Plane and quasi-plane waves

1. Basic Equations for Wave Processes in Fluids and Solids. - 1.1 Sound in Layered Fluids. - 1.1.1 Derivation of Wave Equations. - 1.1.2 Plane Waves and Spherical Waves. - 1.1.3 Boundary Conditions. - 1.2 Harmonic Waves. - 1.2.1 Conditions at Infinity. - 1.2.2 Waves with Harmonical Dependence on Horizontal Coordinates and Time. - 1.2.3 Modified Wave Equations. - 1.3 Elastic Waves in Isotropic Solids. - 1.3.1 General Relations. - 1.3.2 Elastic Waves in Homogeneous Solids. - 1.3.3 Elastic Wave Equations in Layered Solids. - 2. Plane Waves in Discretely Layered Fluids. - 2.1 Inhomogeneous Plane Waves. Energy of Sound Waves. - 2.2 Reflection at the Interface of Two Homogeneous Media. - 2.3 Locally Reacting Surfaces. - 2.4 Reflection from a Plane Layer. - 2.4.1 The Input Impedance of a Layer. - 2.4.2 The Reflection and Transmission Coefficients. - 2.4.3 Another Approach to the Reflection and Transmission Coefficients Calculation. - 2.4.4 Two Special Cases. - 2.4. 5 Penetration of a Wave Through a Layer. - 2.5 Reflection and Transmission Coefficients for an Arbitrary Number of Layers. - 2.6 Moving Layers. Impedance of Harmonic Waves in Moving Media. - 2.6.1 Sound Wave Impedance in a Moving Medium. - 2.6.2 Plane Wave Reflection from Discretely Layered Moving Media. - 2.6.3 Reflection at a Single Interface. - 3. Monochromatic Plane-Wave Reflection from Continuously Layered Media. - 3.1 General Relations. - 3.2 Solvable Profiles k(z) from the Confluent Hypergeometric Equation. - 3.3 Solvable Profiles Obtained from the Hypergeometric Equation. - 3.4 Plane-Wave Reflection from an Epstein Layer. - 3.4.1 Expressions for Reflection and Transmission Coefficients. - 3.4.2 Transition Epstein Layer. - 3.4.3 Symmetrical Epstein Layer. - 3.5 Reflection of a Plane Wave from a Half-Space with a Linear Law for the Squared Refraction Index. - 3.5.1 The Airy Functions. - 3.5.2 The Case when dn2/dz is Positive. - 3.5.3 The Case when dn2/dz is Negative. - 3.6 Other Cases with Exact Solutions for Normal Incidence. - 3.6.1 Smooth k(z) Profiles. - 3.6.2 "Compound" Layered Media. - 3.7 Exact Solutions for Media with Continuous Stratification of Sound Velocity, Density, and Flow Velocity. - 3.7.1 Motionless Fluid with Density Stratification. - 3.7.2 Moving Layered Medium. - 4. Plane-Wave Reflection from the Boundaries of Solids. - 4.1 Plane Waves in Elastic Half-Spaces with a Free Boundary. - 4.2 Reflection from Solid-Solid and Solid-Fluid Interfaces. - 4.2.1 Two Elastic Half-Spaces in Contact. - 4.2.2 Sound Wave Reflection from Solid. - 4.2.3 Elastic Wave Reflection from Fluid Half-Space. - 4.3 Reflection from a System of Solid Layers. - 4.3.1 Matrix Propagator. - 4.3.2 Reflection Coefficient of the Sound Wave. - 4.3.3 Scattering Matrix for the Elastic Waves. - 4.3.4 Some Special Cases. - 4.4 Surface and "Leaky" Waves. - 4.4.1 A Simple Example of Surface Waves. - 4.4.2 Rayleigh Wave. - 4.4.3 Stoneley and Other Waves at Fluid-Solid and Solid-Solid Interfaces. - 5. Reflection of Sound Pulses. - 5.1 General Relations. Law of Conservation of Integrated Pulse. - 5.1.1 Integral Representation of Sound Field. - 5.1.2 Conservation of Integrated Pulse. - 5.1.3 Energy Conservation. - 5.2 Change of Pulse Shape upon Total Internal Reflection from a Boundary Between Two Homogeneous Media. - 5.3 Total