and finite element discretizations. Comp. Mech., 29(1):52-60.

The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations.

Honein, T, & Herrmann, G, 1992: Conservation Laws for Elastic Systems with Dissipation, Mech. Eng. Dept., Stanford University.

Research was concerned with applying the Neutral Action method in order to derive conservation laws for viscoelastic bodies. In contrast with previous attempts, emphasis was put in constructing conservation laws in the space-time domain and the viscoelastic models considered are therefore given in terms of rate equations. The final report summarizes the results of this effort.

Kienzler, R, & Herrmann, G, 2004: On conservation laws in elastodynamics. Int'l J. Solids & Structures, 41(13):3595-3606.

The objective of this investigation is the establishment of governing balance and conservation laws in elastodynamics. The feature of the approach employed here consists in placing time on the same level as the space coordinates, as is done in the theory of relativity, i.e., pursuing a 4 × 4 formalism. Both the Lagrangian and the Eulerian descriptions of the postulated Lagrangian function are formulated. The Euler-Lagrange equations in each of the two descriptions are discussed, as well as the results of the application of the gradient, divergence and curl. The latter two operations are made to act on the product of coordinates and the Lagrangian function, i.e., a four-vector. In this manner a variety of balance and conservation laws are derived, partly well known and partly seemingly novel. In each case the general results for elastodynamics are illustrated for the simple case of a linearly elastic bar.

Cockburn, B, & Shu, C-W, 1989: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comp., 52(186):411-435.

This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws u_t + S{i=1,d}( f_i(u))_xi = 0. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

Cockburn, B, Houm S, & Shu, C-W, 1990: TVB Runge-Kutta local projection discontinuous Galerkin Finite element method for conservation laws IV: the multidimensional case. Math. Comp., 54(190):545-581.

In this paper we study the two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already de- fined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate. Prelimi- nary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.

Gottlieb, S, 2005: On High Order Strong Stability Preserving Runge-Kutta and Multi Step Time Discretizations. J. Scienti?c Comp., 25(1-2):105-128.

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi -discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm-of the spatial discretization coupled with ?rst order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.

Gottlieb, S, & Shu, C-W, 1998: Total variation diminishing Runge-Kutta schemes. Math. Comput., 67(221):73-85.

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

Carpenter, M, & Kennedy, C, 1994: Fourth-order 2N-storage Runge-Kutta schemes. Technical Memorandum 109112, NASA Langley Research Center.

A family of five-stage fourth-order Runge-Kutta schemes is derived; these schemes require only two storage locations. A particular scheme is identified that has desirable efficiency characteristics for hyperbolic and parabolic initial (boundary) value problems. This scheme is competitive with the classical fourth-order method (high-storage) and is considerably more efficient and accurate than existing third -order low-storage schemes.

Hong, J, & Li, C, 2006: Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys., 211(2):448-472.

In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law.

Traas, C R, 2004: Another Approach to Runge-Kutta Methods. Memorandum 1739 (Revised 2006), Department of Applied Mathematics, University of Twente.

The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct numerical optimization, during which simplifying assumptions upon the Runge-Kutta coefficients may or may not be used. Depending on the optimization criterion, different types of optimal Runge-Kutta methods can be pursued. In the present article the emphasis is on rounding minimization.

Liska, R, & Wendroff, B, 1998: Composite schemes for conservation laws. SIAM J. Numer. Anal., 35(6):2250-2271.

Global composition of several time steps of the two-step Lax-Wendroff scheme followed by a Lax -Friedrichs step seems to enhance the best features of both, although only first order accurate. We show this by means of some examples of one-dimensional shallow water flow over an obstacle. In two dimensions we present a new version of Lax-Friedrichs and an associated second order predictor -corrector method. Composition of these schemes is shown to be effective and efficient for some two -dimensional Riemann problems and for Noh's infinite strength cylindrical shock problem. We also show comparable results for composition of the predictor-corrector scheme with a modified second order accurate WENO scheme. That composition is second order but is more efficient and has better symmetry properties than WENO alone. For scalar advection in two dimensions the optimal stability of the new predictor-corrector scheme is shown using computer algebra. We also show that the generalization of this scheme to three dimensions is unstable, but using sampling we are able to show that the composites are sub-optimally stable.

