Dr. Peter Hagedorn Dynamics and Vibrations Group, Department of Mechanical Engineering Technische Universität Darmstadt has over 200 publications which includes papers in international journals (such as ASME Journal of Applied Mechanics, International Journal of Non-linear Mechanics, Journal of Sound and Vibration, Journal of Fluids and Structures, Journal of Vibration and Control Nonlinear Dynamics, Journal of Vibrations and Acoustics, Archive for Rational Mechanics and Analysis, AIAA Journal, Journal of Optimization Theory and Applications, Wind and Structures, ZAMM, ZAMP), refereed conferences, book chapters, lecture notes and books. The books are: Nonlinear oscillations.  Since 1974, he is a Professor at TU Darmstadt (Germany). He has 7 patents to his credit. He has taught various courses such as Vibrations of continuous systems, Machine dynamics, Multi-body dynamics, Statics, Theory of elasticity, and Dynamics.  He has been Visiting Professor at COPPE, Rio de Janeiro (Brasil), Lecturer at University of Karlsruhe (Germany), Research Fellow at Stanford University (US), Visiting Professor at UC Berkeley (US), Universities in Paris (France), Irbid (Jordan), and Christchurch (New Zealand). He has also served as the Director of the Institute of Mechanics, Dean, and Vice-President, all at TU Darmstadt. Dr. Anirvan DasGupta Indian Institute of Technology Kharagpur, Kharagpur - 721302, INDIA obtained his Doctoral degree in Mechanical Engineering from IIT Kanpur (India) in 1999. He has about 35 publications which includes papers in International journals and refereed conferences. He joined the Mechanical Engineering department at IIT Kharagpur (India) in 1999 as an Assistant Professor, and is presently an Associate Professor. He has taught courses such as Mechanics, Dynamics, Kinematics of Machines, Dynamics of Machines, and Machine Vibration Analysis. He has supervised two Doctoral students. He has been at the University of Tokyo (Japan) as a research fellow, and at TU Darmstadt (Germany) as an Alexander von Humboldt research fellow.

• Preface xi1 Vibrations of strings and bars 11.1 Dynamics of strings and bars: the Newtonian formulation 11.1.1 Transverse dynamics of strings 11.1.2 Longitudinal dynamics of bars 61.1.3 Torsional dynamics of bars 71.2 Dynamics of strings and bars: the variational formulation 91.2.1 Transverse dynamics of strings 101.2.2 Longitudinal dynamics of bars 111.2.3 Torsional dynamics of bars 131.3 Free vibration problem: Bernoulli’s solution 141.4 Modal analysis 181.4.1 The eigenvalue problem 181.4.2 Orthogonality of eigenfunctions 241.4.3 The expansion theorem 251.4.4 Systems with discrete elements 271.5 The initial value problem: solution using Laplace transform 301.6 Forced vibration analysis 311.6.1 Harmonic forcing 321.6.2 General forcing 361.7 Approximate methods for continuous systems 401.7.1 Rayleigh method 411.7.2 Rayleigh–Ritz method 431.7.3 Ritz method 441.7.4 Galerkin method 471.8 Continuous systems with damping 501.8.1 Systems with distributed damping 501.8.2 Systems with discrete damping 531.9 Non-homogeneous boundary conditions 561.10 Dynamics of axially translating strings 571.10.1 Equation of motion 581.10.2 Modal analysis and discretization 581.10.3 Interaction with discrete elements 61Exercises 62References 672 One-dimensional wave equation: d’Alembert’s solution 692.1 D’Alembert’s solution of the wave equation 692.1.1 The initial value problem 722.1.2 The initial value problem: solution using Fourier transform 762.2 Harmonic waves and wave impedance 772.3 Energetics of wave motion 792.4 Scattering of waves 832.4.1 Reflection at a boundary 832.4.2 Scattering at a finite impedance 872.5 Applications of the wave solution 932.5.1 Impulsive start of a bar 932.5.2 Step-forcing of a bar with boundary damping 952.5.3 Axial collision of bars 992.5.4 String on a compliant foundation 1022.5.5 Axially translating string 104Exercises 107References 1123 Vibrations of beams 1133.1 Equation of motion 1133.1.1 The Newtonian formulation 1133.1.2 The variational formulation 1163.1.3 Various boundary conditions for a beam 1183.1.4 Taut string and tensioned beam 1203.2 Free vibration problem 1213.2.1 Modal analysis 1213.2.2 The initial value problem 1323.3 Forced vibration analysis 1333.3.1 Eigenfunction expansion method 1343.3.2 Approximate methods 1353.4 Non-homogeneous boundary conditions 1373.5 Dispersion relation and flexural waves in a uniform beam 1383.5.1 Energy transport 1403.5.2 Scattering of flexural waves 1423.6 The Timoshenko beam 1443.6.1 Equations of motion 1443.6.2 Harmonic waves and dispersion relation 1473.7 Damped vibration of beams 1493.8 Special problems in vibrations of beams 1513.8.1 Influence of axial force on dynamic stability 1513.8.2 Beam with eccentric mass distribution 1553.8.3 Problems involving the motion of material points of a vibrating beam 1593.8.4 Dynamics of rotating shafts 1633.8.5 Dynamics of axially translating beams 1653.8.6 Dynamics of fluid-conveying pipes 168Exercises 171References 1784 Vibrations of membranes 1794.1 Dynamics of a membrane 1794.1.1 Newtonian formulation 1794.1.2 Variational formulation 1824.2 Modal analysis 1854.2.1 The rectangular membrane 1854.2.2 The circular membrane 1904.3 Forced vibration analysis 1974.4 Applications: kettledrum and condenser microphone 1974.4.1 Modal analysis 1974.4.2 Forced vibration analysis 2014.5 Waves in membranes 2024.5.1 Waves in Cartesian coordinates 2024.5.2 Waves in polar coordinates 2044.5.3 Energetics of membrane waves 2074.5.4 Initial value problem for infinite membranes 2084.5.5 Reflection of plane waves 209Exercises 213References 2145 Vibrations of plates 2175.1 Dynamics of plates 2175.1.1 Newtonian formulation 2175.2 Vibrations of rectangular plates 2225.2.1 Free vibrations 2225.2.2 Orthogonality of plate eigenfunctions 2285.2.3 Forced vibrations 2295.3 Vibrations of circular plates 2315.3.1 Free vibrations 2315.3.2 Forced vibrations 2345.4 Waves in plates 2365.5 Plates with varying thickness 238Exercises 239References 2416 Boundary value and eigenvalue problems in vibrations 2436.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations 2436.1.1 General properties and expansion theorem 2436.1.2 Green’s functions and integral formulation of eigenvalue problems 2526.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods 2556.2 Forced vibrations 2596.2.1 Equations of motion 2596.2.2 Green’s function for inhomogeneous vibration problems 2606.3 Some discretization methods for free and forced vibrations 2616.3.1 Expansion in function series 2616.3.2 The collocation method 2626.3.3 The method of subdomains 2666.3.4 Galerkin’s method 2676.3.5 The Rayleigh–Ritz method 2696.3.6 The finite-element method 272References 2887 Waves in fluids 2897.1 Acoustic waves in fluids 2897.1.1 The acoustic wave equation 2897.1.2 Planar acoustic waves 2947.1.3 Energetics of planar acoustic waves 2957.1.4 Reflection and refraction of planar acoustic waves 2977.1.5 Spherical waves 3007.1.6 Cylindrical waves 3057.1.7 Acoustic radiation from membranes and plates 3077.1.8 Waves in wave guides 3147.1.9 Acoustic waves in a slightly viscous fluid 3187.2 Surface waves in incompressible liquids 3207.2.1 Dynamics of surface waves 3207.2.2 Sloshing of liquids in tanks 3237.2.3 Surface waves in a channel 330Exercises 334References 3378 Waves in elastic continua 3398.1 Equations of motion 3398.2 Plane elastic waves in unbounded continua 3448.3 Energetics of elastic waves 3468.4 Reflection of elastic waves 3488.4.1 Reflection from a free boundary 3498.5 Rayleigh surface waves 3538.6 Reflection and refraction of planar acoustic waves 357Exercises 359References 361A The variational formulation of dynamics 363References 365B Harmonic waves and dispersion relation 367B.1 Fourier representation and harmonic waves 367B.2 Phase velocity and group velocity 369References 372C Variational formulation for dynamics of plates 373References 378Index 379

Dr. Peter Hagedorn Dynamics and Vibrations Group, Department of Mechanical Engineering Technische Universität Darmstadt has over 200 publications which includes papers in international journals (such as ASME Journal of Applied Mechanics, International Journal of Non-linear Mechanics, Journal of Sound and Vibration, Journal of Fluids and Structures, Journal of Vibration and Control Nonlinear Dynamics, Journal of Vibrations and Acoustics, Archive for Rational Mechanics and Analysis, AIAA Journal, Journal of Optimization Theory and Applications, Wind and Structures, ZAMM, ZAMP), refereed conferences, book chapters, lecture notes and books. The books are: Nonlinear oscillations.  Since 1974, he is a Professor at TU Darmstadt (Germany). He has 7 patents to his credit. He has taught various courses such as Vibrations of continuous systems, Machine dynamics, Multi-body dynamics, Statics, Theory of elasticity, and Dynamics.  He has been Visiting Professor at COPPE, Rio de Janeiro (Brasil), Lecturer at University of Karlsruhe (Germany), Research Fellow at Stanford University (US), Visiting Professor at UC Berkeley (US), Universities in Paris (France), Irbid (Jordan), and Christchurch (New Zealand). He has also served as the Director of the Institute of Mechanics, Dean, and Vice-President, all at TU Darmstadt. Dr. Anirvan DasGupta Indian Institute of Technology Kharagpur, Kharagpur - 721302, INDIA obtained his Doctoral degree in Mechanical Engineering from IIT Kanpur (India) in 1999. He has about 35 publications which includes papers in International journals and refereed conferences. He joined the Mechanical Engineering department at IIT Kharagpur (India) in 1999 as an Assistant Professor, and is presently an Associate Professor. He has taught courses such as Mechanics, Dynamics, Kinematics of Machines, Dynamics of Machines, and Machine Vibration Analysis. He has supervised two Doctoral students. He has been at the University of Tokyo (Japan) as a research fellow, and at TU Darmstadt (Germany) as an Alexander von Humboldt research fellow.

• Preface xi1 Vibrations of strings and bars 11.1 Dynamics of strings and bars: the Newtonian formulation 11.1.1 Transverse dynamics of strings 11.1.2 Longitudinal dynamics of bars 61.1.3 Torsional dynamics of bars 71.2 Dynamics of strings and bars: the variational formulation 91.2.1 Transverse dynamics of strings 101.2.2 Longitudinal dynamics of bars 111.2.3 Torsional dynamics of bars 131.3 Free vibration problem: Bernoulli’s solution 141.4 Modal analysis 181.4.1 The eigenvalue problem 181.4.2 Orthogonality of eigenfunctions 241.4.3 The expansion theorem 251.4.4 Systems with discrete elements 271.5 The initial value problem: solution using Laplace transform 301.6 Forced vibration analysis 311.6.1 Harmonic forcing 321.6.2 General forcing 361.7 Approximate methods for continuous systems 401.7.1 Rayleigh method 411.7.2 Rayleigh–Ritz method 431.7.3 Ritz method 441.7.4 Galerkin method 471.8 Continuous systems with damping 501.8.1 Systems with distributed damping 501.8.2 Systems with discrete damping 531.9 Non-homogeneous boundary conditions 561.10 Dynamics of axially translating strings 571.10.1 Equation of motion 581.10.2 Modal analysis and discretization 581.10.3 Interaction with discrete elements 61Exercises 62References 672 One-dimensional wave equation: d’Alembert’s solution 692.1 D’Alembert’s solution of the wave equation 692.1.1 The initial value problem 722.1.2 The initial value problem: solution using Fourier transform 762.2 Harmonic waves and wave impedance 772.3 Energetics of wave motion 792.4 Scattering of waves 832.4.1 Reflection at a boundary 832.4.2 Scattering at a finite impedance 872.5 Applications of the wave solution 932.5.1 Impulsive start of a bar 932.5.2 Step-forcing of a bar with boundary damping 952.5.3 Axial collision of bars 992.5.4 String on a compliant foundation 1022.5.5 Axially translating string 104Exercises 107References 1123 Vibrations of beams 1133.1 Equation of motion 1133.1.1 The Newtonian formulation 1133.1.2 The variational formulation 1163.1.3 Various boundary conditions for a beam 1183.1.4 Taut string and tensioned beam 1203.2 Free vibration problem 1213.2.1 Modal analysis 1213.2.2 The initial value problem 1323.3 Forced vibration analysis 1333.3.1 Eigenfunction expansion method 1343.3.2 Approximate methods 1353.4 Non-homogeneous boundary conditions 1373.5 Dispersion relation and flexural waves in a uniform beam 1383.5.1 Energy transport 1403.5.2 Scattering of flexural waves 1423.6 The Timoshenko beam 1443.6.1 Equations of motion 1443.6.2 Harmonic waves and dispersion relation 1473.7 Damped vibration of beams 1493.8 Special problems in vibrations of beams 1513.8.1 Influence of axial force on dynamic stability 1513.8.2 Beam with eccentric mass distribution 1553.8.3 Problems involving the motion of material points of a vibrating beam 1593.8.4 Dynamics of rotating shafts 1633.8.5 Dynamics of axially translating beams 1653.8.6 Dynamics of fluid-conveying pipes 168Exercises 171References 1784 Vibrations of membranes 1794.1 Dynamics of a membrane 1794.1.1 Newtonian formulation 1794.1.2 Variational formulation 1824.2 Modal analysis 1854.2.1 The rectangular membrane 1854.2.2 The circular membrane 1904.3 Forced vibration analysis 1974.4 Applications: kettledrum and condenser microphone 1974.4.1 Modal analysis 1974.4.2 Forced vibration analysis 2014.5 Waves in membranes 2024.5.1 Waves in Cartesian coordinates 2024.5.2 Waves in polar coordinates 2044.5.3 Energetics of membrane waves 2074.5.4 Initial value problem for infinite membranes 2084.5.5 Reflection of plane waves 209Exercises 213References 2145 Vibrations of plates 2175.1 Dynamics of plates 2175.1.1 Newtonian formulation 2175.2 Vibrations of rectangular plates 2225.2.1 Free vibrations 2225.2.2 Orthogonality of plate eigenfunctions 2285.2.3 Forced vibrations 2295.3 Vibrations of circular plates 2315.3.1 Free vibrations 2315.3.2 Forced vibrations 2345.4 Waves in plates 2365.5 Plates with varying thickness 238Exercises 239References 2416 Boundary value and eigenvalue problems in vibrations 2436.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations 2436.1.1 General properties and expansion theorem 2436.1.2 Green’s functions and integral formulation of eigenvalue problems 2526.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods 2556.2 Forced vibrations 2596.2.1 Equations of motion 2596.2.2 Green’s function for inhomogeneous vibration problems 2606.3 Some discretization methods for free and forced vibrations 2616.3.1 Expansion in function series 2616.3.2 The collocation method 2626.3.3 The method of subdomains 2666.3.4 Galerkin’s method 2676.3.5 The Rayleigh–Ritz method 2696.3.6 The finite-element method 272References 2887 Waves in fluids 2897.1 Acoustic waves in fluids 2897.1.1 The acoustic wave equation 2897.1.2 Planar acoustic waves 2947.1.3 Energetics of planar acoustic waves 2957.1.4 Reflection and refraction of planar acoustic waves 2977.1.5 Spherical waves 3007.1.6 Cylindrical waves 3057.1.7 Acoustic radiation from membranes and plates 3077.1.8 Waves in wave guides 3147.1.9 Acoustic waves in a slightly viscous fluid 3187.2 Surface waves in incompressible liquids 3207.2.1 Dynamics of surface waves 3207.2.2 Sloshing of liquids in tanks 3237.2.3 Surface waves in a channel 330Exercises 334References 3378 Waves in elastic continua 3398.1 Equations of motion 3398.2 Plane elastic waves in unbounded continua 3448.3 Energetics of elastic waves 3468.4 Reflection of elastic waves 3488.4.1 Reflection from a free boundary 3498.5 Rayleigh surface waves 3538.6 Reflection and refraction of planar acoustic waves 357Exercises 359References 361A The variational formulation of dynamics 363References 365B Harmonic waves and dispersion relation 367B.1 Fourier representation and harmonic waves 367B.2 Phase velocity and group velocity 369References 372C Variational formulation for dynamics of plates 373References 378Index 379

Dr. Peter Hagedorn Dynamics and Vibrations Group, Department of Mechanical Engineering Technische Universität Darmstadt has over 200 publications which includes papers in international journals (such as ASME Journal of Applied Mechanics, International Journal of Non-linear Mechanics, Journal of Sound and Vibration, Journal of Fluids and Structures, Journal of Vibration and Control Nonlinear Dynamics, Journal of Vibrations and Acoustics, Archive for Rational Mechanics and Analysis, AIAA Journal, Journal of Optimization Theory and Applications, Wind and Structures, ZAMM, ZAMP), refereed conferences, book chapters, lecture notes and books. The books are: Nonlinear oscillations.  Since 1974, he is a Professor at TU Darmstadt (Germany). He has 7 patents to his credit. He has taught various courses such as Vibrations of continuous systems, Machine dynamics, Multi-body dynamics, Statics, Theory of elasticity, and Dynamics.  He has been Visiting Professor at COPPE, Rio de Janeiro (Brasil), Lecturer at University of Karlsruhe (Germany), Research Fellow at Stanford University (US), Visiting Professor at UC Berkeley (US), Universities in Paris (France), Irbid (Jordan), and Christchurch (New Zealand). He has also served as the Director of the Institute of Mechanics, Dean, and Vice-President, all at TU Darmstadt. Dr. Anirvan DasGupta Indian Institute of Technology Kharagpur, Kharagpur - 721302, INDIA obtained his Doctoral degree in Mechanical Engineering from IIT Kanpur (India) in 1999. He has about 35 publications which includes papers in International journals and refereed conferences. He joined the Mechanical Engineering department at IIT Kharagpur (India) in 1999 as an Assistant Professor, and is presently an Associate Professor. He has taught courses such as Mechanics, Dynamics, Kinematics of Machines, Dynamics of Machines, and Machine Vibration Analysis. He has supervised two Doctoral students. He has been at the University of Tokyo (Japan) as a research fellow, and at TU Darmstadt (Germany) as an Alexander von Humboldt research fellow.

• Preface xi1 Vibrations of strings and bars 11.1 Dynamics of strings and bars: the Newtonian formulation 11.1.1 Transverse dynamics of strings 11.1.2 Longitudinal dynamics of bars 61.1.3 Torsional dynamics of bars 71.2 Dynamics of strings and bars: the variational formulation 91.2.1 Transverse dynamics of strings 101.2.2 Longitudinal dynamics of bars 111.2.3 Torsional dynamics of bars 131.3 Free vibration problem: Bernoulli’s solution 141.4 Modal analysis 181.4.1 The eigenvalue problem 181.4.2 Orthogonality of eigenfunctions 241.4.3 The expansion theorem 251.4.4 Systems with discrete elements 271.5 The initial value problem: solution using Laplace transform 301.6 Forced vibration analysis 311.6.1 Harmonic forcing 321.6.2 General forcing 361.7 Approximate methods for continuous systems 401.7.1 Rayleigh method 411.7.2 Rayleigh–Ritz method 431.7.3 Ritz method 441.7.4 Galerkin method 471.8 Continuous systems with damping 501.8.1 Systems with distributed damping 501.8.2 Systems with discrete damping 531.9 Non-homogeneous boundary conditions 561.10 Dynamics of axially translating strings 571.10.1 Equation of motion 581.10.2 Modal analysis and discretization 581.10.3 Interaction with discrete elements 61Exercises 62References 672 One-dimensional wave equation: d’Alembert’s solution 692.1 D’Alembert’s solution of the wave equation 692.1.1 The initial value problem 722.1.2 The initial value problem: solution using Fourier transform 762.2 Harmonic waves and wave impedance 772.3 Energetics of wave motion 792.4 Scattering of waves 832.4.1 Reflection at a boundary 832.4.2 Scattering at a finite impedance 872.5 Applications of the wave solution 932.5.1 Impulsive start of a bar 932.5.2 Step-forcing of a bar with boundary damping 952.5.3 Axial collision of bars 992.5.4 String on a compliant foundation 1022.5.5 Axially translating string 104Exercises 107References 1123 Vibrations of beams 1133.1 Equation of motion 1133.1.1 The Newtonian formulation 1133.1.2 The variational formulation 1163.1.3 Various boundary conditions for a beam 1183.1.4 Taut string and tensioned beam 1203.2 Free vibration problem 1213.2.1 Modal analysis 1213.2.2 The initial value problem 1323.3 Forced vibration analysis 1333.3.1 Eigenfunction expansion method 1343.3.2 Approximate methods 1353.4 Non-homogeneous boundary conditions 1373.5 Dispersion relation and flexural waves in a uniform beam 1383.5.1 Energy transport 1403.5.2 Scattering of flexural waves 1423.6 The Timoshenko beam 1443.6.1 Equations of motion 1443.6.2 Harmonic waves and dispersion relation 1473.7 Damped vibration of beams 1493.8 Special problems in vibrations of beams 1513.8.1 Influence of axial force on dynamic stability 1513.8.2 Beam with eccentric mass distribution 1553.8.3 Problems involving the motion of material points of a vibrating beam 1593.8.4 Dynamics of rotating shafts 1633.8.5 Dynamics of axially translating beams 1653.8.6 Dynamics of fluid-conveying pipes 168Exercises 171References 1784 Vibrations of membranes 1794.1 Dynamics of a membrane 1794.1.1 Newtonian formulation 1794.1.2 Variational formulation 1824.2 Modal analysis 1854.2.1 The rectangular membrane 1854.2.2 The circular membrane 1904.3 Forced vibration analysis 1974.4 Applications: kettledrum and condenser microphone 1974.4.1 Modal analysis 1974.4.2 Forced vibration analysis 2014.5 Waves in membranes 2024.5.1 Waves in Cartesian coordinates 2024.5.2 Waves in polar coordinates 2044.5.3 Energetics of membrane waves 2074.5.4 Initial value problem for infinite membranes 2084.5.5 Reflection of plane waves 209Exercises 213References 2145 Vibrations of plates 2175.1 Dynamics of plates 2175.1.1 Newtonian formulation 2175.2 Vibrations of rectangular plates 2225.2.1 Free vibrations 2225.2.2 Orthogonality of plate eigenfunctions 2285.2.3 Forced vibrations 2295.3 Vibrations of circular plates 2315.3.1 Free vibrations 2315.3.2 Forced vibrations 2345.4 Waves in plates 2365.5 Plates with varying thickness 238Exercises 239References 2416 Boundary value and eigenvalue problems in vibrations 2436.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations 2436.1.1 General properties and expansion theorem 2436.1.2 Green’s functions and integral formulation of eigenvalue problems 2526.1.3 Bounds for eigenvalues: Rayleigh’s quotient and other methods 2556.2 Forced vibrations 2596.2.1 Equations of motion 2596.2.2 Green’s function for inhomogeneous vibration problems 2606.3 Some discretization methods for free and forced vibrations 2616.3.1 Expansion in function series 2616.3.2 The collocation method 2626.3.3 The method of subdomains 2666.3.4 Galerkin’s method 2676.3.5 The Rayleigh–Ritz method 2696.3.6 The finite-element method 272References 2887 Waves in fluids 2897.1 Acoustic waves in fluids 2897.1.1 The acoustic wave equation 2897.1.2 Planar acoustic waves 2947.1.3 Energetics of planar acoustic waves 2957.1.4 Reflection and refraction of planar acoustic waves 2977.1.5 Spherical waves 3007.1.6 Cylindrical waves 3057.1.7 Acoustic radiation from membranes and plates 3077.1.8 Waves in wave guides 3147.1.9 Acoustic waves in a slightly viscous fluid 3187.2 Surface waves in incompressible liquids 3207.2.1 Dynamics of surface waves 3207.2.2 Sloshing of liquids in tanks 3237.2.3 Surface waves in a channel 330Exercises 334References 3378 Waves in elastic continua 3398.1 Equations of motion 3398.2 Plane elastic waves in unbounded continua 3448.3 Energetics of elastic waves 3468.4 Reflection of elastic waves 3488.4.1 Reflection from a free boundary 3498.5 Rayleigh surface waves 3538.6 Reflection and refraction of planar acoustic waves 357Exercises 359References 361A The variational formulation of dynamics 363References 365B Harmonic waves and dispersion relation 367B.1 Fourier representation and harmonic waves 367B.2 Phase velocity and group velocity 369References 372C Variational formulation for dynamics of plates 373References 378Index 379