DOCSLIB.ORG

A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 696597, 6 pages http://dx.doi.org/10.1155/2013/696597 Research Article A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order Abdon Atangana1 and Adem KJlJçman2 1 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa 2 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Malaysia Correspondence should be addressed to Adem Kılıc¸man; [email protected] Received 18 February 2013; Accepted 18 March 2013 Academic Editor: Guo-Cheng Wu Copyright © 2013 A. Atangana and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The standard version of acoustic wave equation is modified using the concept of the generalized Riemann-Liouville fractional order derivative. Some properties of the generalized Riemann-Liouville fractional derivative approximation are presented. Some theorems are generalized. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The numerical simulation of solution of the modified equation gives a better prediction than the standard one. 1. Introduction A derivation of general linearized wave equations is discussed by Pierce and Goldstein [1, 2]. However, neglecting Acoustics was in the beginning the study of small pressure the nonlinear effects in this equation, may lead to inaccurate waves in air which can be detected by the human ear: sound. prediction of the propagation of acoustic wave through The possibility of acoustics has been extended to higher the medium. Therefore in order to include explicitly the and lower frequencies: ultrasound and infrasound. Structural effect of corresponding relative density fluctuations in the vibrations are now often included in acoustics. Also the mathematicalformulation,oneneedstoinsertitinthe perception of sound is an area of acoustical research. In our partial differential equation that governs the propagation present paper we will limit ourselves to the original definition of the acoustic wave. Recently, the acoustic equation was andtothepropagationinfluidslikeairandwater.Insucha extended to the concept of fractional order derivative in [3]. caseacousticsisapartoffluiddynamics.Amajorproblemof Therefore in this paper, our concern is the modification of fluid dynamics is that the equations of motion are nonlinear. the previous equation by perturbing the order of the first This implies that an exact general solution of these equations derivative by replacing the first order of the derivative with is not available. Acoustics is a first-order approximation in 1+where is a positive small parameter. Also, when we which nonlinear effects are neglected. The corresponding consider diffusion process in porous medium, if the medium / relative density fluctuations 0 are considered very small structure or external field changes with time, in this situation, [1]. The acoustic wave equation governs the propagation of the ordinary integer order and constant-order fractional acoustic waves through a material medium. The form of the diffusion equation model cannot be used to well characterize equation is a second-order partial differential equation. The such phenomenon (see [3–9]). equation describes the evolution of acoustic pressure or particle velocity as a function of position and time .A simplified form of the equation that describes acoustic waves 2. Definitions and Approximation in only one spatial dimension is considered in this paper To describe the propagation of acoustic waves through 2 1 2 a material medium with coordinate and time-dependent − =0. (1) 2 2 2 perturbed dimension, one must use Riemann-Liouville 2 Mathematical Problems in Engineering fractional order derivative that was introduced and used in (ii) Division a number of works (see [3, 6–8, 10]). These derivatives are If and 1/() are differentiable on the open interval definedas(see[3, 6–8]) I then 1+ [− (1 + ) () + ()] = 1+ 1 +, [ ]≅ () 2 () () (6) =( ) ∫ [ ], −+1 − () () ( ) 0 Γ(− ()(−) ) = − + . 