Liska, R, & Wendroff, B, 1998: Composite centered schemes for multidimensional conservation laws. Proc. 7th Int'l Conf. Hyperbolic Problems - Theory, Numerics, Applications.

The oscillations of a centered second order finite difference scheme and the excessive diffusion of a first order centered scheme can be overcome by global composition of the two, that is by performing cycles consisting of several time steps of the second order method followed by one step of the diffusive method. We show the effectiveness of this approach on some test problems in two and three dimensions.

Kurganov, A, Noelle, S, & Petrova, G, 2001: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput., 23(3):707-740.

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241-282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461-1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720- 742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide varietyof applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton-Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier-Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

Liu, X-D, Osher, S, & Chan, T, 1994: Weighted Essentially Non-oscillatory Schemes. J. Comput. Phys., 115(1):200-212.

In this paper we introduce a new version of ENO (essentially non-oscillatory) shock-capturing schemes which we call weighted ENO. The main new idea is that, instead of choosing the "smoothest" stencil to pick one interpolating polynomial for the ENO reconstruction, we use a convex combination of all candidates to achieve the essentially non-oscillatory property, while additionally obtaining one order of improvement in accuracy. The resulting weighted ENO schemes are based on cell averages and a TVD Runge-Kutta time discretization. Preliminary encouraging numerical experiments are given.

Jiang, G, & Shu, C-W, 1996: Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126(1):202-228.

In this paper, we further analyze, test, modify, and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher, and Chan. It was shown by Liuet al.that WENO schemes constructed from the rth order (in L1 norm) ENO schemes are (r+1)th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a fifth-order WENO scheme for the case r=3, instead of the fourth-order with the original smoothness measurement by Liuet al. This fifth -order WENO scheme is as fast as the fourth-order WENO scheme of Liuet al.and both schemes are about twice as fast as the fourth-order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest computing the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights and work well for problems which do not contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy wave interaction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.

Harten, A, Engquist, B, Osher, S, & Chakravarthy, S, 1987: Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys., 71(2):231-303.

We continue the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws. We present an hierarchy of uniformly high-order accurate schemes which generalizes Godunov's scheme and its second-order accurate MUSCL extension to an arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and, consequently, the resulting schemes are highly nonlinear.

Shu, C-W, 1997b: Preface to the Republication of Uniformly High Order Essentially Non-oscillatory Schemes, III, by Harten, Engquist, Osher, and Chakravarthy. J. Comput. Phys., 131(1):1-2.

Harten, A, & Osher, S, 1987: Uniformly high-order accurate non-oscillatory schemes, I. SIAM J. Numer. Anal., 24(2):279-309.

We begin the construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper we construct a uniformly second-order approximation, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise-linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem and an average of this approximate solution over each cell.

Shu, C-W, 1997a: Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, ICASE Report No. 97-65, Institute for Computer Applications in Science and Engineering.

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non -Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the reader can understand the algorithms and code them up for applications. Sample codes are also available from the author.

Levy, D, Puppo, G, & Russo, G, 2000: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput., 22(2):656-672.

We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes, our method is based on reconstructing a piecewise-polynomial interpolant from cell-averages which is then advanced exactly in time. In the reconstruction step, we introduce a new third-order, compact, central weighted essentially nonoscillatory (CWENO) reconstruction, which is written as a convex combination of interpolants based on different stencils. The heart of the matter is that one of these interpolants is taken as a suitable quadratic polynomial, and the weights of the convex combination are set as to obtain third order accuracy in smooth regions. The embedded mechanism in the WENO-like schemes guarantees that in regions with discontinuities or large gradients, there is an automatic switch to a one-sided second-order reconstruction, which prevents the creation of spurious oscillations. In the one-dimensional case, our new third-order reconstruction is based on an extremely compact three-point stencil. Analogous compactness is retained in more space dimensions. The accuracy, robustness, and high-resolution properties of our scheme are demonstrated in a variety of one- and two-dimensional problems.

