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This chapter discusses how gravitational waves emerge from general relativity, and what their properties are. The most straightforward approach is ‘linearized theory’, where the Einstein equations are expanded around the flat Minkowski metric. It is shown how a wave equation emerges and how the solutions can be put in an especially simple form by an appropriate gauge choice. Using standard tools of general relativity such as the geodesic equation and the equation of the geodesic deviation, how these waves interact with a set of test masses is detailed. The energy and momentum carried by GWs are then computed and discussed. This chapter approaches the problem from a geometric point of view, identifying the energy-momentum tensor of GWs from their effect on the background curvature. Finally, GW propagation in curved space is discussed.

This chapter completes the task of describing linear natural modes in a homogeneous, drift-free cold-plasma and concludes with a brief look at the nonlinear coupling of linear modes. The linear wave analysis here follows the formulation and developments of the previous two chapters, and highlights the analysis of linear waves at any angle. Waves that propagate at an arbitrary angle are of a very large variety and their detailed description requires approximate analytical approaches for various applications. For propagation in infinitely-extended plasmas, it has been found useful to classify their description in a space of normalized magnetic field and density variables where the cutoff and resonance of the principal waves—this is referred to here as the Clemmow, Mullaly, Allis (CMA) diagram. The chapter also analyses approximate wave propagation treatments that have been found useful in various, more specific, wave applications in magnetized plasmas.

This chapter surveys aspects of electromagnetic fields inside of regions bounded by conductors, in scenarios of relevance to accelerator technology. It explains that the discussion was based on general considerations of electromagnetic wave propagation in vacuum, dielectrics, and conductors. It then builds the discussion of resonant cavity modes upon relevant concepts in waveguide theory. It clarifies the forms of fields in couple cavity linacs, devices that are also profitably viewed as TM waveguides in their own right. It uses the view of rf cavities as generalized oscillators to introduce the Slater perturbation theorem, a powerful tool for tuning and measuring cavity characteristics.

This chapter considers mathematical techniques in the analysis of linear wave propagation and parametric interactions in plasma and beams with spatially-nonuniform (inhomogeneous) equilibria in density and confining magnetic fields. These techniques can help in the modeling and analysis of realistic plasma systems. The chapter gives the simplest description of the modifications of collective modes when the plasma equilibrium is no longer homogeneous in space and/or time, but varies sufficiently slowly so that modes maintain their basic character while changing their propagation. This is accomplished by introducing the techniques of geometric optics and ray tracing for a medium with temporal and spatial dispersion such as a plasma. It also describes techniques for analysis of situations in which the basic collective modes are modified and couple to each other due to inhomogeneity in plasma equilibrium, and concludes with a discussion of variational principles which helps in the analysis of bounded plasmas that are also inhomogeneous.

A variety of optical phenomena can be classified under the heading of nonlinear optics. Examples are the Kerr and Pockels electro-optic effects, the Zeeman effect, and the photoelastic effect in which a low-frequency electric or magnetic field or mechanical strain is mixed with the optical wave. Another large group of optical nonlinearities involves the scattering of light from atomic or electronic oscillators or other fluctuations within the optical medium. In this category fall, for instance, second- or higher-order Raman scattering from optic phonons in general (including polariton scattering from infrared active phonons), Brillouin scattering from acoustic phonons, and Rayleigh scattering from entropy or orientation fluctuations. This chapter examines the important role played by ferroelectrics in the areas of electro-optics and optical mixing, as well as phenomenological and microscopic theories which relate the acentric nature of the host crystal to the nonlinear optical susceptibilities. Wave propagation in nonlinear dielectrics is discussed, along with phase matching, electro-optics and nonlinear optic coefficients, nonlinear susceptibility, anharmonic oscillator, bond anharmonic polarisability model, polarization potential, and hyper-Rayleigh and hyper-Raman scattering.

This chapter explores dynamic coordination in the primary sensory cortex of mammals during low-level (non-attention-related) perception. It considers the possible existence of subcortical or cortical supervisors in higher mammals and explains how coordination is generated by the sensory drive and amplified by built-in anisotropies in the network connectivity. Drawing on synaptic functional imaging (at the intracellular level) and real-time voltage-sensitive dye network imaging (at the functional map level), it illustrates the role of intracortical depolarizing waves whose functional features support the hypothesis of a dynamic association field. These waves help propagate synaptic modulation in space and time via lateral (and perhaps feedback) connectivity, which explains the emergence of illusions predicted by Gestalt theory. The chapter also explains how lateral propagation waves are reconstructed from synaptic echoes and looks at the propagation of orientation belief.

Recent magnetic field missions flying in a low Earth orbit (LEO), like Ørsted, ST5, and CHAMP, have enabled us to study in situ the occurrence of ultra low of frequency (ULF) waves in the topside ionosphere. CHAMP satellite has been one of the most successful missions for the study of the Earth’s magnetic field for more than a decade (July 2000–September 2010), allowing for the first time-term statistical studies on the occurrence of ULF wave events in the topside ionosphere. This chapter provides a review of recent studies of ULF waves using spacecraft in low Earth orbit. It combines a topical review of the various subclasses of ULF waves that the amenable to study from LEO with a detailed description of many recent observational studies and of several relevant theoretical efforts. It concludes by focusing on further studies made possible by the Swarm mission (in fruitful combination with other spacecraft and ground data) as well as by recent theoretical developments. The chapter also attempts to point out open questions.

Chapter 4 proposes that interactions between radio scientists and commercial companies for elucidating wave propagation in the upper atmosphere was not conspicuously different from those evolved from seismic prospecting of the Earth’s crust in America. It contends that there was a predominant epistemic paradigm in crustal seismology in the interwar period—simplicity—which was altered because of the strong influence of a particular commercial environment, i.e. the oil industry. To this end, several steps are followed. Firstly, it shows how Harold Jeffreys formulated the ‘simplicity postulate’ as the basis for his probabilistic epistemology, embraced by several seismologists who developed crustal models based on mathematical idealizations. Next, it shows that there was a renunciation of simplicity in the 1930s, emerging too quickly to have been the result of new geological evidence. Finally, it demonstrates that the paradigm shift among seismologists was a result of the significant rise in seismic exploration generated by the oil industry.

Chapter 2 reacquaints the reader with electric and magnetic force fields and their interactions with ponderable media through Maxwell’s equations and the accompanying Lorentz force law. Macroscopic quantities of permittivity and permeability are emphasised, and through the constitutive relations, polarisation and magnetisation fields. Dipole radiation, space-propagating and surface-propagating wave solutions to Maxwell’s equations are all fundamental to understanding energy and momentum transport around, and through, atomic scale and nanoscale structured materials. The chapter ends with a development of plane wave propagation, reflection, and transmission in homogenous media and at dielectric and metallic surfaces.

The need for numerical approximations of the seismic wave-propaga-tion problem in the field of seismology is motivated by the fact that we have to deal with complex 3D structures. The term ‘computational seismology’ is defined and contrasted with other more classical approaches such as ray tracing approaches and quasi-analytical solutions such as normal mode techniques and the reflectivity method. The structure of the volume content is illustrated and guidelines are given how to use the content in combination with the supplementary electronic material.

This book is about recent mathematical, numerical and statistical approaches for elasticity imaging of inclusions and cracks with waves at zero, single or multiple non-zero frequencies. It considers important developments in asymptotic imaging, stochastic modeling, and analysis of both deterministic and stochastic elastic wave propagation phenomena and puts them together in a coherent way. It gives emphasis on deriving the best possible imaging functionals for small inclusions and cracks in the sense of stability and resolution. For imaging extended elastic inclusions, the book develops accurate optimal control methodologies and examines the effect of uncertainties of the geometric or physical parameters on their stability and resolution properties. It also presents an asymptotic framework for vibration testing and a method for identifying, locating, and estimating inclusions and cracks in elastic structures by measuring their modal characteristics.

This chapter deals with the mathematics of ocean acoustics. A number of environmental factors affect the transmission of sound in the ocean, including the depth and configuration of the bottom, the sound velocity structure within the ocean, and the shape of the ocean surface. The depths in the ocean are distributed in a peculiar manner, and the solution of underwater-sound problems may be grouped into two categories that differ mainly in terms of dimension: the average depths of water for deep-water transmission are 10,000 to 20,000 feet, whereas those for shallow-water transmission are less than 300 feet. The chapter first provides an overview of ocean acoustic waveguides before discussing one-dimensional waves in an inhomogeneous medium. It also considers a mathematical model of acoustic wave propagation in a stratified fluid and concludes with an analysis of the one-dimensional time-independent Schrödinger equation for solving the potential well problem.

This chapter discusses light in Newtonian theory. Astronomy (and physics in general) is certainly not a complete science without a theory of light. However, the nature of light and its kinematical properties were not completely understood until the advent of Maxwell’s theory, special and general relativity, and quantum field theory. The answers to these questions provided by Newtonian theory were only partial and sometimes even contradictory. Thus this chapter seeks to present a few aspects of this topic, such as the influences of gravity on light, stellar aberrations, and wave propagation. It also studies the Fizeau and Michelson–Morley experiments.

Currently, the design of most composite components is based on stiffness, and therefore methods for static measurement of stiffness are in wide use. The disadvantages of these methods lie in their destructive nature (the samples must be cut from parts of different orientations), in the difficulty of measuring shear properties, and in the need for extra care when measuring Young’s modulus in off-axis directions. Ultrasonic methods are more accurate and have higher spatial resolution than static measurements. As we showed in Chapter 2, by measuring ultrasonic velocities in several predefined directions, all elastic constants can be determined. The generic method described there is also destructive, however, requiring cutting numerous samples with appropriate fiber orientation. Specialized nondestructive methods for determining the elastic moduli of composite materials are more powerful and they can be applied to composite coupons before, during, and after strength or fatigue testing. It is important to have a fast and inexpensive technique to estimate input parameters for composite design. It is even more important to have a technique to evaluate composites during service to verify that the manufactured elastic stiffnesses match those assumed in the design. Several methods that utilize bulk ultrasonic waves for measurement of composite elastic constants are considered in this chapter. By bulk wave methods, we mean quasilongitudinal and quasitransverse ultrasonic wave velocity measurement methods that are applicable when the sample thickness h is larger than both the ultrasonic pulse space length τV and the wavelength λ (τ is the ultrasonic pulse length in time, and V is the wave speed). Other methods, which are applicable in the range h < τV and which account for wave interference with the boundaries of the specimen, will be considered in the following chapters. The most promising way to evaluate composite elastic properties nondestructively is to measure ultrasonic velocities in different directions in the composite material and reconstruct the elastic constants from these values using some kind of an inversion technique. One possible method has been suggested by Markham in the 1970s, who used ultrasonic waves obliquely incident from water onto a composite plate to measure ultrasonic velocities in various directions and evaluated the results to determine elastic constants.