2 2 2 (2) () () () 1+ = +, (iii) Multiplication () () () If , and are differentiable in the open interval =( ) ∫ [ ]. I then −+1 0 Γ(− ()(−) ) 1+ [ () ⋅()]≅() () +() () Here, Γ is the Euler gamma function; ={}+1,where +( +) () {} is the integer part of for ≥0,thatis,−1≤< (7) and =0for <.Following(2)wehavethat =1+ + ( () ()). and =1+. The integral operator defined previously for fractional exponents and depending on coordinates (iv) Power and time can be expressed in terms of ordinary derivative If and () are differentiable in the open interval I then and integral [11]for|| ≪.Here 1 and are considered 1+ −1 as the corresponding relative density fluctuations /0 that [( ()) ]≅ vary slightly in time and space, respectively. For this matter, (8) −1 generalized Riemann-Liouville fractional derivatives satisfy + + ,≥1. the approximate relations: If and () are two times differentiable in the open 1+ ≅(1+) + , interval I then 1+ 1+ (3) [ [ ()]] ≅ (1 + ) 1+ ≅(1+ ) + . 2 2 (9) ×[(1+) +3 + ] + . Thepreviousrelationsmakeitpossibletodescribethe 2 2 dynamic system, including the effect of the corresponding relative density fluctuations, by means of partial differential 3.1. Clairaut’s Theorem for the Approximation. Assume that 2 and integral equations. (,, ) ,and are functions for which /, 2 2 2 /, /,and / exist and are contin- 2 1+ 1+ 3. Some Properties of the Approximation uous over a domain ⊂R then, [ [(, )]] and 1+ 1+ [ [(, )]] exist and are continuous over the Let us examine some properties of the previous derivative = operator. domain D. If in addition then (i) Addition 1+ 1+ [ [ (, )]] If , (),and() are differentiable in the opened I (10) interval then, 1+ 1+ = [ [ (, )]] . 1+ 1+ 1+ [ () +()]≅ [ ()]+ [ ()]. (4) Proof. If (,, ) ,and are functions for which Proof. We have 2 2 2 2 /, /, /,and / exist and 1+ 2 [ () +()] are continuous over a domain ⊂R then 1+ 1+ [() +()] [ [ (, )]] ≅(1+) 2 2 [()] ≅(1+)[ + + (,)+ ] + [ () +()](1+ ) (5) [()] +(1+ ) + (,). + [ ()]+(1+ ) + [ ()] (11) 1+ 1+ ≅ [ ()]+ [ ()]. 1+ 1+ Now interchanging by we obtain [ [(, )]]. Mathematical Problems in Engineering 3 2 2 If = then /=/ according to Cla- 4. Modification of the Equation 2 2 iraut’s theorem; thus replacing / by / in 1+ 1+ [ [(, )]] In order to include explicitly the possible effect of the corre- ,weobtainthat sponding relative density fluctuations into the mathematical 1+ 1+ 1+ 1+ [ [ (, )]] = [ [ (, )]] . (12) formulation, in this paper, we replace the classical version of the derivative of (1) by the modified Riemann-Liouville fractional derivative approximation (3)toobtain 3.2. Chain-Rule for the Approximation. We have 1+ 1+ 1 1+ 1+ [ ] = [ ] . (19) 2 1+ (∘)≅ (1+) () [ ()]+ (∘) Making use of (3)andrelation(8), the previous equation can be transformed to the following partial differential equation = () [ ()]+ () [ ()] for ≪1: + ( ∘ ) . 2 2 (1 + )[(1+ ) +3 + ] + 2 2 (13) 1 2 = {(1 + )[(1+) 3.3. Rolle’s Theorem for the Approximation. If a real-valued 2 2 (20) functions and are continuous on a closed interval [a, b], (, ) () = () differentiable on the open interval ,and , 2 then there exist a in the open interval (, ) and a small +3 + ] + } . 2 parameter such that 1+ 2 () =() . (14) Omitting the terms of in the previous equation, we obtain Proof. Following Rolle’s theorem, there exists a c in the open the following: (, ) () = 0 interval such that .Forthis we have that 2 2 (1 + 2 ) +3 + 1+ 2 2 () =(1+ ()) () + () () (21) 1 2 2 = () () (15) = {(1 + 2 ) +3 + } . 2 2 2 =() . Now since the small parameters representing the pertur- If (), (),and are differentiable in an open interval bation additions to unity are small, the right- and left-hand I, then there exist 2>>1and >0such that sides of (21) can be divided by 1+2.Inthiscase,weobtain 1+ 1+ () − () ≤ () − () 2 2 (,) = +(,) +(,) . (22) + () −() , ∀. 2 2 (16) Here, We have, 1 Proof. Let ∈;then (,) = (1 + 2 −2 ), 2 1+ 1+ () − () 1 (,) =3( − ), 2 (23) = (1 + ()) () + () () 2 2 1 −(1+ ()) () − () () (,) =( − ). 