Godunov, S K, 1959: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 47-89(3):271-306.

Abstract in Russian.

Lax, P D, 1997: Introduction to "High Resolution Schemes for Hyperbolic Conservation Laws". J. Comput. Phys., 135(2):259.

This paper was a landmark; it introduced a new design principle-total variation diminishing schemes -that led, in Harten's hands, and subsequently in the hands of others, to an efficient, robust, highly accurate class of schemes for shock capturing free of oscillations. The citation index lists 429 references to it, not only in journals of numerical analysis and computational fluid dynamics, but also in journals devoted to mechanical engineering, astronautics, astrophysics, geophysics, nuclear science and technology, space-craft and rockets, plasma physics, sound and vibration, aerothermodynamics, hydraulics, turbo and jet engines, and computer vision and imaging.

Harten, A, 1983: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49(3):357-393.

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so -derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

Harten, A, Lax, P D, & Van Leer, B, 1983: On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws. SIAM Rev., 25(1):35-61.

This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

Haack, J R, & Hauck, C D, 2008: Oscillatory behavior of asymptotic-preserving splitting methods for a linear model of diffusive relaxation. Kinetic and Related Models, 1(4):573-590.

The occurrence of oscillations in a well-known asymptotic preserving (AP) numerical scheme is investigated in the context of a linear model of diffusive relaxation, known as the P1 equations. The scheme is derived with operator splitting methods that separate the P1 system into slow and fast dynamics. A careful analysis of the scheme shows that binary oscillations can occur as a result of a black-red diffusion stencil and that dispersive-type oscillations may occur when there is too little numerical dissipation. The latter conclusion is based on comparison with a modified form of the P1 system. Numerical fixes are also introduced to remove the oscillatory behavior.

Autrique, J-C, & Magoules, F, 2006: Numerical Analysis Of A Coupled Finite-Infinite Element Method For Exterior Helmholtz Problems. J. Comput. Acoust., 14(1):21-43.

Coupled finite-infinite element computations are very efficient for modeling large scale acoustics problems. Parallel algorithms, like sub-structuring and domain decomposition methods, have shown to be very efficient for solving huge linear systems arising from acoustics. In this paper, a coupled finite -infinite element method is described, formulated and analyzed for parallel computations purpose. New numerical results illustrate the efficiency of this method for academic test cases and industrial problems alike.

Nordbrock, U, & Kienzler, R, 2008: Conservation laws - a simple application to the telegraph equation. J. Comput. Elect., 7(2):47-51.

Conservation laws are a recognized tool in physical and engineering sciences. The classical procedure to construct conservation laws makes use of Noether's Theorem. It requires the existence of a Lagrange -function for the system under consideration. Two unknown sets of functions have to be determined. A broader class of such laws is obtained, if Noether's Theorem is applied together with the Bessel-Hagen extension, raising the number of sets of unknown functions to three. The same conservation laws can be obtained by using the Neutral-Action method with the advantage that only one set of unknown functions is required. Moreover, the Neutral-Action method is also applicable in the absence of a Lagrangian, since for this procedure only the governing differential equations are needed. By this, the Neutral-Action method appears to be the most useful tool in constructing conservation laws for systems with dissipation.

The intention of this paper is to give a reference for further research on this topic rather than showing some detailed calculation on a special subject. Thus, the Neutral-Action method is introduced in detail and is applied to a simple example (the telegraph equation) to show the usefulness of this procedure for further applications in electronics.

Dehghan, M, & Shokri, A, 2008: A numerical method for solving the hyperbolic telegraph equation. Numer. Meth. Part. Diff. Equ., 24(4):1080-1093.

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one-dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

Šimek, V, Kraus, M, Kunovskż, J, & Petrek, J, 2008: GPU Based Acceleration of Telegraph Equation. Proc. 10th Int'l Conf. Computer Modeling and Simulation (UKSIM 2008), 629-630.

