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The solid Earth's medium is heterogeneous over a wide range of scales. Seismological observations, including envelope broadening with increasing distance from an earthquake source and the excitation of long-lasting coda waves, provide a means of investigating velocity inhomogeneities in the lithosphere. These phenomena have been studied primarily using radiative transfer theory with random medium modelling. This book presents the mathematical foundations of scalar- and vector-wave scattering in random media, using the Born or Eikonal approximation, which are useful for understanding random inhomogeneity spectra and the scattering characteristics of the solid Earth. A step-by-step Monte Carlo simulation procedure is presented for synthesizing the propagation of energy density for impulsive radiation from a source in random media. Simulation results are then verified by comparison with analytical solutions and finite-difference simulations. Presenting the latest seismological observations and analysis techniques, this is a useful reference for graduate students and researchers in geophysics and physics.
In this chapter, the aim is to visualize wave dynamics in one dimension as dictated by the Schrödinger equation. The necessary numerical tools are introduced in the first part of the chapter. Via discretization, the wave function is represented as a column vector and the Hamiltonian, which enters into the Schrödinger equation, as a square matrix. It is also seen how different approximations behave as the numerical wave function reaches the numerical boundary – where artefacts appear. This numerical framework is first used to see how a Gaussian wave packet would change its width in time and, eventually, spread out. Two waves interfering is also simulated. And wave packets are sent towards barriers to see how they bounce back or, possibly, tunnel through to the other side. In the last part of the chapter, it is explained how quantum measurements provide eigenvalues as answers – for any observable physical quantity. This, in turn, is related to what is called the collapse of the wave function. It is also discussed how a quantity whose operator commutes with the Hamiltonian is conserved in time. Finally, the concept of stationary solutions is introduced in order to motivate the following chapter.
Dynamical and thermodynamic properties of water at room temperature in Ca- and hexade-cyltrimethylammonium- (HDTMA) exchanged bentonite were determined for 4 different water contents (~0.03–0.55 g water g-1 clay). Incoherent quasi-elastic neutron scattering (QENS) was used to measure the translational and rotational mobility of water in the clays, while chilled mirror dewpoint psychrometry measured water activity of the samples, differential scanning calorimetry (DSC) provided information about the temperature of dehydration and X-ray diffraction (XRD) quantified layer spacings for the clays. The neutron scattering data were fit to a jump diffusion model that yielded mean jump lengths, jump diffusion residence times and rotational relaxation times for water in the clays. Mean jump lengths were quite similar for the 2 different cation saturations at equivalent water contents, and decreased with increasing water content. The fitted jump lengths ranged from 0.27–0.5 nm and were 2–4 times larger than that found for bulk water (0.13 nm). Jump diffusion residence times were 3–30 times longer than that for bulk water (1.2 ps) and also decreased with increasing water content. The residence times were somewhat shorter for HDTMA-clay as compared with Ca-clay at equivalent water contents. Rotational motion was less strongly influenced than translational motion by the presence of the clay surface. The energy state of water in the 2 cation saturations were quite different; dehydration temperatures for the HDTMA-clay were approximately 30 °C lower than the Ca-clay at equal water contents, while water activities, as P/P0, were up to 0.6 units higher. A linear relationship was found between water activity and the translational diffusion coefficient, although at the highest water content, the diffusion coefficient of water for the HDTMA-clay was approximately 30% higher than that measured for bulk water.
Chapter 4treats the fundamentals of radio propagation path loss, also known as large-scale fading. A wide range of practical radio propagation models are presented, and the fundamental theories of reflection, scattering, and diffraction are presented with many examples. These propagation mechanisms give rise to level of coverage and interference experienced in any wireless network, and, in urban environments, it is shown how the radar cross-section and ray tracing models can give accurate prediction of large-scale path loss in a mobile communication system. Shadowing is also considered, and the log-normal distribution is found to describe the shadowing about the distance-dependent mean signal level. Statistical approaches to quantifying outage are provided.
Study of the behavior of landfill lining materials (clays) in organic solvents is important because, in waste management, lining prevents groundwater contamination by the adsorption of various pollutants such as chemicals and organic solvents. Although scaling behavior and the self-association property of clays in water-alcohol binary solvents have been studied by many researchers, the anomalous behavior of Laponite XLG in binary solvents requires investigation as suggested by previous studies. In the present study, Laponite® RD, which is structurally similar to Laponite XLG, was used to gain further insight into the reasons for the anomalous viscosity, aggregation, and non-ergodic behavior of clay in a water–methanol binary solvent. Dynamic light scattering (DLS) revealed the emergence of the non-ergodic phase of 3% w/v Laponite® RD in the water–methanol binary solvent, which increased in the presence of a large methanol content as well as with aging time in the binary solvent. Viscosity measurements further indicated that aggregation was responsible for the non-ergodic behavior, and small-angle X-ray scattering (SAXS) revealed that a large methanol content enhanced the aggregation. Moreover, SAXS data also revealed that the surface charge was responsible for anomalous viscosity fluctuations in the binary solvent due to interparticle repulsion within aggregates. Rheological studies showed that the large methanol content in the binary solvent led to frequency-independent behavior of the storage modulus of Laponite® RD.
The reign of Emperor Muhammad Shah (r. 1720−48) ushered in a significant revival of Mughal courtly arts. Right at the centre of this vibrant milieu was the Emperor’s singing teacher and master of his atelier, Khushhal’s grandson Anjha Baras Khan. But posterity has forgotten him—it is his rivals Ni‘mat Khan “Sadarang” and Firoz Khan “Adarang” whom we remember as the greatest musicians of the eighteenth century. Why? As Delhi was repeatedly invaded and sacked 1739−61, Mughal court musicians scattered all over India, and had to seek new strategies to survive. What happened to Delhi’s musicians and their music is documented in a genre new to writing on music at this time: the tazkira (roughly, biographical collection). In this chapter I show that the proliferation of musicians’ biographies and genealogies were both a product of upheaval, dispersal, and enforced diversification; and a record of these things, particularly in anecdotes of rivalry.
We present a catalogue of over 7000 sources from the GLEAM survey which have significant structure on sub-arcsecond scales at 162 MHz. The compact nature of these sources was detected and quantified via their Interplanetary Scintillation (IPS) signature, measured in interferometric images from the Murchison Widefield Array. The advantage of this approach is that all sufficiently compact sources across the survey area are included down to a well-defined flux density limit. The survey is based on
${\sim}250\times 10\hbox{-}\mathrm{min}$
observations, and the area covered is somewhat irregular, but the area within
$1\,\mathrm{h}<\mathrm{RA}<11\,\mathrm{h}$
;
$-10^\circ<\mathrm{Decl.}<+20^\circ$
is covered entirely, and over 85% of this area has a detection limit for compact structure below 0.2 Jy. 7839 sources clearly showing IPS were detected (
${>}5\sigma$
confidence), with a further 5550 tentative (
${>}2\sigma$
confidence) detections. Normalised Scintillation Indices (NSI; a measure of the fraction of flux density coming from a compact component) are reported for these sources. Robust and informative upper limits on the NSI are reported for a further 31081 sources. This represents the largest survey of compact sources at radio frequencies ever undertaken.
In this chapter we describe scattering representations, a signal representation built using wavelet multiscale decompositions with a deep convolutional architecture. Its construction highlights the fundamental role of geometric stability in deep learning representations, and provides a mathematical basis to study CNNs. We describe its main mathematical properties, its applications to computer vision, speech recognition and physical sciences, as well as its extensions to Lie Groups and non-Euclidean domains. Finally, we discuss recent applications to modeling high-dimensional probability densities.
We learn about unbound states and find that the energies are no longer quantized. We learn about momentum eigenstates and superposing momentum eigenstates in a wave packet. We apply unbound states to the problem of scattering from potential wells and barriers in one dimension.
In the interstellar medium, inelastic collisions are so rare that they cannot maintain a local thermodynamical equilibrium (LTE). Atomic and molecular populations therefore do not follow a simple Boltzmann distribution and non-LTE spectra are the rule rather than the exception. In such conditions, accurate state-to-state collisional data are crucial for a quantitative interpretation of spectra. In recent years, considerable progress has been made in quantum calculations of inelastic cross sections for a variety of targets, types of transitions and projectiles. For a few benchmark species, detailed comparisons between theory and experiment were also carried out at the state-to-state level and in the quantum regime. In this article, we highlight such comparisons for three important molecules: CO, H2O and CH+. We also describe current computational efforts to extend these advances to ever larger targets, new transition types, and new environments (e.g. stellar envelopes or cometary atmospheres).
In the research of the natural ecological environment, environmental scattering is an important research content whose results are widely used. In order to fully study the relevant characteristics of the natural grassland environment, this paper established a grassland broadband clutter model, summarized the dominant and recessive laws of the grassland environment broadband clutter, and provided a new perspective for related environmental monitoring which would be helpful to solve the problem of target detection in complex environments. The main work of this paper is as follows: The dielectric constant model of the grass was established, and the curve of the dielectric constant with frequency was obtained. The Monte Carlo method combined with the Gaussian spectrum function was used to generate a two-dimensional Gaussian surface to simulate the actual grass surface. A broadband clutter model was established, and, considering the calculation efficiency and accuracy, the Sparse Matrix/Canonical Grid (SMCG) was used to calculate the surface scattering coefficient. Then, the environmental clutter amplitude with different radar bandwidth (10, 40, 80 MHz), surface roughness (h = 0.1 m, h = 0.2 m, h = 0.4 m) and grazing angle (30°, 60°) were calculated and the probability density function (PDF) was obtained. The results show that the higher the radar resolution, the larger the incident angle, and the rougher the interface, all of which will cause the tail of the PDF to become longer which will not only reduce the detection probability of the radar, but also the tracking accuracy. The research results can be used for civilian and military field.
The scattering by the perfectly electric conducting (PEC) half-plane and PEC zero thickness disk placed on parallel planes is considered. The fields are represented in the spectral domain, i.e. in the domain of Fourier transform. The operator equations with respect to the Fourier amplitudes of the scattered field are obtained. The kernel functions of these equations contain poles. After regularization procedure, which is connected with the elimination of the poles, operator equations are converted to the system of singular integral equations. The convergence of the solution is based on the corresponding theorems. The scattered field consists of the plane wave, reflected by the infinite part of the half-plane, cylindrical waves, which appear as a result of scattering by the edge of the half-plane, and spherical waves, which appear as a result of scattering by the disk and multiple re-scattering by the disk-half-plane. The total near-field distribution and far-field patterns of cylindrical waves are presented.
It presents the unitary time evolution operator and its integral kernel in the space representation, also known as the quantum-mechanical propagator. It introduces the resolvent operator and its analytical properties, and it discusses in some detail scattering theory in one dimension.
We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$.
Knowledge of the output and three dimensional distribution of all constituents of galaxies (stars of all ages, gas, dust, cosmic rays) is a prerequisite for understanding the process of star-formation along the cosmic time, and ultimately the formation and evolution of galaxies. However, what we measure is the spatial and spectral energy distribution (SED) of galaxies. In this chapter we describe self-consistent modeling of the SED involvingradiative transfer (RT) calculations that follow the interaction between stellar photons and dust particles, and make predictions for all emission mechanisms involved. Tracing the energy flow and accounting for the anisotropy of the problem requires modelling of SEDs spanning a broad range in wavelengths and the spatial distribution of the emission. A RT modelaccurately calculates the stellar SEDs emitted by the newly-formed stars by both calculating the effect of dust attenuation throughout the galaxy, and by providing a three dimensional picture of the stellar emission of these stars. This way it produces a solution for the SFR, and the 3D distributions of all stellar components of a galaxy (stars of all ages and from different morphological components, like disks, bulges, and bars) and of the dust distribution, giving us a detailed understanding of the make up of a galaxy, both of its stellar content and of the interstellar medium structure.
In our galaxy, the existence of dust is revealed by the fact that dust grains absorb, scatter, polarize, and emit light. The interaction of dust grains with light depends on the size and shape of the grains, as well as on the index of refraction of the material making up the grains. Observations indicate that the mass of dust in our galaxy is about 1% the mass of interstellar gas. Most grains are either graphite or silicate, with a typical grain radius of ∼0.1 micron. The equilibrium temperature of dust grains is set by the balance between absorbing starlight and emitting thermal radiation; for interstellar grains, the equilibrium is at T ∼ 20 K. Cool stellar winds, like those of Mira variable stars, give rise to circumstellar dust grains. As these grains are spread through interstellar space, they can grow by accretion of atoms or be destroyed by sputtering or be vaporized by shock-heating.
The theoretical foundations of the description of heterogeneity are established for both body waves and modal fields. We first consider perturbations of the wavefield using Born series and show how such concepts can be combined with the use of reflection and transmission operators to provide a flexible treatment of structures with varying heterogeneity in different zones of the model. Although the various modes of surface waves propagate independently in simple structure, the presence of heterogeneity induces cross-coupling that modifies the wavefield. Because the nature of heterogeneity differs in the various parts of the Earth, the interaction of different classes of heterogeneity has an important role in shaping the nature of the full seismic wavefield.
We examine the interaction of seismic waves with heterogeneity at all scales, with an emphasis on the influence of structure on multiple scales.Strong interactions occur when seismic wavelengths are comparable to the size of heterogeneity, producing complex scattering. Because seismic waves span a broad spectrum of frequencies and hence wavelengths, any heterogeneous structures will be perceived in different ways by the various aspects of the wavefield, with significant difference in behaviour between body waves and longer period surface waves. Such complications become most evident when a wide range of heterogeneity scales arepresent simultaneously, as in the lithosphere.
The large-scale structure of the Earth can be extracted with seismic tomography, but the finer scales of variation within the Earth lie beyond any capacity for direct imaging. Nevertheless, the scattered wavefield produced by small-scale heterogeneity contains important information on structure.We consider the representation of variations in Earth structure on scales from the global to the regional, and discuss ways in which numerical simulations and inversions can exploit data with differing station density to provide maximum resolution of structure. We contrast deterministic and stochastic (parametric) representations of heterogeneity, and examine the way in which ensemble results can be exploited for Earth structure that is time invariant. We also consider the way that effective media, with simpler structure, can be extracted from complex models by the process of wavespeed upscaling