2 2 2 = 1+ () () − () + () () −() . It is easy to observe that (22)differsfrom(1)inthree (17) properties. First the velocity of the sound in this case depends on time But () is very small such that |1+()| < 2 and |()| > 0;itfollowsthat and coordinates due to the effect of the corresponding relative density fluctuations. Secondly the force 1+ () () − () + () () −() 1 (18) (,) =3( − ) 2 (24) ≤ () − () + () −() . It is important to observe that if =0,werecoverthe appears due to the coordinate and time dependence of properties of normal derivatives. the corresponding relative density fluctuations, and such 4 Mathematical Problems in Engineering force was considered in [12]. Third there is a derivative- The correction functional for5 ( ) can be approximately free term that depends on both
Load more

Recommended publications
  • Waves and Imaging Class Notes - 18.325
    Waves and Imaging Class notes - 18.325 Laurent Demanet Draft December 20, 2016 2 Preface In the margins of this text we use • the symbol (!) to draw attention when a physical assumption or sim- plification is made; and • the symbol ($) to draw attention when a mathematical fact is stated without proof. Thanks are extended to the following people for discussions, suggestions, and contributions to early drafts: William Symes, Thibaut Lienart, Nicholas Maxwell, Pierre-David Letourneau, Russell Hewett, and Vincent Jugnon. These notes are accompanied by computer exercises in Python, that show how to code the adjoint-state method in 1D, in a step-by-step fash- ion, from scratch. They are provided by Russell Hewett, as part of our software platform, the Python Seismic Imaging Toolbox (PySIT), available at http://pysit.org. 3 4 Contents 1 Wave equations 9 1.1 Physical models . .9 1.1.1 Acoustic waves . .9 1.1.2 Elastic waves . 13 1.1.3 Electromagnetic waves . 17 1.2 Special solutions . 19 1.2.1 Plane waves, dispersion relations . 19 1.2.2 Traveling waves, characteristic equations . 24 1.2.3 Spherical waves, Green's functions . 29 1.2.4 The Helmholtz equation . 34 1.2.5 Reflected waves . 35 1.3 Exercises . 39 2 Geometrical optics 45 2.1 Traveltimes and Green's functions . 45 2.2 Rays . 49 2.3 Amplitudes . 52 2.4 Caustics . 54 2.5 Exercises . 55 3 Scattering series 59 3.1 Perturbations and Born series . 60 3.2 Convergence of the Born series (math) . 63 3.3 Convergence of the Born series (physics) .
    [Show full text]
  • 1 Fundamental Solutions to the Wave Equation 2 the Pulsating Sphere
    1 Fundamental Solutions to the Wave Equation Physical insight in the sound generation mechanism can be gained by considering simple analytical solutions to the wave equation. One example is to consider acoustic radiation with spherical symmetry about a point ~y = fyig, which without loss of generality can be taken as the origin of coordinates. If t stands for time and ~x = fxig represent the observation point, such solutions of the wave equation, @2 ( − c2r2)φ = 0; (1) @t2 o will depend only on the r = j~x − ~yj. It is readily shown that in this case (1) can be cast in the form of a one-dimensional wave equation @2 @2 ( − c2 )(rφ) = 0: (2) @t2 o @r2 The general solution to (2) can be written as f(t − r ) g(t + r ) φ = co + co : (3) r r The functions f and g are arbitrary functions of the single variables τ = t± r , respectively. ± co They determine the pattern or the phase variation of the wave, while the factor 1=r affects only the wave magnitude and represents the spreading of the wave energy over larger surface as it propagates away from the source. The function f(t − r ) represents an outwardly co going wave propagating with the speed c . The function g(t + r ) represents an inwardly o co propagating wave propagating with the speed co. 2 The Pulsating Sphere Consider a sphere centered at the origin and having a small pulsating motion so that the equation of its surface is r = a(t) = a0 + a1(t); (4) where ja1(t)j << a0.
    [Show full text]
  • Implementation of Aeroacoustic Methods in Openfoam
    EXAMENSARBETE I TEKNISK MEKANIK 120 HP, AVANCERAD NIVÅ STOCKHOLM, SVERIGE 2016 Implementation of Aeroacoustic Methods in OpenFOAM ERIKA SJÖBERG KTH KUNGLIGA TEKNISKA HÖGSKOLAN SKOLAN FÖR TEKNIKVETENSKAP TRITA TRITA-AVE 2016:01 ISSN 1651-7660 www.kth.se Abstract A general method is established for external low Mach-number flows where aeroa- coustic analogies are used to decouple the sound generation from the sound prop- agation. The CFD solver OpenFOAM is used to compute the flow induced sound sources and Ffowcs-Williams and Hawkings acoustic analogy is implemented to calculate the propagation of sound. Incompressible and compressible source data is gathered for a test case and upon evaluation of the noise emission the assump- tion of incompressibility prove to be valid for a low Mach-number flow. Fur- thermore the advantage of non-reflecting boundary conditions in OpenFOAM is appraised and found to be effective. Lastly the method is tested on a more com- plicated test case in terms of a generic side mirror and results are found to agree well with previous studies. 3 Acknowledgments I want to extend my warmest thank Creo Dynamics for giving me the opportunity to do my master thesis at their company. I have felt like a part of Creo from day one and could not have wished for better colleges; your help and expertise have made this thesis possible. Moreover I want to extend a special thanks to Johan Hammar who has guided me through this process and always put time aside for me no matter how busy of a schedule he has had.
    [Show full text]
  • Fundamentals of Acoustics Introductory Course on Multiphysics Modelling
    Introduction Acoustic wave equation Sound levels Absorption of sound waves Fundamentals of Acoustics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELINSKI´ bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland 3 Sound levels Sound intensity and power Decibel scales Sound pressure level 2 Acoustic wave equation Equal-loudness contours Assumptions Equation of state Continuity equation Equilibrium equation 4 Absorption of sound waves Linear wave equation Mechanisms of the The speed of sound acoustic energy dissipation Inhomogeneous wave A phenomenological equation approach to absorption Acoustic impedance The classical absorption Boundary conditions coefficient Introduction Acoustic wave equation Sound levels Absorption of sound waves Outline 1 Introduction Sound waves Acoustic variables 3 Sound levels Sound intensity and power Decibel scales Sound pressure level Equal-loudness contours 4 Absorption of sound waves Mechanisms of the acoustic energy dissipation A phenomenological approach to absorption The classical absorption coefficient Introduction Acoustic wave equation Sound levels Absorption of sound waves Outline 1 Introduction Sound waves Acoustic variables 2 Acoustic wave equation Assumptions Equation of state Continuity equation Equilibrium equation Linear wave equation The speed of sound Inhomogeneous wave equation Acoustic impedance Boundary conditions 4 Absorption of sound waves Mechanisms of the acoustic energy dissipation A phenomenological approach
    [Show full text]
  • The Acoustic Wave Equation and Simple Solutions
    Chapter 5 THE ACOUSTIC WAVE EQUATION AND SIMPLE SOLUTIONS 5.1INTRODUCTION Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluido In addition to the audible pressure fields of modera te intensity, the most familiar, there are also ultrasonic and infrasonic waves whose frequencies lie beyond the limits of hearing, high-intensity waves (such as those near jet engines and missiles) that may produce a sensation of pain rather than sound, nonlinear waves of still higher intensities, and shock waves generated by explosions and supersonic aircraft. lnviscid fluids exhibit fewer constraints to deformations than do solids. The restoring forces responsible for propagating a wave are the pressure changes that oc­ cur when the fluid is compressed or expanded. Individual elements of the fluid move back and forth in the direction of the forces, producing adjacent regions of com­ pression and rarefaction similar to those produced by longitudinal waves in a bar. The following terminology and symbols will be used: r = equilibrium position of a fluid element r = xx + yy + zz (5.1.1) (x, y, and z are the unit vectors in the x, y, and z directions, respectively) g = particle displacement of a fluid element from its equilibrium position (5.1.2) ü = particle velocity of a fluid element (5.1.3) p = instantaneous density at (x, y, z) po = equilibrium density at (x, y, z) s = condensation at (x, y, z) 113 114 CHAPTER 5 THE ACOUSTIC WAVE EQUATION ANO SIMPLE SOLUTIONS s = (p - pO)/ pO (5.1.4) p - PO = POS = acoustic density at (x, y, Z) i1f = instantaneous pressure at (x, y, Z) i1fO = equilibrium pressure at (x, y, Z) P = acoustic pressure at (x, y, Z) (5.1.5) c = thermodynamic speed Of sound of the fluid <I> = velocity potential of the wave ü = V<I> .
    [Show full text]
  • Wave Propagation Problems with Aeroacoustic Applications
    Universitat Politècnica de Catalunya BarcelonaTech Escola Tècnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona Doctoral Program in Civil Engineering Doctoral Thesis Wave Propagation Problems with Aeroacoustic Applications Supervisors: Author: Prof. Ramon Codina Hector Espinoza Prof. Santiago Badia 2015 Wave Propagation Problems with Aeroacoustic Applications 2015 Universitat Politécnica de Catalunya Carrer de Jordi Girona 1-3, 08034, Barcelona, España www.upc.edu IMPRESO EN ESPAÑA-PRINTED IN SPAIN to my family Acknowledgement I would like to acknowledge the help and constant support of my advisors Ramon Codina and Santiago Badia. Their guidance, scientific rigor and valuable suggestions made this work possible. Additionally, I appreciate the opportune help from my workmates Joan, Matías and Christian in various code implementation aspects. In addition, I would like to mention Ernesto, Fermin, Ester, Arnau and Alexis with whom I shared office during my doctoral period. Likewise, I would like to mention Oscar, Alberto, Martí and Luis from CIMNE-TTS where I worked in many interesting projects. In addition, I would like to thank the support received from the EUNISON project (EU-FET grant 308874), specially from Oriol Guasch and Marc Arnela regarding the voice simulation cases. Moreover, I would like to acknowledge Johan and Cem for their support during my short stay at Kungliga Tekniska Högskolan (Royal Institute of Technology) in Stockholm, Sweden. Additionally, I would like to mention Aurélien, Jeannette, Bärbel and Rodrigo whom I met at KTH. Not to mention the support of my family that despite the distance is always with me in many ways. Finally, I would like to acknowledge the Formación de Profesorado Universitario pro- gram from the Spanish Ministry of Education for their support through the doctoral grant AP2010-0563.
    [Show full text]
  • Relativistic Acoustic Geometry in flat Space-Time [3, 7] Have Their Relativistic Counterparts
    Relativistic Acoustic Geometry Neven Bili´c Rudjer Boˇskovi´cInstitute, 10001 Zagreb, Croatia E-mail: [email protected] February 7, 2008 February 7, 2008 Abstract Sound wave propagation in a relativistic perfect fluid with a non-homogeneous isentropic flow is studied in terms of acoustic geometry. The sound wave equation turns out to be equivalent to the equation of motion for a massless scalar field propagating in a curved space-time geometry. The geometry is described by the acoustic metric tensor that depends locally on the equation of state and the four-velocity of the fluid. For a relativistic supersonic flow in curved space-time the ergosphere and acoustic horizon may be defined in a way analogous to the non-relativistic case. A general-relativistic expression for the acoustic analog of surface gravity has been found. 1 Introduction Since Unruh’s discovery [1] that a supersonic flow may cause Hawking radiation, the analogy between the theory of supersonic flow and the black hole physics has been extensively dis- arXiv:gr-qc/9908002v1 31 Jul 1999 cussed [2, 3, 4, 5, 6, 7]. All the works have considered non-relativistic flows in flat space-time. In a comprehensive review [7] Visser defined the notions of ergo-region, acoustic apparent horizon and acoustic event horizon, and investigated the properties of various acoustic ge- ometries. In this paper we study the acoustic geometry for relativistic fluids moving in curved space-time. Under extreme conditions of very high density and temperature, the velocity of the fluid may be comparable with the speed of light c.
