Acoustic Waves (acoustic + wave)

Distribution by Scientific Domains

Kinds of Acoustic Waves

  • surface acoustic wave


  • Selected Abstracts


    Dispersion of Dust Acoustic Modes and Perturbations of Plasma Flux Balance

    CONTRIBUTIONS TO PLASMA PHYSICS, Issue 3 2007
    V. Tsytovich
    Abstract Previous considerations of dust acoustic waves is demonstrated to be inconsistent - the required equilibrium state for perturbations was not defined since balance of plasma fluxes was neglecting. The self-consistent treatment shows that plasma flux perturbations are accompanying any collective waves propagating in dusty plasmas and can play an important role in wave dispersion, wave damping and can create instabilities. This is illustrated by the derivation of dispersion relation for dust acoustic modes taking into account the plasma flux balances and plasma flux perturbations by waves. The result of this approach shows that the dust acoustic waves with linear dependence of wave frequency on the wave number exist only in restricted range of the wave numbers. Only for wave numbers larger than some critical wave number for low frequency modes the frequency can be have approximately a linear dependence on wave number and can be called as dust acoustic wave but the phase velocity of these waves is different from that which can be obtained neglecting the flux balance and depends on grain charge variations which are determined by the balance of fluxes. The presence of plasma fluxes previously neglected is the main typical feature of dusty plasmas. The dispersion relation in the range of small wave numbers is found to be mainly determined by the change of the plasma fluxes and is quite different from that of dust acoustic type, namely it is found to have the same form as the well known dispersion relation for the gravitational instability. This result proves in general way the existence of the collective grain attractions of negatively charged grains for for large distances between them and for any source of ionization. The attraction of grains found from dispersion relation of the dust acoustic branch coincides with that found previously for pair grain interactions using some models for the ionization source. For the existing experiments the effective Jeans length for such attraction is estimated to be about 8 , 10 times larger than the ion Debye length and the effective gravitational constant for the grain attraction is estimated to be several orders of magnitude larger than the usual gravitational constant. The grain attraction at large inter-grain distances described by the gravitationlike grain instability is considered as the simplest explanation for observed dust cloud clustering, formation of dust structures including the plasma crystals. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Robust optimum design of SAW filters by the penalty function method

    ELECTRICAL ENGINEERING IN JAPAN, Issue 3 2007
    Kiyoharu Tagawa
    Abstract In order to increase the reliability of surface acoustic wave (SAW) filters, a robust optimum design technique is presented. The frequency response characteristics of SAW filters are governed primarily by their geometrical structures, that is, the configurations of the interdigital transducers (IDTs) and reflectors fabricated on piezoelectric substrates. To choose desirable structures of SAW filters through computer simulation, conventional design techniques utilize the equivalent circuit model of the IDT. However, they have rarely considered the accuracy of the underlying model, which may be degraded by the dispersion of the circuit parameters. In this paper, considering the errors of these parameters, the robust optimum design of SAW filters is formulated as a constrained optimization problem. Then, a penalty function method combined with an improved variable neighborhood search is proposed and applied to the problem. Computational experiments conducted on a practical design problem of a resonator type SAW filter demonstrate the usefulness of the proposed method. © 2006 Wiley Periodicals, Inc. Electr Eng Jpn, 158(3): 45,54, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20469 [source]


    An efficient method for solving the nonuniqueness problem in acoustic scattering

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2006
    A. Mohsen
    Abstract The problem of acoustic wave scattering by closed objects via second kind integral equations, is considered. Based on, combined Helmholtz integral equation formulation (CHIEF) method, an efficient method for choosing and utilizing interior field relations is suggested for solving the non- uniqueness problem at the characteristic frequencies. The implementation of the algorithm fully utilizes previous computation and thus significantly reduces the CPU time compared to the usual least-squares treatment. The method is tested for acoustic wave scattering by both acoustically hard and soft spheres. Accurate results compared to the known exact solutions are obtained. Copyright © 2006 John Wiley & Sons, Ltd. [source]


    Lagrangian finite element treatment of transient vibration/acoustics of biosolids immersed in fluids

