This book [1] was written at an important point in the development of applications of electromagnetic (radio) waves to communications, navigation, and remote sensing. Such applications require accurate propagation predictions for a variety of path conditions, and this book provides the theoretical basis for such predictions. The book is based on fundamental research in electromagnetic wave propagation that James R. Wait performed in the Central Radio Propagation Laboratory (CRPL) of NBS from 1956 to 1962. The mathematical theory in the book is very general, and the “stratified media” models are applicable to the earth crust, the troposphere, and the ionosphere. The frequencies of the communication, navigation, and remote sensing applications treated in this book range all the way from extremely low frequencies (ELF) to microwaves. The mathematical theory of electromagnetic wave propagation is based on Maxwell’s equations [2], formulated by James Clerk Maxwell in the 1860s. Experimental propagation studies in free space [3] and over the earth [4] also go back over 100 years. Research in radio science, standards, and measurements began in NBS in the early 1900s, and the long history of radio in NBS has been thoroughly covered by Snyder and Bragaw [5]. CRPL was moved to Boulder in 1954, and Wait joined the organization in 1955. The mathematics of electromagnetic wave propagation in stratified (layered) media is very complicated, and progress in propagation theory in the early 1900s was fairly slow. Wait’s book [1] included the most useful theory (much of which he developed) and practical applications that were available in 1962. A hallmark

Electromagnetic Waves in Stratified Media

Preface to the second edition. 1 Plane waves in isotropic fluids and solids. 1.1 Introduction. 1.2 Notation - vector operators. 1.3 Strain in a deformable medium. 1.4 Stress in a deformable medium. 1.5 Stress-strain relations for an isotropic elastic medium. 1.6 Equations of motion. 1.7 Wave equation in a fluid. 1.8 Wave equations in an elastic solid. References. 2 Acoustic impedance at normal incidence of fluids. Substitution of a fluid layer for a porous layer. 2.1 Introduction. 2.2 Plane waves in unbounded fluids. 2.3 Main properties of impedance at normal incidence. 2.4 Reflection coefficient and absorption coefficient at normal incidence. 2.5 Fluids equivalent to porous materials: the laws of Delany and Bazley. 2.6 Examples. 2.7 The complex exponential representation. References. 3 Acoustic impedance at oblique incidence in fluids. Substitution of a fluid layer for a porous layer. 3.1 Introduction. 3.2 Inhomogeneous plane waves in isotropic fluids. 3.3 Reflection and refraction at oblique incidence. 3.4 Impedance at oblique incidence in isotropic fluids. 3.5 Reflection coefficient and absorption coefficient at oblique incidence. 3.6 Examples. 3.7 Plane waves in fluids equivalent to transversely isotropic porous media. 3.8 Impedance at oblique incidence at the surface of a fluid equivalent to an anisotropic porous material. 3.9 Example. References. 4 Sound propagation in cylindrical tubes and porous materials having cylindrical pores. 4.1 Introduction. 4.2 Viscosity effects. 4.3 Thermal effects. 4.4 Effective density and bulk modulus for cylindrical tubes having triangular, rectangular and hexagonal cross-sections. 4.5 High- and low-frequency approximation. 4.6 Evaluation of the effective density and the bulk modulus of the air in layers of porous materials with identical pores perpendicular to the surface. 4.7 The biot model for rigid framed materials. 4.8 Impedance of a layer with identical pores perpendicular to the surface. 4.9 Tortuosity and flow resistivity in a simple anisotropic material. 4.10 Impedance at normal incidence and sound propagation in oblique pores. Appendix 4.A Important expressions. Description on the microscopic scale. Effective density and bulk modulus. References. 5 Sound propagation in porous materials having a rigid frame. 5.1 Introduction. 5.2 Viscous and thermal dynamic and static permeability. 5.3 Classical tortuosity, characteristic dimensions, quasi-static tortuosity. 5.4 Models for the effective density and the bulk modulus of the saturating fluid. 5.5 Simpler models. 5.6 Prediction of the effective density and the bulk modulus of open cell foams and fibrous materials with the different models. 5.7 Fluid layer equivalent to a porous layer. 5.8 Summary of the semi-phenomenological models. 5.9 Homogenization. 5.10 Double porosity media. Appendix 5.A: Simplified calculation of the tortuosity for a porous material having pores made up of an alternating sequence of cylinders. Appendix 5.B: Calculation of the characteristic length LAMBDA'. Appendix 5.C: Calculation of the characteristic length LAMBDA for a cylinder perpendicular to the direction of propagation. References. 6 Biot theory of sound propagation in porous materials having an elastic frame. 6.1 Introduction. 6.2 Stress and strain in porous materials. 6.3 Inertial forces in the biot theory. 6.4 Wave equations. 6.5 The two compressional waves and the shear wave. 6.6 Prediction of surface impedance at normal incidence for a layer of porous material backed by an impervious rigid wall. Appendix 6.A: Other representations of the Biot theory. References. 7 Point source above rigid framed porous layers. 7.1 Introduction. 7.2 Sommerfeld representation of the monopole field over a plane reflecting surface. 7.3 The complex sin theta plane. 7.4 The method of steepest descent (passage path method). 7.5 Poles of the reflection coefficient. 7.6 The pole subtraction method. 7.7 Pole localization. 7.8 The modified version of the Chien and Soroka model. Appendix 7.A Evaluation of N. Appendix 7.B Evaluation of p r by the pole subtraction method. Appendix 7.C From the pole subtraction to the passage path: Locally reacting surface. References. 8 Porous frame excitation by point sources in air and by stress circular and line sources - modes of air saturated porous frames. 8.1 Introduction. 8.2 Prediction of the frame displacement. 8.3 Semi-infinite layer - Rayleigh wave. 8.4 Layer of finite thickness - modified Rayleigh wave. 8.5 Layer of finite thickness - modes and resonances. Appendix 8.A Coefficients r ij and M i,j. Appendix 8.B Double Fourier transform and Hankel transform. Appendix 8.B Appendix .C Rayleigh pole contribution. References. 9 Porous materials with perforated facings. 9.1 Introduction. 9.2 Inertial effect and flow resistance. 9.3 Impedance at normal incidence of a layered porous material covered by a perforated facing - Helmoltz resonator. 9.4 Impedance at oblique incidence of a layered porous material covered by a facing having cirular perforations. References. 10 Transversally isotropic poroelastic media. 10.1 Introduction. 10.2 Frame in vacuum. 10.3 Transversally isotropic poroelastic layer. 10.4 Waves with a given slowness component in the symmetry plane. 10.5 Sound source in air above a layer of finite thickness. 10.6 Mechanical excitation at the surface of the porous layer. 10.7 Symmetry axis different from the normal to the surface. 10.8 Rayleigh poles and Rayleigh waves. 10.9 Transfer matrix representation of transversally isotropic poroelastic media. Appendix 10.A: Coefficients T i in Equation (10.46). Appendix 10.B: Coefficients A i in Equation (10.97). References. 11 Modelling multilayered systems with porous materials using the transfer matrix method. 11.1 Introduction. 11.2 Transfer matrix method. 11.3 Matrix representation of classical media. 11.4 Coupling transfer matrices. 11.5 Assembling the global transfer matrix. 11.6 Calculation of the acoustic indicators. 11.7 Applications. Appendix 11.A The elements T ij of the Transfer Matrix T ]. References. 12 Extensions to the transfer matrix method. 12.1 Introduction. 12.2 Finite size correction for the transmission problem. 12.3 Finite size correction for the absorption problem. 12.4 Point load excitation. 12.5 Point source excitation. 12.6 Other applications. Appendix 12.A: An algorithm to evaluate the geometrical radiation impedance. References. 13 Finite element modelling of poroelastic materials. 13.1 Introduction. 13.2 Displacement based formulations. 13.3 The mixed displacement-pressure formulation. 13.4 Coupling conditions. 13.5 Other formulations in terms of mixed variables. 13.6 Numerical implementation. 13.7 Dissipated power within a porous medium. 13.8 Radiation conditions. 13.9 Examples. References. Index.

Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials

Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

Seismic Wave Propagation in Stratified Media

A model for an imperfectly bonded interface between two elastic media is proposed. Displacement across this surface is not required to be continuous. The displacement discontinuity, or slip, is taken to be linearly related to the stress traction which is continuous across the interface. For isotropic interface behavior, there are two complex frequency dependent interface compliances, ηN and ηT, where the component of the slip normal to the interface is given by ηN times the normal stress and the component tangential to the interface is given by ηT times the shear stress and is in the same direction. Reflection and transmission coefficients for harmonic plane waves incident at arbitrary angles upon a plane linear slip interface are computed in terms of the interface compliances. These coefficients are frequency dependent even when the compliances are real and frequency independent. Examples of the effects of buried slip interfaces on reflection coefficient spectra and on Love‐wave dispersion relations are ...

https://www.math.purdue.edu/~santos/research/fracture_references/jasa_shoenberg_80.pdf

Elastic wave behavior across linear slip interfaces

This paper reviews analytical methods in electromagnetic scattering theory (i.e., geometrical and physical optics, perturbation, iteration, and integral-equation) which are applicable to the problems of remote sensing of the ocean. In dealing with Earth's surface (in this case, the weakly non-linear ocean), it is not possible to have a complete and exact description of its spatial and temporal statistics. Only the first few moments are generally available; and in the linear approximation the statistics are assumed homogeneous, stationary and Gaussian. For this case, the high-frequency methods (geometrical and physical optics) and perturbation (Rayleigh-Rice), or a combination of them, provide tractable analytical results (i.e., the specular-point, the slightly-rough Bragg scattering and the composite-surface models). The applicability and limitations of these models are discussed. At grazing incidence and for higher frequencies, other scattering mechanisms become significant; and shadowing, diffraction and trapping must be considered. The more exact methods (integral-equation and Green's function) have not been as successful in yielding tractable analytical solutions, although they have the potential to provide improved theoretical scattering results in the future.

F. N. Frenkiel

Papers

Waves in layered media