    [Show full text]
  • 3D Finite-Difference Time-Domain Modeling of Acoustic Wave
    3D finite-difference time-domain modeling of acoustic wave propagation based on domain decompostion UMR G´eosciences Azur CNRS-IRD-UNSA-OCA Villefranche-sur-mer Supervised by: Dr. St´ephaneOperto Jade Rachele S. Garcia July 21, 2009 Abstract This report mainly focuses on three-dimensional acoustic wave propagation modeling in the time domain through finite difference method which comprises the forward modeling part of the 3D acoustic Full Waveform Inversion (FWI) method. An explicit, time-domain, finite-difference (FD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. Dependent vari- ables are stored on staggered spatial and temporal grids, and centered FD operators possess 2nd-order and 4th-order time/space accuracy. The algorithm is designed to execute on parallel computational platforms by utilizing a classical spatial domain-decomposition strategy. FDTD modeling in 3D acoustic media Contents 1 Introduction 2 1.1 Seismic Exploration . 2 1.2 Full Waveform Inversion . 2 2 Scope and Aim of the Work 3 3 The Forward Problem: Seismic Wave Propagation Modeling 4 3.1 The Acoustic Wave Equation: The Earth as a Fluid . 4 3.2 Numerical Solution of the Wave Equation . 7 3.2.1 Finite Difference Method versus Finite Element Method . 7 3.2.2 Time Domain Modeling versus Frequency Domain Modeling . 8 3.3 The Finite Difference Discretization of the 3D Acoustic Wave Equation . 8 3.3.1 Staggered Grid Stencil . 8 3.3.2 Discretized 3D Acoustic Wave Equation . 9 3.3.3 Numerical Dispersion and Stability . 11 3.3.4 Absorbing Boundary Condition .
    [Show full text]
  • 6.013 Electromagnetics and Applications, Chapter 13
    Chapter 13: Acoustics 13.1 Acoustic waves 13.1.1 Introduction Wave phenomena are ubiquitous, so the wave concepts presented in this text are widely relevant. Acoustic waves offer an excellent example because of their similarity to electromagnetic waves and because of their important applications. Beside the obvious role of acoustics in microphones and loudspeakers, surface-acoustic-wave (SAW) devices are used as radio-frequency (RF) filters, acoustic-wave modulators diffract optical beams for real-time spectral analysis of RF signals, and mechanical crystal oscillators currently control the timing of most computers and clocks. Because of the great similarity between acoustic and electromagnetic phenomena, this chapter also reviews much of electromagnetics from a different perspective. Section 13.1.2 begins with a simplified derivation of the two main differential equations that characterize linear acoustics. This pair of equations can be combined to yield the acoustic wave equation. Only longitudinal acoustic waves are considered here, not transverse or “shear” waves. These equations quickly yield the group and phase velocities of sound waves, the acoustic impedance of media, and an acoustic Poynting theorem. Section 13.2.1 then develops the acoustic boundary conditions and the behavior of acoustic waves at planar interfaces, including an acoustic Snell’s law, Brewster’s angle, the critical angle, and evanescent waves. Section 13.2.2 shows how acoustic plane waves can travel within pipes and be guided and manipulated much as plane waves can be manipulated within TEM transmission lines. Acoustic waves can be totally reflected at firm boundaries, and Section 13.2.3 explains how they can be trapped and guided in a variety of propagation modes closely resembling those in electromagnetic waveguides, where they exhibit cutoff frequencies of propagation and evanescence below cutoff.
    [Show full text]
  • Waves and Imaging Class Notes - 18.367
    Waves and Imaging Class notes - 18.367 Laurent Demanet Draft April 13, 2021 2 Preface In the margins of this text we use • the symbol (!) to draw attention when a physical assumption or sim- plification is made; and • the symbol ($) to draw attention when a mathematical fact is stated without proof. The exercises at the end of the chapters have \star rankings": ? for easy, ?? for medium, ??? for hard. Thanks are extended to the following people for discussions and sug- gestions: William Symes, Thibaut Lienart, Nicholas Maxwell, Pierre-David Letourneau, Russell Hewett, and Vincent Jugnon. These notes are accompanied by computer exercises in Python, that show how to code the adjoint-state method in 1D, in a step-by-step fash- ion, from scratch. They are provided by Russell Hewett, as part of our software platform, the Python Seismic Imaging Toolbox (PySIT), available at http://pysit.org. 3 4 Contents 1 Wave equations 9 1.1 Acoustic waves . .9 1.2 Elastic waves . 13 1.3 Electromagnetic waves . 17 1.4 Exercises . 20 2 Special waves 23 2.1 Plane waves, dispersion relations . 23 2.2 Traveling waves, characteristic equations . 27 2.3 Spherical waves, Green's functions . 33 2.4 The Helmholtz equation . 37 2.5 Reflected waves . 39 2.6 Exercises . 42 3 Geometrical optics 47 3.1 Traveltimes and Green's functions . 47 3.2 Rays . 51 3.3 Amplitudes . 54 3.4 Caustics . 56 3.5 Exercises . 57 4 Scattering series 61 4.1 Perturbations and Born series . 62 4.2 Convergence of the Born series (math) .
    [Show full text]
  • Acoustic Wave Equation
    ACOUSTIC WAVE EQUATION Contents INTRODUCTION BULK MODULUS AND LAMÉ’S PARAMETERS INTRODUCTION As we try to visualize the earth seismically we make certain physical simplifications that make it easier to make and explain our observations. Liner (2004, p.33) considers three types of simplified descriptions for the earth materials: fluid, porous and solid. Fluids, e.g. gases and liquids characteristically can deform only by compression and decompression. Only acoustic, or sound, or P-wave or compressional body waves are the natural wave vibrations that can propagate through fluid earth materials. In solids we can simplify P-waves via an acoustic model so as to ignore the effect of coupling to shear stresses (p.55, Ilke and Amundsen). Note that in this case the shear modulus still remains as one of the Lamé’s parameters . For liquids the shear modulus is 0 and for solids it is non-zero. In a fluid, seismic data can be collected by hydrophones in the form of pressure measurements. The pressure field is a scalar field, which is simpler to deal with mathematically, than a tensor field. From the following mathematical derivations we can reach several accurate concepts to help us visualize particle strain during acoustic wave transmission in a fluid. • First, within a fluid, and at a given point, particle motion increases as the pressure gradient increases. • Second, the larger the particle density the slower the particle acceleration. Note that an unweathered piece of granite is denser than a piece of slate from a blackboard and so based ON ONLY the property of its density particle acceleration will be smaller in granite than in slate.
    [Show full text]
  • Basics of Aeroacoustics)
    Lecture notes Grundlagen der Aeroakustik (Basics of Aeroacoustics) WS 2016/2017 at Technische Universitat¨ Braunschweig Prof. Dr.-Ing. Jan Delfs Institut fur¨ Aerodynamik und Stromungstechnik¨ Abteilung Technische Akustik DLR – Deutsches Zentrum fur¨ Luft- und Raumfahrt e.V. Lilienthalplatz 7 38108 Braunschweig, Germany October 2016 Dieses von mir, Prof. Dr.-Ing. Jan Delfs, verfasste Skript unterliegt dem deutschen Urheberrecht. Es darf nur zum personlichen¨ Gebrauch zu Zwecken des Studiums verwendet werden. Insbesondere ist es nicht erlaubt, das Skript oder Teile daraus zu bearbeiten, zu ubersetzen,¨ zu kopieren oder in elektronischer Form zu speichern und an andere Personen weiterzugeben, weder in Kopie, noch auf elektronischem Wege per Email, auf Speichermedien (z. B. CD, USB-Stick usw.), uber¨ Datenbanken oder andere Medien und Systeme. Lediglich die Herstellung von Kopien und Downloads fur¨ den personlichen,¨ privaten und nicht kommerziellen Gebrauch ist erlaubt. i Introduction Among the various disciplines of classical mechanics, ”Aeroacoustics” is a comparatively young subject area. This is mainly because of the fact that it took until 1952, when Lighthill formulated his famous aeroacoustic wave equation and thus ”founded” the field of modern aeroacoustics. With the help of the source terms in Lighthills equation it became possible to physically under- stand for the first time the origin of sound produced by free turbulence. The derivation of this very equation initiated many developments extremely important for the understanding of aeroacoustic sound generation in general. These notes are an attempt to provide a very brief overview about these essential aspects of the theoretical foundations of aeroacoustics, at least from the author’s point of view.
    [Show full text]