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008
    P. Krysl
    Abstract Superposition principle is used to separate the incident acoustic wave from the scattered and radiated waves in a displacement-based finite element model. An absorbing boundary condition is applied to the perturbation part of the displacement. Linear constitutive equation allows for inhomogeneous, anisotropic materials, both fluids and solids. Displacement-based finite elements are used for all materials in the computational volume. Robust performance for materials with limited compressibility is achieved using assumed-strain nodally integrated simplex elements or incompatible-mode brick elements. A centered-difference time-stepping algorithm is formulated to handle general damping accurately and efficiently. Verification problems (response of empty steel cylinder immersed in water to a step plane wave, and scattering of harmonic plane waves from an elastic sphere) are discussed for assumed-strain simplex and for voxel-based brick finite element models. A voxel-based modeling scheme for complex biological geometries is described, and two illustrative results are presented from the bioacoustics application domain: reception of sound by the human ear and simulation of biosonar in beaked whales. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Actively Q-switched all-fiber lasers

    LASER PHYSICS LETTERS, Issue 2 2008
    M.V. Andrés
    Abstract Q-switching of fiber lasers using bulk elements has important drawbacks as reduced mechanical stability and high insertion losses. The development of efficient all-fiber modulation techniques is the key to obtain robust, compact and efficient Q-switched all-fiber lasers. Certainly, the development of fiber Bragg gratings (FBG) has been crucial to make progress on fiber lasers. FBGs permit a simple way to assemble all-fiber laser cavities and can be written in the active fiber itself. The Q-factor of this type of cavities is determined by the reflectivity of the FBGs and the losses of the fiber. Here, we focus on the use of magnetostrictive materials and the acousto-optic interaction to develop efficient Q-factor modulators. Most of these modulators include an FBG and take advantage of the specific interaction of the magnetostrictive materials or the acoustic wave with the FBG itself. Fiber optic technologies permit the development of a rather unique type of fiber lasers, i.e., actively Q-switched distributed feedback (DFB) fiber lasers. In this case, both the use of magnetostrictive materials and the acousto-optic interaction permit the generation of dynamic defects in an FBG that has been previously written in a highly Er-doped fiber. (© 2007 by Astro Ltd., Published exclusively by WILEY-VCH Verlag GmbH & Co. KGaA) [source]


    Ladder-type filter based on bulk acoustic wave resonators with improved out-of-band rejection

    MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 1 2008
    J. Verdú
    Abstract Filters based on bulk acoustic wave (BAW) resonators are called to replace actual surface acoustic wave filters. The transmission response of BAW-based ladder-type filters presents a steep rejection near the passband, but a poor out-of-band (OoB) rejection. This article describes how modifying the basic configuration of this type of filters, its OoB rejection can be improved. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 103,107, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23015 [source]


    Progress in SWNT based, SAW driven charge pumps

    PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 13 2006
    Viktor Siegle
    Abstract The work presented gives a short overview of existing implementations for single electron pumps. A proposal by Talyanskii et al. [Phys. Rev. Lett. 87, 276802 (2001)] for a single walled carbon nanotube (SWNT) based device is discussed. SWNT is suggested to function as one-dimensional channel for electrons, which are driven by the surface acoustic wave (SAW). The SAW is created on the surface of a piezoelectric by lithographically defined structures. This work addresses the details in manufacturing such a device. This work investigates the possibilities of growing SWNTs by chemical vapor deposition (CVD) on the surface of LiNbO3, which is known for its excellent piezoelectric properties. The results of this study are reported together with the progress in fabrication of such a device. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Coherent spin transport by acoustic fields in GaAs quantum wells

    PHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 12 2006
    O. D. D. Couto
    Abstract We review processes for long-range spin transport and manipulation in GaAs quantum wells using mobile potentials created by the piezoelectric field of a surface acoustic wave. By reducing spin dephasing mechanisms associated with the spin orbit-coupling, these potentials can coherently transport spins over distances on the order of 100 µm. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Lattice dynamics and the electromechanical properties of BaTiO3 in tetragonal phase

    PHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 9 2006
    Ali Khalal
    Abstract A lattice dynamical formalism using the shell model (R. A. Cowley, Phys. Rev. 134, A981 (1964) [1]) is applied to ferroelectric crystal BaTiO3, in the tetragonal phase. This microscopic model includes the short-range interactions of axially symmetric type and the long-range Coulomb interactions takes into account the electronic polarizability of constituent ions. The stability conditions of crystal are used to determine several first-derivatives potential for the crystals. Zone center phonons are used for fitting. In general, we obtained the best agreement between the calculated and observed frequencies. A calculation of the oblique phonons and electromechanical coupling are presented and compared with the available experimental data. It can be seen that, the electromechanical coupling is at a maximum when the acoustic wave is polarized parallel to the polar c -axis, and there is zero coupling when the polarization is perpendicular to this axis. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    A survey of the 2001 to 2005 quartz crystal microbalance biosensor literature: applications of acoustic physics to the analysis of biomolecular interactions

    JOURNAL OF MOLECULAR RECOGNITION, Issue 3 2007
    Matthew A. Cooper
    Abstract The widespread exploitation of biosensors in the analysis of molecular recognition has its origins in the mid-1990s following the release of commercial systems based on surface plasmon resonance (SPR). More recently, platforms based on piezoelectric acoustic sensors (principally ,bulk acoustic wave' (BAW), ,thickness shear mode' (TSM) sensors or ,quartz crystal microbalances' (QCM)), have been released that are driving the publication of a large number of papers analysing binding specificities, affinities, kinetics and conformational changes associated with a molecular recognition event. This article highlights salient theoretical and practical aspects of the technologies that underpin acoustic analysis, then reviews exemplary papers in key application areas involving small molecular weight ligands, carbohydrates, proteins, nucleic acids, viruses, bacteria, cells and lipidic and polymeric interfaces. Key differentiators between optical and acoustic sensing modalities are also reviewed. Copyright © 2007 John Wiley & Sons, Ltd. [source]


    Dispersion of Dust Acoustic Modes and Perturbations of Plasma Flux Balance

    CONTRIBUTIONS TO PLASMA PHYSICS, Issue 3 2007
    V. Tsytovich
    Abstract Previous considerations of dust acoustic waves is demonstrated to be inconsistent - the required equilibrium state for perturbations was not defined since balance of plasma fluxes was neglecting. The self-consistent treatment shows that plasma flux perturbations are accompanying any collective waves propagating in dusty plasmas and can play an important role in wave dispersion, wave damping and can create instabilities. This is illustrated by the derivation of dispersion relation for dust acoustic modes taking into account the plasma flux balances and plasma flux perturbations by waves. The result of this approach shows that the dust acoustic waves with linear dependence of wave frequency on the wave number exist only in restricted range of the wave numbers. Only for wave numbers larger than some critical wave number for low frequency modes the frequency can be have approximately a linear dependence on wave number and can be called as dust acoustic wave but the phase velocity of these waves is different from that which can be obtained neglecting the flux balance and depends on grain charge variations which are determined by the balance of fluxes. The presence of plasma fluxes previously neglected is the main typical feature of dusty plasmas. The dispersion relation in the range of small wave numbers is found to be mainly determined by the change of the plasma fluxes and is quite different from that of dust acoustic type, namely it is found to have the same form as the well known dispersion relation for the gravitational instability. This result proves in general way the existence of the collective grain attractions of negatively charged grains for for large distances between them and for any source of ionization. The attraction of grains found from dispersion relation of the dust acoustic branch coincides with that found previously for pair grain interactions using some models for the ionization source. For the existing experiments the effective Jeans length for such attraction is estimated to be about 8 , 10 times larger than the ion Debye length and the effective gravitational constant for the grain attraction is estimated to be several orders of magnitude larger than the usual gravitational constant. The grain attraction at large inter-grain distances described by the gravitationlike grain instability is considered as the simplest explanation for observed dust cloud clustering, formation of dust structures including the plasma crystals. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2010
    Qianlong Liu
    Abstract Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2- and 3-D compressible Navier,Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1-D characteristics-based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Toward accurate hybrid prediction techniques for cavity flow noise applications

