Inhomogeneous Waves in Solids and Fluids (Series in Theoretical and Applied Mechanics)

The first volume of Frontiers of Computational Fluid Dynamics was published in 1994 and it was dedicated to Prof Antony Jameson. The present volume is dedicated to Prof Earl Murman in appreciation of his original contributions to this field 1. Inhomogeneous waves. 1.1. Introduction to inhomogeneous waves -- 1.2. Geometric aspects of inhomogeneous waves -- 1.3. Damping of inhomogeneous waves -- 1.4. Properties from Fourier analysis -- 2. Modelling of dissipative media. 2.1. Preliminaries on deformation and motion -- 2.2. Balance laws -- 2.3. Elementary models of dissipative bodies -- 2.4. Viscoelastic solids -- 2.5. Dissipative fluids -- 2.6. Equation of motion in prestressed dissipative bodies -- 3. Inhomogeneous waves in unbounded media. 3.1. Helmholtz representation and viscoelastic potentials -- 3.2. Inhomogeneous waves in viscoelastic solids -- 3.3. Inhomogeneous waves in dissipative fluids -- 3.4. Rate of energy and energy flux -- 3.5. Energy flux at inhomogeneous waves in solids -- 3.6. Energy flux at inhomogeneous waves in fluids -- 3.7. Waves in constrained media -- 3.8. Body force effects on waves -- 4. Reflection and refraction. 4.1. Coordinate representation for displacement and traction -- 4.2. Generalized Snell's law -- 4.3. Displacement and traction at interfaces -- 4.4. Reflection at a free surface -- 4.5. Boundary between viscoelastic half-spaces -- 4.6. Reflection and refraction coefficients -- 4.7. AppDcations and numerical results -- 5. Surface waves. 5.1. Surface waves on viscous fluids -- 5.2. Rayleigh waves on elastic solids -- 5.3. Rayleigh waves on viscoelastic half-spaces -- 5.4. Stoneley waves -- 5.5. The limit case of a rarefied medium -- 5.6. Admissible roots of Stoneley equation -- 5.7. Surface waves on prestressed half-spaces -- 6. Wave propagation in multilayered media. 6.1. Discretely layered media -- 6.2. Thick layers -- 6.3. Layers with singular transfer matrices -- 6.4. Scalar fields in continuously layered media -- 6.5. Traction-displacement vector in continuously layered media -- 6.6. Reflection and transmission matrices for a layer -- 6.7. Remarks about reflection and transmission matrices -- 7. Scattering by obstacles. 7.1. The scalar theory of scattering -- 7.2. Integral representations for displacement -- 7.3. Radiation condition -- 7.4. The scattered field -- 7.5. Uniqueness theorems -- 7.6. Scattering cross-section -- 7.7. High-frequency far field and curvature effects -- 7.8. Boundary integral equations 223 Appendix. Asymptotic behaviour via the method of stationary phase -- 8. Perturbation methods in heterogeneous media. 8.1. The Born approximation -- 8.2. Perturbation field generated by small heterogeneities -- 8.3. The WKB method -- 8.4. Turning points and extended WKB method -- 8.5. Separable variables and successive approximations -- 9. Ray method for heterogeneous dissipative media. 9.1. Ray method for the Helmholtz equation -- 9.2. Rays in viscoelastic solids -- 9.3. Amplitude evolution and energy partition -- 9.4. Rays in viscoelastic fluids -- 9.5. Reflection and refraction of rays -- 9.6. Remarks on rays in solids The book may be viewed as an introduction to time-harmonic waves in dissipative bodies, notably viscoelastic solids and fluids. The inhomogeneity of the waves, which is due to the fact that planes of constant phase are not parallel to planes of constant amplitude, is shown to be strictly related to the dissipativity of the medium. A preliminary analysis is performed on the propagation of inhomogeneous waves in unbounded media and of reflection and refraction at plane interfaces. Then emphasis is given to those features that are of significance for applications. In essence, they regard surface waves, scattering by (curved) obstacles, wave propagation in layered heterogeneous media, and ray methods. The pertinent mathematical techniques are discussed so as to make the book reasonably self-contained. The book may be viewed as an introduction to time-hadronic waves in dissipative bodies, notably viscoelastic solids and fluids. The inhomogeneity of the waves is shown to be strictly related to the dissipativity of the medium. The wave propagation problems developed in this book involve material behaviours expressed by linear, or linearized, constitutive equations.