Reflection of a Pulse in Continuously Layered Media. - 6. Universal Properties of the Plane-Wave Reflection and Transmission Coefficients. - 6.1 Symmetry with Respect to Reversion of the Wave Path. - 6.1.1 Sound Waves in Fluids. - 6.1.2 Elastic Waves in Solids. - 6.2 Analytic Properties. - 6.2.1 Poles of the Reflection and Transmission Coefficients. - 6.2.2 Branch Points. - 6.3 Nonreflecting Layers. - 6.3.1 For Any Fixed Frequency and Angle of Incidence a Reflectionless Layer Exists. - 6.3.2 An Example of a Reflectionless Layer for Arbitrary Angle of Incidence. - 6.3.3 An Example of a Reflectionless Layer for Arbitrary Frequency. - 7. Acoustic Waves in Absorbing Anisotropic Media. - 7.1 Absorption of Sound. - 7.1.1 Waves in Dissipative Fluids. - 7.1.2 Absorption in Solids. - 7.2 Anisotropic Elastic Media. Gulyaev-Bluestein Waves. - 7.2.1 The Christoffel Equation. - 7.2.2 Elastic Waves in Transversally Isotropic Solids. Matrix Exponent. - 7.2.3 Harmonic Waves in Piezocrystals. - 7.3 Elastic Properties of Finely Layered Media. - 7.3.1 Matrix Propagator for Inhomogeneous Solids. - 7.3.2 An Effective Medium. - 7.3.3 The Most Important Special Cases. - 8. Geometrical Acoustics. WKB Approximation. - 8.1 The WKB Approximation and Its Range of Validity. - 8.1.1 Asymptotic Solution of the Wave Equation. - 8.1.2 WKB Approximation's General Conditions of Use. - 8.1.3 Regions of Applicability in Vicinities of Turning Points and Horizons of Resonance Interaction. - 8.2 Physical Meaning of the Approximate Solutions. - 8.2.1 Medium at Rest. - 8.2.2 Moving Medium. - 8.3 Another Approach to the Ray Acoustics Approximation. - 9. The Sound Field in the Case of Turning Horizons and Resonance Interaction with a Flow. - 9.1 Reference Equation Method. - 9.1.1 High-Frequency Solution of the Wave Equation. - 9.1.2 An Estimation of the Asymptotic Solution Accuracy. - 9.1.3 The Simplest Example. - 9.2 Sound Field in the Vicinity of a Turning Point. - 9.3 Reflection from a "Potential Barrier". - 9.3.1 Uniform Asymptotics of the Sound Field. - 9.3.2 Relation to the WKB Approximation. - 9.3.3 Reflection and Transmission Coefficients. - 9.4 Amplification of Sound in an Inhomogeneous Flow. - 9.4.1 Reference Problem. - 9.4.2 General Flow. Well-Separated Horizon of Resonance Interaction and Turning Points. - 9.4.3 General Flow. Arbitrary Separated Horizon of Resonance Interaction and Turning Points. - 9.4.4 Discussion of the Results. - 10. Sound Reflection from a Medium with Arbitrarily Varying Parameters. - 10.1 Differential Equations for Reflection Coefficient and Impedance of a Sound Wave. - 10.1.1 Riccati Equation. - 10.1.2 Two Properties of the Reflection Coefficient in Inhomogeneous Media. - 10.1.3 On Separation of the Wave Field into Direct and Inverse Waves. - 10.2 Reflection from a Thin Inhomogeneous Layer. - 10.2.1 Reduction of the Problem to an Integral Equation. - 10.2.2 Iterative Solution of the Integral Equation. - 10.2.3 Physical Consequences. - 10.3 Method of Successive Approximations for Weakly Reflecting Layers. - 10.3.1 Integral Equation for the Reflection Coefficient. - 10.3.2 The Born Approximation. - 10.3.3 Numerical Example. - 10.4 Reflection at Interfaces in Continuously Layered Media. - 10.4.1 General Approach. - 10.4.2 Sound Reflection at a Weak Interface. - 10.4.3 Ray Interpretation. - References.