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2009
    W. De Roeck
    Abstract A large variety of hybrid computational aeroacoustics (CAA) approaches exist differing from each other in the way the source region is modeled, in the way the equations are used to compute the propagation of acoustic waves in a non-quiescent medium, and in the way the coupling between source and acoustic propagation regions is made. This paper makes a comparison between some commonly used numerical methods for aeroacoustic applications. The aerodynamically generated tonal noise by a flow over a 2D rectangular cavity is investigated. Two different cavities are studied. In the first cavity (L/D=4, M=0.5), the sound field is dominated by the cavity wake mode and its higher harmonics, originating from a periodical vortex shedding at the cavity leading edge. In the second cavity (L/D=2, M=0.6), shear-layer modes, due to flow-acoustic interaction phenomena, generate the major components in the noise spectrum. Source domain modeling is carried out using a second-order finite-volume large eddy simulation. Propagation equations, taking into account convection and refraction effects, are solved using high-order finite-difference schemes for the linearized Euler equations and the acoustic perturbation equations. Both schemes are compared with each other for various coupling methods between source region and acoustic region. Conventional acoustic analogies and Kirchhoff methods are rewritten for the various propagation equations and used to obtain near-field acoustic results. The accuracy of the various coupling methods in identifying the noise-generating mechanisms is evaluated. In this way, this paper provides more insight into the practical use of various hybrid CAA techniques to predict the aerodynamically generated sound field by a flow over rectangular cavities. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Towards a transparent boundary condition for compressible Navier,Stokes equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001
    C. H. Bruneau
    Abstract A new artificial boundary condition for two-dimensional subsonic flows governed by the compressible Navier,Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 John Wiley & Sons, Ltd. [source]


    Invertibility of Helmholtz operators for nonhomogeneous medias

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2010
    Vladimir Rabinovich
    Abstract The paper is devoted to the investigation of the Helmholtz operators (1) describing the propagation of acoustic waves in non-homogeneous space. We consider the operator A with a wave number k such that where k0 is a positive function, k± are complex constants with ,(k)>0. The Helmholtz operator A with such wave number describes the propagation of acoustic waves in the waveguides being no homogeneous layer between two absorbing half-space. We prove that the operator A has an inverse operator A - 1 bounded in the Hilbert space L2(,n). Our proof is based on the limit operators method. We also consider the construction of the inverse operator for the Helmholtz operator A, with the density , = ,(xn) depending on xn only and wave number k0 = k0(,x,, xn) depending on a small parameter ,>0 which characterizes the slowness of variation of the wave number in the horizontal direction. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Fiber-optic acoustic transducer utilizing a dual-core collimator combined with a reflective micromirror

    MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 9 2006
    Ju-Han Song
    Abstract A photonic acoustic transducer utilizing a dual-core fiber collimator and a membrane type micromirror was proposed and demonstrated. The collimator and the mirror serve as a compact optical head and a reflective diaphragm, respectively. The micromirror diaphragm is suspended by a silicon bar connected to a frame, allowing for displacement induced by acoustic waves. The optical head incorporating dual collimators integrated in a single housing provides light to and receives it from the diaphragm. It facilitates the initial adjustment of the distance between it and the diaphragm, thanks to its slowly varying beam profiles. For the proposed acoustic transducer, the static characteristics were measured to find the operation point defined as the optimum distance between the head and the diaphragm, and a frequency response with a variation of ,±5 dB was achieved for the range of up to 3 kHz. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1833,1836, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21789 [source]


    Particle Size of Pneumatically Conveyed Powders Measured Using Impact Duration

    PARTICLE & PARTICLE SYSTEMS CHARACTERIZATION, Issue 6 2007
    Peter J. Coghill
    Abstract CSIRO Minerals has developed a technique for measuring particle size in pneumatically conveyed powders [1] by measurement of the acoustic waves produced by particle impacts upon a specially designed transducer. Previous work has focused on using the peak acoustic wave amplitude to determine particle size. This produces a spectrum that is hard to determine the particle size from, as the peak amplitude is a non-linear function of particle diameter, and is strongly affected by angle of incidence and velocity of the impacting particle. In this paper impact duration measurements are used to overcome these difficulties while retaining the advantages of being able to measure in high solids loadings of up to at least 0.5,kg/m3 of powder. In laboratory tests the impact size monitor's (ISM) results have been correlated with optical diffraction measurements of the mean (by number) powder size with a correlation coefficient of 0.985 and a relative error of 5.5,%. The ISM operated successfully in the laboratory at a loading of 0.5,kg/m3 of powder and measured particles down to 50,microns in size. [source]


    Fabrication and properties of InSb films with ion-beam sputtering for use in the amplification of magneto-surface-acoustic waves

    PHYSICA STATUS SOLIDI (A) APPLICATIONS AND MATERIALS SCIENCE, Issue 8 2004
    N. Obata
    Abstract The magnetic surface acoustic wave (MSAW) device can be manipulated by an external magnetic field. However the MSAW is attenuated largely at high frequencies above MHz. Therefore, we proposed a MSAW device having a hybrid structure consisting of FeB amorphous thin film/InSb semiconductor thin film/LiNbO3 substrate. This device is utilized for amplification of surface acoustic waves by the interaction between surface acoustic waves and the carriers in the InSb semiconductor. We prepared (111) InSb thin films by ion beam sputtering. We obtained InSb thin films having Hall mobilities of about 2000 cm2/Vs by subsequent annealing. This showed the possibility of MSAW amplification by low voltages. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Nondestructive testing of polyaramide cables by longitudinal wave propagation: Study of the dynamic modulus

    POLYMER ENGINEERING & SCIENCE, Issue 7 2000
    M. Ferreira
    It has been observed that cables at different states of fatigue had their own speed of longitudinal propagation of acoustic waves (1). This speed can be measured with piezoelectric captors and is proportional to the square root of the sonic modulus. Our experiments, which have been carried out on Technora cables of diameter 2 mm, show that the modulus obtained from the wave speed has the same behavior in fatigue as the modulus obtained from tensile tests. Furthermore, our experiments also show that the residual strength in the cable is proportional to the modulus. A nondestructive control of cables can hence be made from these sonic modulus measurements. [source]


    Radiative upper-boundary conditions for a non-hydrostatic atmosphere

    THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 582 2002
    R. James Purser
    Abstract A non-hydrostatic compressible model supports vertically propagating acoustic modes in addition to the modes of meteorological significance, such as the quasi-geostrophically balanced, and gravity modes. The acoustic modes are stimulated inadvertently, either by initial conditions incompatibly balanced for the model discretization, or by physical processes injecting abrupt impulses of heat or motion into the model during the integration. A standard method for removing unwanted acoustic energy is through the inclusion in the model of three-dimensional divergence damping. However, an alternative approach is to employ an upper-boundary condition designed to radiate acoustic waves, at least partially, as they impinge on the model top. This note explores this latter option through the use of an upper-boundary condition that incorporates a time filter to facilitate the selective absorption of both gravity and acoustic waves at the top. Copyright © 2002 Royal Meteorological Society [source]


    Acoustic matrix microseeding: improving protein crystal growth with minimal chemical bias

    ACTA CRYSTALLOGRAPHICA SECTION D, Issue 5 2010
    Armando G. Villaseñor
    A crystal seeding technique is introduced that uses acoustic waves to deliver nanolitre volumes of seed suspension into protein drops. The reduction in delivery volume enables enhanced crystal growth in matrix-seeding experiments without concern for bias from chemical components in the seed-carrying buffer suspension. Using this technique, it was found that while buffer components alone without seed can marginally promote crystal growth in some cases, crystal seeding is far more effective in boosting the number of sparse-matrix conditions that yield protein crystals. [source]


    On the observation of traveling acoustic waves in the solar atmosphere using a magneto-optical filter

    ASTRONOMISCHE NACHRICHTEN, Issue 3-4 2007
    M. Haberreiter
    Abstract In contrast to low-frequency waves that are trapped in the cavity of the Sun, high-frequency waves can travel freely in the solar atmosphere. By modelling the observed intensity signal in the red and blue wings of K I 7699 Å and Na I 5890 Å, we aim to better understand the measurements carried out with the Magneto-Optical Filter at Two Heights (MOTH) experiment. We model the observed intensity signal with radiative transfer calculations carried out with the COde for Solar Irradiance (COSI). Furthermore, we derive the formation height of the lines in order to analyze to what extent the contribution functions are modulated by the acoustic waves. We find a phase lag between the red and blue filter for acoustic waves with a frequency above ,7 mHz and conclude that a frequency dependent data analysis is required for higher frequencies. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    A representation of acoustic waves in unbounded domains,

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2005
    Bradley K. Alpert
    Compact, time-harmonic, acoustic sources produce waves that decay too slowly to be square-integrable on a line away from the sources. We introduce an inner product, arising directly from Green's second theorem, to form a Hilbert space of these waves and present examples of its computation.1 © 2005 Wiley Periodicals, Inc. [source]