In a matter of just a few years, the programmable graphics processor unit has evolved into an absolute computing workhorse. With multiple cores driven by very high memory bandwidth, today's GPUs offer incredible resources for both graphics and non-graphics processing. An original mathematical method "Modern Taylor Series Method" (MTSM) which uses the Taylor series method for solving differential equations in a nontraditional way has been developed and implemented in TKSL software [3]. Even though this method is not much preferred in the literature, experimental calculations have shown and theoretical analyses have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. It is the aim of the paper to illustrate GPU and MTSM for numerical solutions of a telegraph line.

Hagen, T R, Henriksen, M O, & Hjelmervik, J M, & K-A Lie, 2007: How to solve systems of conservation laws numerically using the graphics processor as a high-performance computational engine. ed: Hasle, G, Lie, K-A, & Quak, E, Geometrical Modeling, Numerical Simulation, and Optimization: Industrial Mathematics at SINTEF, 211-264, Springer Verlag.

The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax-Friedrichs scheme and the Lax-Wendroff scheme. Using a simple example, we show how these two schemes fail to give accurate approximations to solutions containing discontinuities. We then introduce a general class of semi-discrete finite-volume schemes that are designed to produce accurate resolution of both smooth and nonsmooth parts of the solution. Using this special class we wish to introduce the reader to the basic principles used to design modern high-resolution schemes. As examples of systems of conservation laws, we consider the shallow-water equations for water waves and the Euler equations for the dynamics of an ideal gas.

The second theme in the paper is how programmable graphics processor units (GPUs or graphics cards) can be used to efficiently compute numerical solutions of these systems. In contrast to instruction driven micro-processors (CPUs), GPUs subscribe to the data-stream-based computing paradigm and have been optimised for high throughput of large data streams. Most modern numerical methods for hyperbolic conservation laws are explicit schemes defined over a grid, in which the unknowns at each grid point or in each grid cell can be updated independently of the others. Therefore such methods are particularly attractive for implementation using data-stream-based processing.

Dokken, T, Hagen, T R, & Hjelmervik, J M, 2007: An introduction to general-purpose computing on programmable graphics hardware. ed: Hasle, G, Lie, K-A, & Quak, E, Geometrical Modeling, Numerical Simulation, and Optimization: Industrial Mathematics at SINTEF, 123-161, Springer Verlag.

Using graphics hardware for general-purpose computations (GPGPU) has for selected applications shown a performance increase of more than one order of magnitude compared to traditional CPU implementations. The intent of this paper is to give an introduction to the use of graphics hardware as a computational resource. Understanding the architecture of graphics hardware is essential to comprehend GPGPU-programming. This paper first addresses the fixed functionality graphics pipeline, and then explains the architecture and programming model of programmable graphics hardware. As the CPU is instruction driven, while a graphics processing unit (GPU) is data stream driven, a good CPU algorithm is not necessarily well suited for GPU implementation. We will illustrate this with some commonly used GPU algorithms. The paper winds up with examples of GPGPU-research at SINTEF within simulation, visualization, image processing, and geometry processing.

Lyne, O D, & Williams, D, 2001: Weak solutions for a simple hyperbolic system. Elec. J. Probab., 6:1-21.

The model studied concerns a simple first-order hyperbolic system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as weak solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical 'exact' solution which is everywhere defined. The direct method used is guided by the theory of measure-valued diffusions. The method is more effective than the method of characteristics, and has the advantage that it leads immediately to the McKean representation without recourse to Itō's formula.
We then conduct computer studies of our model, both by integration schemes (which do use characteristics) and by 'random simulation'.

Fedotov, S, 2001: Front Propagation into an Unstable State of Reaction-Transport Systems. Phys. Rev. Lett., 86(5):926-929.

We studied the propagation of traveling fronts into an unstable state of the reaction-transport systems involving integral transport. By using a hyperbolic scaling procedure and singular perturbation techniques, we determined a Hamiltonian structure of reaction-transport equations. This structure allowed us to derive asymptotic formulas for the propagation rate of a reaction front. We showed that the macroscopic dynamics of the front are "nonuniversal" and depend on the choice of the underlying random walk model for the microscopic transport process.

Xiu, D, 2008: Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys., 5(2-4):242-272.

This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

Xiu, D, & Karniadakis, G E, 2002: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput., 24(2):619-644.

We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs.