Contents -- Preface -- 1. Introduction -- 2. Wavelet Analysis -- 2.1 Wavelet and Wavelet Analysis. Preliminary Notion -- 2.1.1 The space L 2 (R) -- 2.1.2 The spaces L p (R) (p = 1) -- 2.1.3 The Hardy spaces H p ( R) (p = 1) -- 2.1.4 The sketch scheme of wavelet analysis -- 2.2 Rademacher, Walsh and Haar Functions -- 2.2.1 System of Rademacher functions -- 2.2.2 System of Walsh functions -- 2.2.3 System of Haar functions -- 2.3 Integral Fourier Transform. Heisenberg Uncertainty Principle -- 2.4 Window Transform. Resolution -- 2.4.1 Examples of window functions -- 2.4.2 Properties of the window Fourier transform -- 2.4.3 Discretization and discrete window Fourier transform -- 2.5 Bases. Orthogonal Bases. Biorthogonal Bases -- 2.6 Frames. Conditional and Unconditional Bases -- 2.6.1 Wojtaszczyk's definition of unconditional basis (1997) -- 2.6.2 Meyer's definition of unconditional basis (1997) -- 2.6.3 Donoho's definition of unconditional basis (1993) -- 2.6.4 Definition of conditional basis -- 2.7 Multiresolution Analysis -- 2.8 Decomposition of the Space L 2 (R) -- 2.9 Discrete Wavelet Transform. Analysis and Synthesis -- 2.9.1 Analysis: transition from the fine scale to the coarse scale -- 2.9.2 Synthesis: transition from the coarse scale to the fine scale -- 2.10 Wavelet Families -- 2.10.1 Haar wavelet -- 2.10.2 Strömberg wavelet -- 2.10.3 Gabor wavelet -- 2.10.4 Daubechies-Jaffard-Journé wavelet -- 2.10.5 Gabor-Malvar wavelet -- 2.10.6 Daubechies wavelet -- 2.10.7 Grossmann-Morlet wavelet -- 2.10.8 Mexican hat wavelet -- 2.10.9 Coifman wavelet - coiflet -- 2.10.10 Malvar-Meyer-Coifman wavelet -- 2.10.11 Shannon wavelet or sinc-wavelet -- 2.10.12 Cohen-Daubechies-Feauveau wavelet -- 2.10.13 Geronimo-Hardin-Massopust wavelet -- 2.10.14 Battle-Lemarié wavelet -- 2.11 Integral Wavelet Transform -- 2.11.1 Definition of the wavelet transform.

2.11.2 Fourier transform of the wavelet -- 2.11.3 The property of resolution -- 2.11.4 Complex-value wavelets and their properties -- 2.11.5 The main properties of wavelet transform -- 2.11.6 Discretization of the wavelet transform -- 2.11.7 Orthogonal wavelets -- 2.11.8 Dyadic wavelets and dyadic wavelet transform -- 2.11.9 Equation of the function (signal) energy balance -- 3. Materials with Micro- or Nanostructure -- 3.1 Macro-, Meso-, Micro-, and Nanomechanics -- 3.2 Main Physical Properties of Materials -- 3.3 Thermodynamical Theory of Material Continua -- 3.4 Composite Materials -- 3.5 Classical Model of Macroscopic (Effective) Moduli -- 3.6 Other Microstructural Models -- 3.6.1 Bolotin model of energy continualization -- 3.6.2 Achenbach-Hermann model of effective stiffness -- 3.6.3 Models of effective stiffness of high orders -- 3.6.4 Asymptotic models of high orders -- 3.6.5 Drumheller-Bedford lattice microstructural models -- 3.6.6 Mindlin microstructural theory -- 3.6.7 Eringen microstructural model. Eringen-Maugin model -- 3.6.8 Pobedrya microstructural theory -- 3.7 Structural Model of Elastic Mixtures -- 3.7.1 Viscoelastic mixtures -- 3.7.2 Piezoelastic mixtures -- 3.8 Computer Modelling Data on Micro- and Nanocomposites -- 4. Waves in Materials -- 4.1 Waves Around the World -- 4.2 Analysis of Waves in Linearly Deformed Elastic Materials -- 4.2.1 Volume and shear elastic waves in the classical approach -- 4.2.2 Plane elastic harmonic waves in the classical approach -- 4.2.3 Cylindrical elastic waves in the classical approach -- 4.2.4 Volume and shear elastic waves in the nonclassical approach -- 4.2.5 Plane elastic harmonic waves in the nonclassical approach -- 4.3 Analysis of Waves in Nonlinearly Deformed Elastic Materials -- 4.3.1 Basic notions of the nonlinear theory of elasticity. Strains -- 4.3.2 Forces and stresses.

4.3.3 Balance equations -- 4.3.4 Nonlinear elastic isotropic materials. Elastic Potentials -- 4.4 Nonlinear Wave Equations -- 4.4.1 Nonlinear wave equations for plane waves. Methods of solving -- 4.4.1.1 Method of successive approximations -- 4.4.1.2 Method of slowly varying amplitudes -- 4.4.2 Nonlinear wave equations for cylindrical waves -- 4.5 Comparison of Murnaghan and Signorini Approaches -- 4.5.1 Comparison of some results for plane waves -- 4.5.2 Comparison of cylindrical and plane wave in the Murnaghan model -- 5. Simple and Solitary Waves in Materials -- 5.1 Simple (Riemann) Waves -- 5.1.1 Simple waves in nonlinear acoustics -- 5.1.2 Simple waves in fluids -- 5.1.3 Simple waves in the general theory of waves -- 5.1.4 Simple waves in mechanics of electromagnetic continua -- 5.2 Solitary Elastic Waves in Composite Materials -- 5.2.1 Simple solitary waves in materials -- 5.2.2 Chebyshev-Hermite functions -- 5.2.3 Whittaker functions -- 5.2.4 Mathieu functions -- 5.2.5 Interaction of simple waves. Self-generation -- 5.2.6 The solitary wave analysis -- 5.3 New Hierarchy of Elastic Waves in Materials -- 5.3.1 Classical harmonic waves (periodic, nondispersive) -- 5.3.2 Classical arbitrary elastic waves (D'Alembert waves) -- 5.3.3 Classical harmonic elastic waves (periodic, dispersive) -- 5.3.4 Nonperiodic elastic solitary waves (with the phase velocity depending on the phase) -- 5.3.5 Simple elastic waves (with the phase velocity depending on the amplitude) -- 6. Solitary Waves and Elastic Wavelets -- 6.1 Elastic Wavelets -- 6.2 The Link between the Trough Length and the Characteristic Length -- 6.3 Initial Profiles as Chebyshev-Hermite and Whittaker Functions -- 6.4 Some Features of the Elastic Wavelets -- 6.5 Solitary Waves in Mechanical Experiments -- 6.6 Ability of Wavelets in Detecting the Profile Features -- Bibliography -- Index.